Dr. Ananthram Swami and Dr. Bruce J. West

Editors-in-Chief
Editorial Board

Dr. Ananthram Swami

Editor-in-Chief

Dr. Bruce J. West

Editor-in-Chief

Advisory Board

Dr. Kwong K. Choi

Dr. Brad E. Forch

Dr. Piotr J. Franaszczuk

Dr. Shashi P. Karna

Dr. Stephen J. Lee

Dr. Tomasz R. Letowski

Dr. Joseph N. Mait

Dr. James W. McCauley

Dr. Nasser M. Nasrabadi

Dr. Peter J. Reynolds

Dr. Paul H. Shen

Dr. Ananthram Swami

Dr. Bruce J. West

Dr. Piotr J. Franaszczuk Dr. Joseph N. Mait

 Research@ can be accessed electronically at www.arl.army.mil/ResearchARL
Network Sciences • 2013

Research@

Introduction to ARL ......................................................................................................................................................................2

Foreword........................................................................................................................................................................................3

John Pellegrino, Director (A)

Introduction to ARL Research in Network Sciences..................................................................................................................4

Bruce J. West, Ananthram Swami, Joseph N. Mait, and Piotr J. Franaszczuk

Physical Sciences

A New Measure of Network Efficiency..................................................................................................................................... 13

Nicholas W. Hollingshad, Malgorzata Turalska, Paolo Allegrini, Bruce J. West, and Paolo Grigolini 

Optimal Topology Control and Power Allocation for Minimum Energy Consumption in Consensus Networks................ 21

Stefania Sardellitti, Sergio Barbarossa, and Ananthram Swami

Efficient Extraction of Drainage Networks from Massive, Radar-Based Elevation Models 

with Least Cost Path Search .................................................................................................................................................... 39

M. Metz, H. Mitasova, and R. S. Harmon

Connectivity of Heterogeneous Wireless Networks ............................................................................................................... 53

Wei Ren, Qing Zhao, and Ananthram Swami

Life Sciences

Epileptic Seizures from Abnormal Networks: Why Some Seizures Defy Predictability...................................................... 73

William S. Anderson, Feraz Azhar, Pawel Kudela, Gregory K. Bergey, and Piotr J. Franaszczuk

Evolving Communicative Complexity: Insights from Rodents and Beyond.......................................................................... 87

Kimberly A. Pollard and Daniel T. Blumstein

Combining the Finite Element Method with Structural Connectome-Based Analysis for Modeling Neurotrauma: 

Connectome Neurotrauma Mechanics ................................................................................................................................... 99

Reuben H. Kraft, Phillip Justin Mckee, Amy M. Dagro, and Scott T. Grafton

On Allometry Relations ...........................................................................................................................................................115

Damien West and Bruce J. West

Social Sciences 

Distributed Algorithms for Learning and Cognitive Medium Access with Logarithmic Regret........................................173

Animashree Anandkumar, Nithin Michael, Ao Kevin Tang, and Ananthram Swami

A Survey on Trust Management for Mobile Ad Hoc Networks ............................................................................................189

Jin-Hee Cho, Ananthram Swami, and Ing-Ray Chen

Temporal Complexity of the Order Parameter at the Phase Transition .............................................................................213

Malgorzata Turalska, Bruce J. West, and Paolo Grigolini

Information Sciences

Noncoherent Physical-Layer Network Coding with FSK Modulation: Relay Receiver Design Issues...............................221

Matthew C. Valenti, Don Torrieri, and Terry Ferrett

Transmission of Information Between Complex Systems: 1/ f Resonance ......................................................................233

Gerardo Aquino, Mauro Bologna, Bruce J. West, and Paolo Grigolini

Robust Control for Mobility and Wireless Communication in Cyber–Physical Systems 

With Application to Robot Teams .......................................................................................................................................... 247

Jonathan Fink, Alejandro Ribeiro, and Vijay Kumar

Biographies of ARL Authors ...................................................................................................................................................263

contents
U.S. ARMY RESEARCH LABORATORY

Introduction to ARL

The Army Research Laboratory of the U.S. Army Research, Development and Engineering Command (RDECOM) is the Army’s 

corporate laboratory. ARL’s research continuum focuses on basic and applied research (6.1 and 6.2) and survivability/lethality 

and human factors analysis (6.6). ARL also applies the extensive research and analysis tools developed in its direct mission 

program to support ongoing development and acquisition programs in the Army Research, Development and Engineering 

Centers (RDECs), Program Executive Offices (PEOs)/Program Manager (PM) Offices, and Industry. ARL has consistently provided 

the enabling technologies in many of the Army’s most important weapons systems.

The Soldiers of today and tomorrow depend on us to deliver the scientific discoveries, technological advances, and the analyses 

that provide Warfighters with the capabilities to execute full-spectrum operations. ARL has Collaborative Technology Alliances in 

Micro Autonomous Systems and Technology, Robotics, Cognition and Neuroergonomics, and Network Science, an International 

Technology Alliance, and new Collaborative Research Alliances in Multiscale Multidisciplinary Modeling of Electronic Materials 

and Materials in Extreme Dynamic Environments. ARL’s diverse assortment of unique facilities and dedicated workforce of 

government and private sector partners make up the largest source of world class integrated research and analysis in the Army. 

ARL Mission

The mission of ARL is to “Provide the underpinning science, technology, and analysis that enable full-spectrum operations.” 

Our Vision

America’s Laboratory for the Army: Many Minds, Many Capabilities, Single Focus on the Soldier

ARL’s Organization

•Army Research Office (ARO) - Initiates the scientific and far reaching technological discoveries in extramural organizations: 

educational institutions, nonprofit organizations, and private industry.

•Computational and Information Sciences Directorate (CISD) - Scientific research and technology focused on information 

processing, network and communication sciences, information assurance, battlespace environments, and advanced computing 

that create, exploit, and harvest innovative technologies to enable knowledge superiority for the Warfighter. 

•Human Research and Engineering Directorate (HRED) - Scientific research and technology directed toward optimizing Soldier 

performance and Soldier-machine interactions to maximize battlefield effectiveness and to ensure that Soldier performance 

requirements are adequately considered in technology development and system design.

•Sensors and Electron Devices Directorate (SEDD) - Scientific research and technology in electro-optic smart sensors, 

multifunction radio frequency (RF), autonomous sensing, power and energy, and signature management for reconnaissance, 

intelligence, surveillance, target acquisition (RISTA), fire control, guidance, fuzing, survivability, mobility, and lethality.

•Survivability/Lethality Analysis Directorate (SLAD) - Integrated survivability and lethality analysis of Army systems and 

technologies across the full spectrum of battlefield threats and environments as well as analysis tools, techniques, and 

methodologies.

•Vehicle Technology Directorate (VTD) - Scientific research and technology addressing propulsion, transmission, aeromechanics, 

structural engineering, and robotics technologies for both air and ground vehicles.

•Weapons and Materials Research Directorate (WMRD) - Scientific research and technology in the areas of weapons, protection, 

and materials to enhance the lethality and survivability of the Nation’s ground forces.

Unique ARL facilities and modeling capabilities 

provide our scientists and engineers access to

world-class research centers.

ARL Workforce in 2013

•1,971 Civilians - 41 Military

•1,181 Research Performing Workforce

•578 (49%) hold PhDs

•13 STs / 26 ARL Fellows

ARL’s Primary Sites 

•Aberdeen Proving Ground, MD

•Adelphi Laboratory Center, MD

•White Sands Missile Range, NM

•Raleigh-Durham, NC

•Orlando, FL

Visit ARL’s web site at www.arl.army.mil
3

Welcome to the second edition of the Research @ARL monograph series. In this 

edition, we share ARL’s leading edge research in the Network Sciences. Today’s 

society is more networked than ever before, and today’s Army reflects that reality. 

However, the tools needed to understand the science of our interconnected global 

society are not yet fully developed. Many of the networks that touch our everyday 

lives, both in the Army and outside, are complex networks. They are often governed 

by local interactions and robust self-organization principles, leading to complex 

dynamics. And networks do not live in isolation; they are interconnected in surprising 

ways. A modern Army cannot function—any more than a modern city can—without the 

latticework of transportation systems, utilities, water and food distribution systems, 

communication webs, health care delivery organizations, and a myriad of other 

interconnected, complex networks. These layers of interpenetrating networks paint 

a picture of today’s Army that overarches the traditional physical sciences, social 

sciences, life sciences, and information sciences. No one discipline or collection of 

disciplines can fully embrace the spectrum of complex problems that the engineered 

networking of humanity presents. 

ARL’s research program in the Network Sciences capitalizes on ARL’s expertise in basic and applied research, ranging across 

a multitude of science areas, while simultaneously drawing upon the talents of others, nationally and internationally, through 

collaborative alliances with industry, laboratories, and academia. We seek to discover the underlying mathematical principles 

and universal laws that govern the behavior of co-evolving multi-genre networks. We also seek to understand emergent behavior 

in such networks, leading to mechanisms for thwarting or accelerating the emergence of interesting behaviors. Our research 

teams are examining the complex neurological networks within the human brain to determine how individuals make decisions 

and perform other cognitive tasks under varying levels of stress and uncertainty. The insight gained from this research is 

being applied to other biological networks to alleviate the injuries and pathologies encountered by the Warfighter. Exploration 

of cooperative behavior of social networks, such as swarms, is a significant research opportunity to control the mobility of 

collections of autonomous platforms. In addition to networks in the life and social sciences, ARL teams are examining the 

connectedness within and between communications and information networks to determine how their complex structures can 

best facilitate the gathering of intelligence and the formation of collective decisions. 

As we investigate the fundamental science addressing these critical applications, we search for laws and principles of Network 

Science that parallel and compliment those in physics. Tentative principles are presently guiding the development of the 

mathematical infrastructure necessary for understanding those aspects of complex networks common to multiple phenomena 

appearing in a variety of scientific disciplines. 

ARL’s long term commitment to encompass interdisciplinary sciences in such diverse areas as vulnerability assessment, 

cognitive sciences, human sciences, electronic devices, weapons and materials sciences, information sciences, survivability, 

and lethality, uniquely positions our teams to investigate and discover previously unrealized commonalities to provide innovative 

solutions. Our discoveries enhance our country’s technological capabilities, our national security, and, ultimately, enable the 

invincible Soldier of the future. 

FOREWORD

Dr. John Pellegrino

Director (A), U.S. Army Research Laboratory
4

Introduction to ARL Research in Network Sciences

B.J.West, A. Swami, J.N. Mait, and P.J. Franaszczuk

1 Army Research Laboratory: Network Science Program

The objective of the Network Science Program of the U.S. Army Research Laboratory (ARL) is to perform foundational research 

on network science (NS) leading to a fundamental understanding of the interplay within and among the physical, social/

cognitive, information, and communication networks. This research is expected to lead to insights on how processes and 

parameters in one network affect and are affected by those in other networks. The underlying long-term goal is to optimize 

human performance and to greatly enhance speed and precision for complex networked military operations.

We define Network Science as the study of complex systems whose behavior and responses are determined by exchanges 

and interactions between subsystems across a possibly dynamic and usually poorly defined set of pathways. The fundamental 

components of a network are its structure (nodes and links or pathways) and its dynamics, which together specify the network’s 

properties (functions and behaviors).

The Army is embarked upon an information age transformation loosely referred to as Network Centric Warfare (NCW) or 

Network Centric Operations (NCO). NCW/NCO seeks to dramatically increase mission effectiveness via robust networking for 

information sharing leading to shared situational awareness, improved collaboration and self-synchronization, and enhanced 

sustainability and speed of command [1, 2]. Central to this effort is an understanding of the interaction of networks in the 

physical, informational, cognitive, and social domains; thus, ARL’s NS program directly supports NCW/NCO.

Society, as pointed out in the 2005 National Research Council (NRC) report Network Science [3], is more interconnected 

than it has been at any time in world history. A modern city could not function without garbage collection, sewers, electricity, 

transportation, water, health care, and food and fuel distribution, and would have a much different form without networks of 

education, banking, telephone service, and the Internet. Some of these activities form local physical or social networks within 

the city and their forms have been evolving for millennia. Part of that evolution was the development of their interoperability such 

that these networks are all interconnected and in one way or another they connect to ever-expanding global networks. These 

interconnections can lead to robustly coevolving networks, but under some conditions, can also lead to cascading failures. This 

corresponds to the first finding of the NRC report: “Networks enable the necessities and conveniences of modern life.” These 

are the engineered networks of humanity, but there are comparable structures in the biosphere and ecosphere involving plant 

and animal networks of tremendous variety.

Moreover, our internal world is also an interlacing and interacting collection of networks. The neuronal network carrying the 

brain’s signals to the body’s physiological networks is even more complex than the modern city. The bio/eco-networks are 

certainly as difficult to understand as the physical/social-networks. This led to another finding in the NRC report: “Social and 

biological networks bear important similarities to engineered networks.” Herein we present a number of studies carried out by 

ARL scientists in collaboration with the broader scientific community to further our understanding of NS. These studies suggest 

how these similarities might be exploited for the development of NS that is of value to society in general and to the United States 

Army in particular.

The insights and mathematical characterizations resulting from ARL’s NS research will enable us to achieve this underlying 

goal, by providing the ability to predict and control the individual and composite behavior of the complex interacting physical, 

communications, information, and social/cognitive networks. Network science is one of ARL’s strategic technology investment 

areas, with a strong program based on the modern research pillars of theory, simulation, emulation, and experimentation. It 

encompasses a variety of transdisciplinary basic and applied programs across ARL’s core competency areas that include close 

collaborations with academic, military, and industrial research partners throughout the US and internationally. These include 

ARL’s Network Science Collaborative Technology Alliance (CTA), the US-UK Network and Information Sciences International 

Technology Alliance (ITA), the Micro Autonomous Systems & Technology CTA, the Cognition & Neuroergonomics CTA, and several 

Multidisciplinary University Research Initiatives (MURIs). The Institute for Collaborative Biotechnology is a University Affiliated 

Research Center (UARC) that approaches networks from the perspective of biological systems, and ARL’s Mobile Network 

Modeling Institute (MNMI) is an applied research effort that seeks to further the science by enabling full fidelity at-scale 

modeling and simulation of large scale heterogeneous networks, incorporating live assets in the emulation environment. More 

information on these programs may be found by navigating ARL’s Collaborative Technology and Research Alliances website at 

http://www.arl.army.mil/www/default.cfm?page=93.
5

The taxonomy of the ARL research efforts in NS can take a number of forms, two of which are given here. To set the stage for 

the scientific problems, we summarize four grand challenges and identify the Army programs conducting research to address 

these challenges. A separate and distinct way of organizing the research is via the network category addressed. The individual 

research papers are identified in this latter grouping according to discipline.

2 Grand Challenges

The ARL research effort in NS seeks to make fundamental advances of and contributions to the science so as to enable the a 

priori prediction of the behaviors of complex interacting networks in diverse and dynamic environments and an understanding 

of the design trade-offs and the impacts of various technologies under a wide variety of dynamic and adverse conditions. 

Moreover, it is intended to quantify the impact of network technologies both technically and operationally to make informed 

acquisition decisions.

2.1 Underpinning Mathematical Framework: Dynamic Multiple-Scale, Cross-Genre Coupling

Multiple-scale and cross-scale couplings within and across networks are recurrent themes in the study of complex physical, 

communication, informational, social, and biological networks. Existing formalisms address one, two, or a continuum of scales, 

but the multi-node problem remains unsolved. Existing mathematical approaches offer insights into the quantitative aspects 

of complex phenomena with multiple scales (no characteristic scale) in stationary and near equilibrium networks, but what 

is needed now is a way to describe the dynamics of such complex phenomena/networks when the underlying processes 

are neither stationary nor near equilibrium [4]. Network characteristics cannot be deduced from the properties of individual 

components; they emerge during the formation and growth of the network. Consequently a mathematical framework is needed 

to characterize the interactions between the dynamic network components, the temporal evolution of the network, and its 

response to external stimuli, including attacks. The framework must take into account heterogeneity, non-stationarity, even 

non-ergodicity, as well as conflicting constraints and objectives.

One way to characterize the interaction of networks is by understanding how information is propagated within and across 

complex adaptive networks — how information, in a general sense, flows across networks so as to maximize utility (e.g., as 

perceived by a decision maker) under constraints, such as timeliness and resource usage. Tools from statistical physics such 

as continuum percolation theory, stochastic geometry, and modern statistics are used to deal with non-Markovian behaviors, 

heavy-tailed phenomena, and the mutual interactions between networks. To cope with multiple potentially conflicting constraints 

and objectives, a generalized control theory of locally reacting multiple agents needs to be developed, perhaps through the 

mimicking of self-repair and replication identified in biological networks.

2.2 Human Metrics in Complex Networks

A critical element in realizing the vision of Network-Centric Operations (NCO) is that the distributed decision making process 

must take into account the human dimension in its elements including decision-makers, data and information sources, 

processed information, communication network elements, etc. The measure or metric of the human dimension must be 

derived in a distributed fashion; in time-critical and stressful situations; in environments where node capture and subversion 

are likely; where the underlying communications network is resource-constrained, mobile, and dynamic; and where decisionmakers’ reliance on and compliance with an information network are subject to numerous internal and external influences. 

Key barriers include the limited understanding of network-of-networks, composability of heterogeneous networks, and a limited 

understanding of how the context (e.g., trust and risk) is analytically derived from variously distributed sources of evidence.

A key aspect of a common quantitative framework is an approach for representing uncertainty (probability theory, possibility 

theory, evidence theory, subjective logic, argumentation theory, or other). Development of metrics is a key issue: metrics that 

take into account the value of information and human intentions are important in information and C2 networks. Relations 

between and among these metrics need to be clarified along with the development of a mathematics of network classes. A 

specific example is the trust metric. The scientific challenge is to understand the different definitions and dimensions of trust, 

for example, in socio-cognitive and communications networks, and from that understanding develop a composite trust metric. 

Rather than merely treating the interaction via a simple interface or utility function offered by one network to another, network 

interactions must be taken into account, as well as the context- and time-varying nature of the trust metric components. 

Encapsulation of these sub-components as a scalar metric is challenging due to the conflicting nature of the components and 

the non-convexity of useful metrics. To quantify human performance in networks, measurable features of data from different 

networks need to be identified to enable the modeling of network complexity and the interactions between different networks.
6

2.3 Abstracting Common Concepts Across Fields

A set of common concepts for network science that is defined consistently across the various scientific disciplines does not 

exist. There are discipline-specific nomenclatures that have been developed for specialized needs, but the equivalence of 

these terminologies for a discipline-independent characterization of networks has not been established. The mathematical 

formulation of these empirically-based concepts will provide a nascent language for network science.

Mathematical rigor across the component disciplines is promoted by focusing on ubiquitous aspects of the underlying complex 

networks, such as the appearance of non-stationary, non-ergodic, and non-renewal statistical processes. These properties 

are manifest through the empirical inverse power-law (IPL) probability density functions (pdf) that appear in physical, social, 

biological, economic, and physiological networks. Tools from non-equilibrium statistical physics and non-equilibrium information 

theory are used as well. To understand and facilitate network agility and adaptability, new techniques, methodologies, and 

analyses from a multi-disciplinary approach have been and will continue to be combined with the application of powerful 

modeling and simulation to explore the dynamics of networks. High-fidelity simulation, emulation, and experiments for largescale dynamic networks — based on complexity-performance trades and incorporation of live assets in an emulation environment 

— will provide us the ability to understand the complicated interactions between large-scale heterogeneous networks.

3 Taxonomy by Discipline

NS is unlike traditional scientific disciplines in that it identifies a particular structure, that of a network, as the basis for the 

science. Since structural webbing exists within and among all scientific disciplines, the over-arching goal of NS is to develop 

a body of principles that will enable modeling, design, analysis, prediction, and control of the behavior of the underlying 

phenomena in the physical (sensors, wireless web), information (communications, knowledge management), social (people, 

organizations, cultures), and cognitive (perceptions, beliefs, decision-making) sciences, to name only a few. The emphasis is 

not restricted to the network structure within a given discipline, but is focused on networked phenomena that encompass many 

disciplines as well as how such complex networks interact with one another. As a nascent science, the underlying principles 

are still missing, so the grand challenge for this effort is to create and organize the fundamental knowledge upon which to 

base these principles. Moreover, NS is intended to bridge the knowledge gap between disciplines and breakdown the artificial 

barriers that have been created between disciplines.

We have partitioned the scientific papers into four sections, each corresponding to a different category of disciplines: the 

physical sciences, the life sciences, the social sciences and the information sciences. This partitioning is intended to direct the 

reader to that area of investigation with which s/he is most comfortable. On the other hand, this opening narrative is intended 

to suggest that NS is more than just a thematic thread that weaves its way through each of the sections; it is intended as an 

over-arching perspective that encompasses the traditional disciplines. However, in the rush to discuss the tangible aspects of 

science, the fieldable products of technology, and the immediate needs of the Army, the network view so crucial to the scientific 

enterprise is often overlooked. It is emphasized here because, like the scientific barrier of complexity, it often represents the 

problem we would like to solve, but do not know how.

3.1 Physical Sciences

Physics has been the paradigm of science since the time of Newton. Consequently, when investigators wanted to emphasize 

the use of quantitative methods in their studies, they appended the suffix physics. In the 19th century this strategy resulted in 

sociology giving rise to sociophysics, psychology spawning psychophysics and biology producing biophysics. Of course, there 

were also a number of physicists speculating on the use of their techniques to understand non-physical phenomena. As a 

result, new disciplines evolved at the boundary between previously established disciplines. This way of extending the frontiers of 

disciplinary science proceeded into the 20th century, producing biochemistry, bioengineering, social engineering, econophysics, 

and so on. But it was the successes of the mathematical methods of physics that attracted the most attention.

The existence of an empirical law or principle from which the dynamics of a physical system can be determined appears almost 

mystical when trying to understand cognition or decision making. The 19th century physicists and applied mathematicians were 

able to take the conservation of energy and from that statement determine the laws of motion for any mechanical system. Such 

variational calculations in the hands of more practical people subsequently produced control theory and the dynamics of nonmechanical systems consistent with a set of imposed constraints.

Sardellitti and Barbarossa of Sapienza–University of Rome collaborated with Swami of ARL on, “Optimal Topology Control 

and Power Allocation for Minimum Energy Consumption in Consensus Networks” (page 21), to design a network topology 

that optimizes the spread of information, exemplified by the computation of globally relevant statistics by exploiting only local 
7

communication among the sensors in a wireless network. As in all design problems, there is a function to be carried out by the 

network while simultaneously satisfying a given constraint such as maximizing efficiency or minimizing energy consumption. Is 

the optimal topology sparse or dense? The research of Sardellitti et al. was concerned with minimizing energy use; the paper 

of Hollingshad et al., discussed later, was concerned with network efficiency, “A New Measure of Network Efficiency” (page 

13). Sardellitti et al. developed a method for optimizing the network topology and the power allocation across every link in 

order to minimize the energy necessary to achieve consensus across the sensor network. Since the topology is the result of 

optimization, this is a non-deterministic polynomial time (NP) combinatorial problem which the authors circumvent by means 

of a clever relaxation procedure.

Another approach to handling network complexity is by recognizing that the elements comprising a fighting force are interlinked 

to such a high degree that changing a single component influences and modifies the response of every other component in 

the network. Furthermore, each element in the network is itself a network and must be treated as such. This observation does 

not in itself contribute anything new to the discussion. It has been known since the first recorded wars that small changes in 

tactics and strategy can lead to overwhelming changes in outcome. This recognition has sometimes led to inaction, because the 

outcome of any action is unpredictable. What is new about the networking concept is the attempt to formalize the interactions 

so as to make the response to change predictable. New methods, such as nonlinear dynamics theory and chaos, have enabled 

us to determine the boundaries of predictability in apparently random dynamic processes, so we can realistically assess what 

we can and cannot know about a particular network. But it all goes back to the individual Soldier, situational awareness, and 

leadership involving the synthesis of information in real time.

The research group consisting of Hollingshad, Turalska, and Grigolini at the University of North Texas; Allegrini from the University 

of Pisa; and West from ARL collaborated on constructing, “A New Measure of Network Efficiency” (page 13). They address the 

dynamical origin of the scale-free probability distribution function (pdf) and, consequently, the related issue of network topology. 

Here again, the local interactions are found to determine the global properties of the network, using a nonlinear mathematical 

model with nearest neighbor interactions on a two-dimensional lattice. The network dynamics undergo a phase transition at 

a critical value of the control parameter, thereby inducing long-range correlations among the members of the network. The 

researchers determine that the distribution of links favoring long-range connections increases the efficiency of information 

transfer, which becomes optimally efficient when the IPL index for the degree pdf is one. They attribute this to the role of network 

leadership being transient, with the rank differences between the most-linked and least-linked nodes being temporary and over 

time the highest rank is shared by all members of the network. The full implications of this insight remain to be explored.

A fundamental difficulty with the concept of networking networks, such as nature has done within the human body or as applied 

to material structures, is related to the multiple scale and multi-physics requirements, as seen, for example, in electronic 

networks. If all physics were known and we had the necessary mathematical tools, we could model the ultimate performance 

of any physical network given the atomic composition of the parts. Quantum physics would enable atomic and nano-scale 

properties to be captured and parameterized yielding the electronic, thermal, and mechanical attributes of materials. From this 

ability to effectively couple across scales, we could eventually determine the capabilities and performance of multifunctional 

networks interrelating sensor, information, and communication capabilities. We could do this without fabricating network 

components, and we could achieve optimization at a level of sophistication that is beyond our wildest dreams using current 

technology and network development. On the other hand, it seems unreasonable to model every atom and electron in a physical 

network, yet the atomic level properties affect the composite performance and, therefore, cannot be neglected. At criticality, 

the smallest fluctuations are often amplified to the macroscopic level and, therefore, contain information on the nature of the 

large-scale network.

Ren and Zhao of the University of California, Davis have collaborated with Swami of ARL on, “Connectivity of Heterogeneous 

Wireless Networks” (page 53). This research results in a better understanding of the spatial connectivity of one heterogeneous 

network (secondary) embedded within another (primary); the heterogeneity is modeled using Poisson statistics for the ad hoc 

network. The analysis is done using percolation theory, which was initiated in statistical physics and has since developed into 

its own branch of mathematics. Percolation theory is a natural method for answering the question of whether the secondary 

network can be connected, i.e., when can a message be transmitted across the entire network without disruption? The 

researchers establish that when the density of the primary transmitters is beyond a certain level, there are simply not enough 

spectrum opportunities for any secondary network to be connected. The design parameters are the interference tolerances 

of the primary and secondary Poisson distributed networks and are used to design the optimal transmission power of the 

secondary network based on that of the primary. Consequently, the connectivity of large-scale ad hoc heterogeneous wireless 

networks are expressed in terms of the occurrence of the percolation phenomenon. It turns out that matching the interference 

ranges of the primary and secondary networks maximizes the tolerance of the secondary network to the primary traffic load. 

Since the tolerable interference range is an indicator of the receiver’s sophistication, this can be interpreted as an instance of 

the complexity matching principle discussed in, “Transmission of Information Between Complex Systems: l/f Resonance” (page 
8

233), and “Temporal Complexity of the Order Parameter at the Phase Transition” (page 213 ). 

A systematic procedure based on multiple models at different scales with representations at adjacent scales linked to each 

other by defined parameterized interfaces that can be calibrated to measurement is one reasonable approach to multi-scale 

modeling. This is done today through a combination of deterministic and stochastic modeling, where the smaller scales give rise 

to statistical uncertainty at the larger scales. Thus multi-scale modeling involves multiple organizations and disciplines leading 

to multiple, seemingly incompatible approaches. One school of thought believes it essential that an object-oriented approach 

be followed, meaning in this context that there are well defined interfaces to models capturing a particular scale and that the 

various scales are captured, most likely, by stand-alone programs. This multi-scale modeling is a grand scientific/engineering 

challenge for Army research.

Metz at the University of Ulm; Mitasova at North Carolina State University; and Harmon at ARL collaborated on, “Efficient 

Extraction of Drainage Networks from Massive, Radar-Based Elevation Models with Least Cost Path Search” (page 39). They 

use a least-cost drainage path (LCP) search methodology to significantly improve the accuracy of stream mapping, especially in 

remote, hard to reach regions and those that can be mapped with massive data sets. The LCP search methods were designed 

to find the shortest or fastest route from a starting point to a given destination. The success in extracting reliable drainage 

networks from massive radar-based data sets indicates that nature must optimize a cost function over geophysical time scales 

in direct analogy to the variation in total energy determining the dynamics of mechanical systems. Here the cost is interpreted 

in terms of local elevation and the least-cost path is minimal elevation contour.

3.2 Life Sciences

Complex networks in the life sciences generate patterns that have been summarized in a number of empirical relations. In 

physics we have Ohm’s law, the perfect gas law, and other empirical relations among the variables describing the physical 

process. These “laws” are, of course, phenomenological rules developed to summarize what has been observed experimentally 

and that were eventually derived using fundamental theory. In more complex living networks, equally simple rules have been 

uncovered that typically relate a particular function to the size of an organism, as first observed in biology some two hundred 

years ago. The first such expression related the mass of an animal’s brain to its total body mass, using mass as a measure of 

size and giving rise to the nomenclature of allometry relation (AR). Allometry, literally meaning by a different measure, has been 

defined as the study of body size and its consequences both within a given organism and between species in a given taxon.

The history of ARs is presented by West at Renesselaer Polytechnique Institute and West at ARL, who critiqued the various 

attempts to construct the theory underlying the scaling observed in ARs in, “On Allometry Relations” (page 115). ARs are 

critically important because they set the boundary conditions that network models must satisfy in order to be compatible with 

existing data. For example, the fractal network nutrient transport model used to successfully derive the metabolic AR is one of 

the network applications reviewed.

Another emerging application of network theory is in analyzing interactions and relationships in groups of animals and/or 

humans. The network science methodology allows for better understanding of the complexity of interactions and communications 

between individuals in groups. The review by Pollard, formerly at the University of California, Los Angeles and now with ARL, 

and Blumstein at ARL, “Evolving Communicative Complexity: Insights from Rodents and Beyond” (page 87), discusses the 

relationship between social interactions and communication patterns, and how focused studies of communicative complexity 

can better untangle the underlying evolutionary relationships. The functional relationship between attributes of social complexity 

and attributes of communicative complexity allow for a better understanding of how socialization and communication evolve.

The importance of interactions in neural networks for understanding of the operation of the brain has been recognized for a 

long time. However, only recently have advances in theoretical, experimental, and computational approaches to the study of 

neural networks begun to reveal fundamental principles of brain structure and function. There is emerging consensus among 

neuroscientists that traditional approaches concentrating on identifying brain structures responsible for specific functions and 

behavior need to be expanded to include investigations of interactions between these structures. New brain imaging techniques 

allow for more precise identification of neural tracts connecting different regions of the brain, providing information necessary 

for creation of the structural connectome network of an individual brain. This allows for analyzing and interpreting characteristic 

features in different individuals as well as identifying abnormalities caused by neurotrauma. The article by Kraft, Mckee, and 

Dagro at ARL in collaboration with Grafton at the University of California, Santa Barbara, “Combining the Finite Element Method 

with Structural Connectome-Based Analysis for Modeling Neurotrauma: Connectome Neurotrauma Mechanics” (page 99), 

presents a novel approach to analysis and interpretation of structural connectome obtained from MRI images by constructing a 

finite element model (FEM) informed by experimental data at a cellular level. This approach integrates brain injury biomechanics 

and graph theoretical analysis of neuronal connections to form a computational model capturing spatiotemporal characteristics 
9

of trauma. Time evolving simulations of the trauma investigated in this study show how brain network measures of global 

and local efficiency are degrading in time after trauma. This approach provides a basis for quantitative analysis tools for 

interpretation of neuroimaging data. Computational models based on theoretical analysis of neural network connections could 

help inform subsequent analysis of functional networks.

Neural network models based on neurophysiological properties of neurons provide insight into functional relationships between 

different brain structures. In recent years, due to increased computational capabilities, it has become possible to model larger 

networks composed of more realistic neurons. This allows for more realistic simulations of normal and/or pathological brain 

function, which can be directly compared with experimental or even clinical measurements. The article by Anderson, Azhar, 

Kudela, and Bergey at Johns Hopkins University in collaboration with Franaszczuk at ARL, “Epileptic Seizures from Abnormal 

Networks: Why Some Seizures Defy Predictability” (page 73), illustrates how this modeling approach can provide some insights 

into observed but difficult to interpret phenomena within the human brain. Epileptic seizures are the pathological activity of 

the brain disrupting the normal activity of an afflicted subject. While a seizure usually lasts only for a relatively short period 

of time (1-2 minutes), its unpredictability causes significant disruption in the life of the subject and may even cause serious 

injury due to associated loss of consciousness and body control. There is limited opportunity to record data associated with 

epileptic seizures. Even during hospitalization, patients may have as few as one or two seizures during an entire week. The 

computationally realistic modeling allows for investigation of multiple simulated seizures with full control of critical parameters. 

In Anderson et al., the authors created a computational model of the cortex consisting of more than 60,000 spiking neurons 

of several types based on histological data. Analysis of simulations of neuronal activity in this model suggest that, in the region 

of the cortex, with abnormal connectivity analogous to seizure focus in the human brain, it is possible to initiate seizure activity 

with random fluctuations of input from the surrounding cortical regions. This mechanism helps explain the difficulty in predicting 

partial seizures in some patients and suggests that intervention not only in the seizure focus area, but also in surrounding 

areas, may be necessary to prevent seizures from occurring.

The continuing research at ARL is directed toward combining the structural modeling of brain networks, “Combining the Finite 

Element Method with Structural Connectome-Based Analysis for Modeling Neurotrauma: Connectome Neurotrauma Mechanics” 

(page 99), with neurophysiological models similar to those presented in “Epileptic Seizures from Abnormal Networks: Why Some 

Seizures Defy Predictability” (page 73) to create a more complete model of brain neural networks.

The diverse applications of network-based or inspired studies in life sciences presented here show that the network sciences 

concepts and methodology provide a useful conceptual platform for description and analysis of various phenomena and 

processes. However, we are in the early stages of applying NS concepts to the various life sciences. Development of more 

specific mathematical and computational tools is required, but the future applications are limitless.

3.3 Social Sciences

In tactical networks, distributed decision making should take into account trust in the elements: the sources of information, the 

processors of information, the elements of the communications network (both radio and other) across which the information 

is transmitted, etc. This trust must often be derived under time-critical conditions, and in a distributed way. It must account for 

the uncertain provenance of the data, including the fact that information sources and relays in a tactical network are subject 

to subversion. Trust is a critical element of networked interactions, including interactions between humans and agents, that 

may be direct or computer-mediated. Trust and reputation are becoming increasingly important due to the explosive growth 

in electronically mediated social interactions, often with sometimes correspondingly weakened social control. Trust is a multifaceted concept and depends upon the context, the trustor, the environment, the infrastructure, and the mediating agents or 

relays. Trust is a critical element of modeling team, group, squad, and company behaviors. It must be taken into account in 

promoting group cohesiveness and effective problem-solving, and it must be adequately modeled in quantifying the tension and 

trade-off between individual utility and group welfare. Technology can enhance as well as weaken trust relationships. Currently 

used models of trust do not adequately capture critical elements of human trust (such as lack of transitivity, symmetry, and 

reciprocity, which calls for novel mathematical tools). Managing trust in such an environment is challenging. Cho and Swami 

from ARL collaborated with Chen from Virginia Tech on the paper, “A Survey of Trust Management in Mobile Ad Hoc Networks” 

(page 189), which addresses some of these challenges. It discusses the concepts and properties of trust pertinent to the 

constituent elements of a tactical network and proposes composite trust metrics that take into account the interaction between 

the constituent networks, the resource constraints, and the mission goals. This comprehensive survey discusses the differences 

between trust and trustworthiness, trust and risk, and trust management schemes.

How can one maximize rewards in an unknown environment? Learning is clearly a crucial element here. There are two important 

performance criteria to assess the efficacy of learning: consistency and regret bounds. A learning algorithm is consistent if the 

learned estimates asymptotically converge (in some sense) to the true values. Regret is a measure of the speed of convergence, 
10

the time-averaged loss in performance compared with that obtained by a genie. Anandkumar from the University of California, 

Irvine, and Michael and Tang from Cornell University collaborated with Swami from ARL on aspects of this problem in the paper, 

“Distributed Algorithms for Learning and Cognitive Medium Access with Logarithmic Regret” (page 173). The scenario they 

considered is related to the problem studied in “Connectivity of Heterogeneous Wireless Networks” (page 53). They consider the 

problem of so-called secondary users learning the unknown rewards (perhaps rates) obtained by accessing a set of channels. 

They consider a distributed framework where there is no information exchange or prior agreement among the secondary users. 

This introduces additional challenges: First, one can expect loss of performance due to collisions among users. Second, there 

is competition since all the secondary users are vying to access channels with the highest rewards. Using the theory of multiarmed bandits, they analyze the trade-off between exploration and exploitation and between cooperation and competition. They 

propose an order optimal distributed learning and access policy that minimizes the regret. An important notion here is that 

one must learn from mistakes (collisions and sensing that may be imperfect) and that one can never stop learning (one must 

continue to explore).

Cooperative interactions can lead to phase transitions and the emergence of spatial and network complexity in many systems. 

The Ising model has been used to model phase transitions in a variety of phenomena, including ferromagnetism, biological 

and neurophysiological processes, the flocking of birds, econophysics, etc. But the temporal complexity of such networks had 

not been explored until recently. Turalska and Grigolini from the University of North Texas collaborated with West of ARL on 

the paper, “Temporal Complexity of the Order Parameter at the Phase Transition” (page 213), which explores the temporal 

complexity issue. They study a decision making model in which the time intervals between significant events has a power-law 

distribution, which they show is equivalent to a two-dimensional Ising model. The underlying process is a renewal process, 

but it is non-stationary and exhibits long-term correlations. Thus the fundamental property of ergodicity no longer holds, and 

new mathematical tools are required to analyze these non-Poisson renewal processes. The researchers establish that phase 

transition leads to temporal complexity as manifested by non-stationary and non-ergodic fluctuations. They argue that this 

is a basis for complexity matching, a notion elaborated on in, “Transmission of Information Between Complex Systems: 1/f 

Resonance” (page 233). The pair of papers, “Temporal Complexity of the Order Parameter at the Phase Transition” (page 213) 

and “Transmission of Information Between Complex Systems: l/f Resonance” (page 233) indicate that information can be 

transferred between networks via extremely weak coupling, provided that the complexity of the two networks is matched (in 

the sense that a form of resonance exists). The concepts of complexity management and complexity matching are likely to find 

wide applicability in NS.

3.4 Information Sciences

In the social and life sciences, network discovery has led to increased understanding of how the brain works and how 

information propagates in communities of people. In the information sciences, one wishes to create a physical network that 

enables communication between entities, either person-to-person, person-to-machine, or machine-to-machine. Design of the 

physical network must account for the manner and the environment in which the network will be used.

In most non-military situations, the backbone of this network is fixed, for example, through cell towers. Senders and receivers 

are tethered to the backbone, that is, fixed spatially and hard-wired, or untethered, physically mobile, and wirelessly connected. 

However, in the environments in which the military operates, designers cannot expect a wireless infrastructure to be available for 

use. Thus, military units must be capable of establishing and maintaining their own network, which is a challenge during mobile 

operations. It is even more of a challenge when nodes on the network are small, autonomously mobile platforms operating on 

city streets and inside buildings, where floor plans and building materials affect the viability of links between nodes.

The use of multiple autonomous platforms is inspired by bionetworks in nature such as insect swarms. Although each single 

entity in a swarm has limited intelligence, the swarm appears to behave with intent when the entities function in concert. 

Consider the behavior of social insects, such as bees and ants. Fish and birds are other examples of animals whose collective 

behavior belies their limited intelligence. Although this analogy is intellectually appealing, differences exist between bionetworks 

and networks of autonomous mobile platforms that make it difficult to implement. Bionetworks, for example, swarm for survival, 

either to seek food or provide protection. In our grandest vision, swarms of autonomous platforms will exhibit complex behavior, 

such as intelligence or sentience through “situational awareness.” Further, communication within animal swarms occurs 

through proximity, e.g., ants touching antennae or following a chemical trail, line-of-sight vision, or sound. Put simply, biological 

swarms communicate simple messages primitively to perform primitive behaviors, whereas our vision of autonomous platforms 

requires complex messaging to achieve complex behaviors.

Given the present state of technology, a better analogy may be a shepherd herding sheep using a sheep dog. That is, a single 

intelligent entity controls a layered hierarchical network wherein each layer contains more entities of lesser intelligence than 

the layer above it. Whether a swarm or a pack is the better analogy, in any case, researchers have only scratched the surface 
11

of issues concerned with designing a mobile network created by a collective of autonomous platforms. Controlling the mobility 

of a pack of autonomous platforms and communicating within it remain unsolved problems.

How platforms move to achieve their mission, for example, generating the floor plan of a building previously unexplored by US 

units, yet maintain connectivity is the focus of the paper, “Robust Control for Mobility and Wireless Communication in CyberPhysical Systems with Application to Robot Teams,” by ARL researcher Fink and colleagues Riberio and Kumar at the University 

of Pennsylvania. The key element of the paper, “Robust Control and Mobility and Wireless Communication in Cyber-Physical 

Systems With Application to Robot Teams” (page 247), is the authors’ statement that “successful deployment of an autonomous 

team of robots requires joint cyber-physical controllers that determine (physical) trajectories for the robots while ensuring 

(cyber) availability of communication resources.” The authors’ approach to solving this problem is to consider communication 

parameters (bandwidth, attenuation, noise) as virtual obstacles that restrict platform movement. However, in an iterative 

control cycle, after platforms move, the network readjusts communications parameters to provide reliable communication (e.g., 

establishes new links, removes old ones, increases transmitted power, or reduces transmission bandwidth).

The basis for control is minimizing a potential function subject to the constraint that data transmission in every established link 

is above some threshold data rate. The strength of this approach is its global perspective of all links. Previous methods consider 

mobility from the perspective of local links only. Simulations and experiments with five mobile ground platforms validate the 

viability of this approach.

Analogies to nature inform the design of these experiments. In addition, they provide a benchmark against which one can 

measure performance. However, the experiments provide a perspective for reinterpreting and understanding nature. Does 

the collection of individual movement and communication minimize a global potential? If so, what potential? The total energy 

expended by the network to achieve goal? How does the mode of communication impact an individual’s movement if the 

communication is non-line-of-sight versus line-of-sight? The answers to these questions fall back into the realm of network 

science theory.

Designers of the physical networks that enable the transfer of electronic information need to consider means for simple, rapid, 

and robust data transfer. In applications where the network is stable and network operators have considerable opportunity to 

monitor and control the network, for example, commercial networks, message traffic can be handled sequentially. However, in 

complex, less stable, environments, parallel transmission of messages provides more robust operation. In this case, messages 

between two channels are transmitted contemporaneously to a relay. The relay combines the messages and retransmits the 

same composite message to both channels. Since each channel knows the signal they transmitted, it is a simple matter for 

them to extract the unknown message from the other channel.

However, parallel messaging dictates that messages between channels must arrive within certain time blocks, which requires 

timing control to minimize transmission delays. As the likelihood of large delays increases, the likelihood of missed messages 

also increases. Although coherent detection, in which all channels operate on the same time base, minimizes timing delays, 

it requires precise control of phase. Such control is unlikely in unstable environments. In these situations, designers trade the 

precision of coherent detection for the simplicity of non-coherent detection, which does not require a frequency or phase base.

In their implementation of a non-coherent system, Valenti and Ferrett of West Virginia University and Torrieri of ARL, “Noncoherent Physical-Layer Network Coding with FSL Modulation: Relay Receiver Design Issues” (page 221), use frequency shift 

keying (FSK) to encode signals. In FSK, the network needs to detect only changes in frequency, not measure absolute frequency, 

to demodulate signals. Because the encoding does not require a reference frequency, it reduces network requirements on 

frequency control and on power control.

In this paper, Valenti, Torrieri and Ferrett present the theory, design, and simulated operation of FSK for non-coherent physicallayer network coding. They show that the encoding provides high throughput with low energy efficiency, which is typical of noncoherent systems. To overcome power losses, they employ low rate turbo codes. The authors show that, for the same energy 

efficiency, their digital network coding (DNC) approach achieves a higher throughput rate in comparison to more traditional 

link-layer network coding (LNC). In one example, the authors show a one-third rate increase. That is, four messages can be sent 

using the authors’ approach for every three messages transmitted using LNC.

Aquino of the Imperial College London, Bologna of the Instituo de Alta Investigacion Chile, Grigolini of the University of North 

Texas, and West of ARL investigated the transport of information between two complex networks to determine the conditions 

under which the information transport is maximized in, “Transmission of Information Between Complex Systems: l/f Resonance” 

(page 233). If network theory is ever to become Network Science, there must exist general principles that transcend the level of 

disciplinary mechanisms and that remain valid across the physical, biological, and social phenomena described above. In this 

paper, Aquino et al. use arguments from non-equilibrium statistical physics to make the case that if the complexity of a network 
12

can be quantified, then the information transferred is maximized through the matching of the complexity of the interacting 

networks.

The complexity of networks, from avalanches of neuronal firings within the human brain to the blinking times in colloidal 

quantum dot fluorescence, is manifest in IPL distributions. The level of complexity of these networks is quantified by the value 

of the IPL index. Of particular importance is the manner in which information is shuttled back and forth between such networks 

and whether or not there exists a general principle that guides the flow of information. The mathematician Norbert Wiener 

speculated that such a principle existed for information-dominated processes. A half century later, Aquino et al. in, “Transmission 

of Information Between Complex Systems: l/f Resonance” (page 233), proved Wiener’s conjecture to be true by establishing 

the Principle of Complexity Management in which information transport is mitigated by the relative values of the networks’ IPL 

indices. They establish the principle by generalizing the fluctuation-dissipation theorem from non-equilibrium statistical physics 

to non-ergodic processes. In this way, it is possible for a network with lesser energy, but greater information, to organize a 

network with greater energy, but lesser information; this is an information-dominated process.

On one hand, a speaker who delivers a lecture standing motionless in front of an audience, talking in a monotone, transfers 

little or no information. The group very quickly drifts off and stops listening almost independently of the content of the lecture. 

On the other hand, another speaker with a modulated voice, striding back and forth across the front the room while gesturing 

dramatically, can hold the audience’s attention. The complexity of the presentation matches that of the typical cognitive activity 

of the brain and information finds its way into the consciousness of the audience member. The success of the second speaker, 

as well as a myriad of other social and psychological phenomena, are explained by the Principle of Complexity Management and 

the relative complexity of the two interacting networks.

The following collection of recently published journal articles is a select sample of the broad spectrum of ARL’s network science 

research investigating physical, social/cognitive, information, and communication networks, and their complex interactions. 

They represent foundational knowledge that will enable optimized human performance and enhanced speed and precision in 

complex networked environments.

References

[1] D.S. Alberts and R.E. Hayes, Power to the Edge: Command and Control in the Information Age, US DoD CCRP Publishing 

Services, June 2003.

[2] The Networking Information Technology Research and Development (NITRD) Program: 2012 Stategic Plan; Executive 

Office of the President, National Science and Technology Council, July 2012.

[3] Network Science, National Research Council of the National Academies, Washington DC, www.nap.edu (2005).

[4] See, for example: J. G. Andrews, N. Jindal, M. Haenggi, R. Berry, S. Jafar, D. Guo, S. Shakkottai, R. W. Heath, Jr., M. Neely, 

S. Weber, and A. Yener, “Rethinking Information Theory for Mobile Ad Hoc Networks”, IEEE Communications Magazine, 

December 2008
Research@

13

A New Measure of Network Efficiency

Nicholas W. Hollingshad, Malgorzata Turalska, 

Paolo Allegrini, Bruce J. West, and Paolo Grigolini

Physica A: Statistical Mechanics and its Applications, 391(4), 1894–1899 (2012)
14

Physica A 391 (2012) 1894–1899

Contents lists available at SciVerse ScienceDirect

Physica A

journal homepage: www.elsevier.com/locate/physa

A new measure of network efficiency

Nicholas W. Hollingshada,∗, Malgorzata Turalska a, Paolo Allegrini b, Bruce J. West c,

Paolo Grigolini a

a Center for Nonlinear Science, University of North Texas, P.O. Box 311427, Denton, TX 76203-1427, USA

b Istituto di Fisiologia Clinica (IFC-CNR), Via Moruzzi 1, 56124 Pisa, Italy

c Information Science Directorate, US Army Research Office, Durham, NC 27709, USA

article info

Article history:

Received 25 July 2011

Received in revised form 3 October 2011

Available online 15 November 2011

Keywords:

Network

abstract

We address the issue of the dynamical origin of scale-free link distributions. We study

a two-dimensional lattice of cooperatively interacting units. Although the units interact

only with the four nearest neighbors, a sufficiently large cooperation strength generates

dynamically a scale-free network with the power law index ν approaching 1. We explain

this result by using a new definition of network efficiency determined by the Euclidean

distance between correlated units. According to this definition the link distribution

favoring long-range connections makes efficiency increase. We embed an ad hoc scale-free

network with power index ν ≥ 1 into a Euclidean two-dimensional space and show that

the network efficiency becomes maximal as ν approaches 1. We therefore conclude that

ν = 1 emerging from the cooperative interaction of units may be a consequence of the

principle of network maximal efficiency.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

The work of Albert and Barabási (AB) [1] prompted interest in scale-free networks, namely networks with the inverse

power law distribution of links

p(k) ∝

1

kν , (1)

with k denoting the number of links per node. The AB model assumed the preferential attachment of the Yule process and

gave rise to the power-law index ν = 3. The ubiquity of these networks has led to the research into the dynamical origin

of the scale-free property. Fraiman et al. [2] has recently studied the two-dimensional Ising model at criticality and using

a suitable threshold ρ+ established a link between the nodes with a dynamical correlation exceeding that threshold. The

interesting result of that procedure was the emergence of the scale-free distribution density of Eq. (1) with a power index

ν that, with the proper choice of average connectivity ⟨k⟩, gives values close to ν = 2 [3]. It is remarkable that the same

procedure, applied to the brain, yields results very close to those afforded by the Ising model [2].

Like the work of Ref. [2], we begin with a regular two-dimensional network, but then take an entirely different approach.

Rather than using the Ising model, we adopt the Decision Making (DM) model of Refs. [4,5] for the creation of the dynamically

induced network. The key result is that while we also find the emergence of a scale-free network at criticality, our network

has an exponent ν close to 1.

∗ Corresponding author.

E-mail address: nwh0012@unt.edu (N.W. Hollingshad).

0378-4371/$ – see front matter © 2011 Elsevier B.V. All rights reserved.

doi:10.1016/j.physa.2011.11.017
15

N.W. Hollingshad et al. / Physica A 391 (2012) 1894–1899 1895

This finding suggests that the cooperative interaction between the units in the network generated from the DM model

should make the network very efficient. However, we find herein that the conventional measures of network efficiency [6]

are relatively insensitive to ν < 2. The second result of this paper is a new measure of network efficiency, the perception

length, that overcomes this limitation, and shows that network efficiency increases as ν approaches 1.

The outline of this paper is as follows. We devote Section 2 to proving the dynamical emergence of the scale-free network

with ν = 1. Section 3 shows that ν approaching 1 maximizes the perception length while realizing a statistical condition

equivalent to the dynamically generated network. We devote Section 4 to concluding remarks.

2. Dynamically induced scale-free network with ν = 1

The question of whether the network emerging from dynamical correlations is scale-free is not trivial. For this paper, we

adopt the Decision Making (DM) model of Refs. [4,5]. Although the DM model yields a phase transition very similar to that

of the conventional Ising model [7], the DM model does not have a Hamiltonian foundation. Furthermore, while the Ising

model rests on the action of a thermal bath at a finite temperature, and the single spins in isolation and with no thermal

driving would be dynamically frozen, the units of the DM model, in isolation, are driven by Poisson dynamics. This difference

may lead to a dynamically created scale-free network different from that of Ref. [2].

The DM model considers N discrete variables located at the nodes of a two-dimensional square lattice. Each unit si is a

stochastic oscillator and can be found in either of two states, ‘‘yes’’ (+1) or ‘‘no’’ (−1). The cooperation among the units is

realized by setting the transition rates between two states to the time-dependent form:

g12(t) = geK M2(t)−M1(t)

M (2)

and

g21(t) = ge−K M2(t)−M1(t)

M (3)

where M denotes the total number of nearest neighbors, which in the case of a two-dimensional lattice results in M = 4.

M1(t) and M2(t) are the nearest neighbors who are making the decision ‘‘yes’’ and ‘‘no’’, respectively. The single unit in

isolation, K = 0, fluctuates between ‘‘yes’’ and ‘‘no’’, with the rate g. When coupling constant K > 0, a unit in the state ‘‘yes’’

(‘‘no’’) makes a transition to the state ‘‘no’’ (‘‘yes’’) faster or slower according to whether M2 > M1 (M1 > M2) or M2 < M1

(M1 < M2), respectively. The quantity KC is the critical value of the control parameter K, at which point a phase-transition

to a global majority state occurs. It can be shown that for a lattice of infinite size KC = 2 ln(1 + √2) ≈ 1.7627.

All simulations are implemented on a lattice of N = 100×100 nodes with periodic boundary conditions. In a single time

step, a run over the entire lattice is performed and for every node an appropriate transition rate (g12 or g21) is calculated

according to which a node is given possibility to change its state. Under those conditions, setting parameter g = 0.01, the

phase transition to the global majority case occurs at KC ≈ 1.70. To study the dynamically induced network topology, we

considered the DM model with this critical value of coupling constant. After an initial 106 time steps, we record 2000 lattice

configurations, obtaining the dynamics of each node {si(t)}. In the next step, we evaluate the linear correlation coefficient

between the i-th and the j-th node:

r(i, j) = ⟨si(t)sj(t)⟩−⟨si(t)⟩⟨sj(t)⟩



⟨s

2

i (t)⟩−⟨si(t)⟩2



⟨s

2

j (t)⟩−⟨sj(t)⟩2

, (4)

where ⟨· · ·⟩ stands for the time average. Nodes i and j of the dynamically induced network are connected by a link when the

correlation between nodes i and j of the two-dimensional lattice is greater than a given positive threshold ρ+.

We investigated a wide range of positive thresholds ρ+ = [0.10 : 0.90]. As expected, correlation networks constructed

with increasing value of the threshold ρ+ are characterized by a decreasing number of links, which results in significant

changes in the corresponding degree distribution. Networks obtained with low values of the threshold ρ+ result in narrow,

peaked degree distributions and relatively high values of the mean degree ⟨k⟩. Topologies resulting from very strong

correlations, however, are characterized by a distribution of links that has a mean close to the minimum value of ⟨k⟩ = 1

and drops off very rapidly. Interestingly, in between those conditions there exists a correlation threshold that creates a

dynamically induced network with a power-law distribution of degrees that is shown in Fig. 1. The connectivity of a network

obtained under this particular condition is inspected and only nodes that are part of a giant component are considered to be

a part of new topology. The resulting network consists of 6740 nodes with a mean degree of ⟨k⟩ = 38. For a correlation

threshold ρ+ = 0.61, we find that the distribution of links follows an inverse power law with the scaling parameter

ν = 1.20 ± 0.23 (R2 = 0.9556).

How can the dynamical origin of this extremal condition be explained? From an intuitive point of view there exists a

connection between the scale-free distribution of links and Zipf’s law [8,9]. In fact, if we rank the nodes moving from the

richest to the poorest, according to Zipf [10] we should obtain the relation between the connectivity k and rank r:

k(r) = C

rα , (5)
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1896 N.W. Hollingshad et al. / Physica A 391 (2012) 1894–1899

Fig. 1. Degree distribution p(k)for correlation network obtained with a positive threshold ρ+ = 0.61. The line follows estimated power-law relation with

the exponent of ν = 1.20.

with the power-law index α being very close to 1. If we select randomly a given rank r with probability π (r)dr of being in

the interval [r,r + dr], this should be equal to the probability p(k)dk, of having k links in the interval [k, k + dk],

p(k)dk = π (r)dr. (6)

Assuming that π (r) is constant, we obtain after some algebra

p(k) ∝

1

k1+ 1

α

, (7)

and thus the two power-law indices are related by

ν = 1 +

1

α . (8)

This leads us to conclude how an efficient complex network is organized if Zipf’s law applies. For example, this suggests

that the intelligence of human brain should be characterized by ν = 2. The recent research work of Fraiman et al. [2] seems

to confirm this prediction. The authors of Ref. [2] studied the two-dimensional Ising model at criticality and using a suitable

threshold ρ+ established a link between the nodes with a dynamical correlation exceeding that threshold. The interesting

result of that procedure was the emergence of the scale-free distribution density of Eq. (1) with a power index ν that, with

the proper choice of average connectivity ⟨k⟩, gives values close to ν = 2 [3]. It is remarkable that the same procedure,

applied to the brain, yields results very close to those afforded by the Ising model [2].

It is also interesting to notice that Boettcher and Percus [11] applied the method of extremal optimization to a Isinglike model to evaluate the rank distribution of Eq. (5) and found that when the number of units tend to infinity the rank

distribution tends to α = 1. This result implies on the basis of the earlier remarks that the cooperation of units in an Isinglike model is expected to confirm Zipf’s prediction and consequently ν = 2. For results confirming this prediction, see also

Refs. [12,13].

We have found, therefore, that the DM model generates a scale-free network at criticality, but with ν approaching 1

which violates Zipf’s condition of α = 1. The emergence of α ≫ 1 corresponds to the number of links k dropping very

quickly when we move from r = 1 to subsequent ranks.

3. Embedding scale-free networks in a two-dimensional regular lattice

The efficiency of a network has been mainly studied on the basis of topological length L [6]. The topological distance

between two nodes is the minimum number of steps needed to move from one node to another. The quantity L is the

corresponding mean value of the minimum topological distance between nodes. As a consequence, the emergence of scalefree networks can be explained by making the conjecture that real networks evolve in time so as to realize maximal

efficiency. In the case of the scale-free network this efficiency is measured by the minimal value of L.

In 2003, Cohen and Havlin [14] determined that scale-free networks are very efficient. They calculated that networks

with N nodes have a topological length L given by

L ∼ ln(ln N), (9)

for a power-law index ν < 3, which is smaller than in the AB model.

As a consequence of Eq. (9), the topological length L remains small even if N is very large. This relation between L and N

suggests that the complex networks with ν < 3 are very efficient.

The region ν < 2 has not been studied by many authors. As an interesting example of earlier work in this region we

refer to the work of Refs. [15,16]. In both cases the authors adopt a kind of generalization of the perspective of Albert and
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N.W. Hollingshad et al. / Physica A 391 (2012) 1894–1899 1897

Fig. 2. Perception length P as a function of ν for ad hoc scale-free networks. The circles represent networks with constant ⟨k⟩ = 4.95; the squares represent

networks with constant N = 1313.

Barabási [1], namely, they grow a network with special prescriptions allowing them to overcome the limiting condition

ν > 3 and yielding ν < 2.

More recently there has been some interest in the efficiency of hierarchical networks with ν < 2 [17] and ν ∼ 1 [18,19].

These networks are characterized by very small topological lengths. In the case of Ref. [19], for N → ∞ the authors predict

the value L = 2. Our numerical calculations show that L is insensitive to ν in the range 1 <ν< 2. Thus, there are good

reasons to believe that the prediction of Eq. (9) applies with no significant dependence on ν.

There are also good reasons to believe that the dynamic source of scale-free networks has a probabilistic rather than

a deterministic origin, as we shall demonstrate. The work of Fraiman et al. [2], although fitting Zipf’s condition, allows us

to define the concept of perception length, which is then used to shed light onto the condition ν = 1. The nodes of a

two-dimensional regular lattice, where each unit interacts with only its four nearest neighbors, correspond to a network

whose topological efficiency is very low. However, if we let the units cooperatively interact, and we set a link between two

nodes, then when their dynamical correlation is large enough, the resulting dynamical network may have a much lower

L. We record the Euclidean distance between each pair of correlated nodes, and we define the network efficiency as the

corresponding mean value. More precisely, we define the network efficiency by means of the quantity

P = 1

N

−N

1

λi, (10)

where

λi = −

ki

j=1

dij (11)

and dij is the Euclidean distance between node i and its nearest neighbor j. We refer to P as the perception length of the

system, this being the measure of the network efficiency replacing 1/L. For numerical analysis, the perception length P is

calculated as follows.

1. Create a 2-dimensional square lattice, where a is the length (in units of the lattice) of the side of the square, and it is

necessary that √N < a.

2. Embed the network on the lattice by randomly assigning the network nodes to lattice points.

3. Using periodic boundary conditions for the lattice, calculate the Euclidean length of each link.

4. These lengths are then used to calculate P in accordance with Eq. (10).

This procedure was then applied to scale-free networks generated according to the method of Catanzaro et al. [20] with

⟨k⟩ = 4.95 and values of ν ranging from 1 to 2. The nodes were embedded randomly in a square lattice of size just sufficient

to contain the network with the largest number of nodes. The perception lengths were calculated, and the results show that

over this range the perception length increases as ν approaches 1 (Fig. 2).

We now compare the sensitivity of the conventional network efficiency (defined as the reciprocal of the network length

1/L) and the perception length P to the variation in ν. These results are shown in Fig. 3. Note that the perception length P and

1/L have been normalized to allow the presentation in a convenient scale. These results show that the perception length, as

a measure of network efficiency, is more sensitive to changes in ν than the conventional 1/L.

Applying this approach to the DM network discussed in Section 2 above yields a mean perception length = 1382. To

provide a basis for comparison, we calculated the perception length for a comparably sized ad hoc scale-free network, also

prepared in accordance with the algorithm in Ref. [20]. The results are compared in Table 1, which shows a very good

agreement.
18

1898 N.W. Hollingshad et al. / Physica A 391 (2012) 1894–1899

Fig. 3. Network efficiency as a function of ν. In both figures, the circles represent the normalized perception length, and the squares represent the

normalized 1/L. Error bars are not shown; standard deviations for the normalized perception lengths were very small (of the order of 0.005).

Table 1

Comparison of perception lengths for the DM network and a comparable

ad hoc scale-free network.

DM network Ad Hoc network

Degree distribution exponent, ν 1.20 1.19

Number of nodes, N 6740 6744

Number of links, k 128,448 128,413

Perception length, P 1382 1373

4. Concluding remarks

The main result of this paper is the proof that the perception length P defined by Eq.(10)is more sensitive to ν in the range

1 <ν< 2, and may therefore be a more useful measure of network efficiency than the topological length L [6]. This new

definition is based on the assumption that the scale-free networks, even those derived using deterministic and hierarchical

arguments [17–19], are compatible with the dynamical derivation. The apparent conflict between the two kinds of network

is explained by noticing that the correlation between different nodes evaluated according to the numerical prescription

of Section 2 rests on time windows of finite size and moving these windows to different time regions generate different

networks with the same hierarchical topology. An alternative but probably more attractive way to explain this is that the

leadership moves in time from some nodes to others. The nodes of the two-dimensional regular lattice that we used for the

dynamical derivation of the free-scale distribution of Eq. (1) are totally equivalent, and the rank differences between the

richest and the poorest nodes are temporary and in the long time run the same high rank is shared by all the nodes.

There are some questionable aspects of the dynamical derivation of ν = 1, realized in Section 2. First of all this result

depends on an arbitrarily selected threshold ρ+. In addition, the power index ν derived from the data in Fig. 1 is affected

by the large fluctuations of the distribution density p(k) for large values of k, thereby explaining why we derive ν = 1

rather than ν = 2, as in the work of Ref. [3]. However, on one hand, this observation led us to the discovery of the new

measure of network efficiency of Eq. (10) and, on the other hand, as already mentioned in Section 1, the DM model is not

quite equivalent to the Ising model.

Furthermore, the interesting fact that α = 1 (Zipf’s law) is related to ν = 2 is not totally convincing. The authors of

Ref. [9] established a connection between topological and dynamical properties by running a random walker on a scale-free

network with the power index ν. They found that the waiting time distribution density for returns of the walker to a given

node are described by an inverse power law with power indexμ = 3−ν. Using a generalized central limit theorem, they also

proved that the same index μ also describes the inverse power law form of the probability density of relative frequencies

p(f) ∼ f −μ, so that, in analogy with Eq. (8), a rank-frequency law f ∼ r−α yields μ = 1 + 1/α, hence α = 1/(2 − ν),

namely α = 1 for ν = 1. It is interesting to note that ν = 1 yields μ = 2, and that the value μ ≈ 2 emerges from the recent

neurophysiological work of Refs. [21,22].

Thus, in addition to the new efficiency measure of Eq. (10), which is the unquestionable result of this paper, we are also

led to make the plausible conjecture that the DM model may be an adequate tool to study the emergence of intelligence
19

N.W. Hollingshad et al. / Physica A 391 (2012) 1894–1899 1899

in the sense pointed out by Couzin [23] and Cavagna et al. [24]. According to these authors the flocks of birds, as well as

the brain [25,26], are systems at criticality, with a correlation length as large as the size of these systems. We are also led

to make the conjecture that the dynamical generation of a scale-free network with ν = 1 may be the manifestation of the

principle of network maximal efficiency: a network of cooperatively interacting units evolves naturally so as to maximize

P, the new measure of network efficiency.

References

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[2] D. Fraiman, P. Balenzuela, J. Foss, D.R. Chialvo, Ising-like dynamics in large-scale functional brain networks, Phys. Rev. E 79 (1–10) (2009) 061922.

[3] E. Tagliazucchi, D.R. Chialvo, The collective brain, in: P. Grigolini, B.J. West (Eds.), Decision Making: A Psychophysics Application of Network Science,

World Scientific, Singapore, 2011.

[4] S. Bianco, E. Geneston, P. Grigolini, M. Ignaccolo, Renewal aging as emerging property of phase synchronization, Physica A 387 (2008) 1387–1392.

[5] M. Turalska, M. Lukovic, B.J. West, P. Grigolini, Complexity and synchronization, Phys. Rev. E 80 (1–12) (2009) 021110.

[6] S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, D.U. Hwang, Complex networks: structure and dynamics, Phys. Rep. 424 (2006) 175–308.

[7] M. Turalska, B.J. West, P. Grigolini, Temporal complexity of the order parameter at the phase transition, Phys. Rev. E 83 (1–6) (2011) 061142.

[8] L.A. Adamic, B.A. Huberman, Zipf’s law and the Internet, Glottometrics 3 (2002) 143–150.

[9] P. Allegrini, P. Grigolini, L. Palatella, Intermittency and scale-free networks: a dynamical model for human language complexity, Chaos Solitons Fractals

20 (2004) 95–105.

[10] G.K. Zipf, Selected Studies of the Principle of Relative Frequency in Language, Harvard University Press, Cambridge, MA, 1932; Human Behavior and

the Principle of Least Effort, Addison-Wesley, Cambridge, MA, 1949.

[11] S. Boettcher, A.G. Percus, Optimization with extremal dynamics, Phys. Rev. Lett. 86 (2001) 5211–5214.

[12] S. Boettcher, Acquiring long-range memory through adaptive avalanches, in: P. Grigolini, B.J. West (Eds.), Decision Making: A Psychophysics

Application of Network Science, World Scientific, Singapore, 2011.

[13] It is important to notice that in the work of Ref. [12] the role of Eq. (5) is played by Eq. (4) that refers to the probability that a node of rank r is updated.

However, if we divide Eq. (5) by the maximum number of links that a node may have, k is replaced by quantity that can be interpreted as a probability

[14] R. Cohen, S. Havlin, Scale-free networks are ultrasmall, Phys. Rev. Lett. 90 (1–4) (2003) 058701.

[15] H. Zhou, Scaling exponent and clustering coefficients of a growing random network, Phys. Rev. E 66 (1–6) (2002) 016125.

[16] H. Seyed-allaei, G. Bianconi, M. Marsili, Scale-free networks with an exponent less than two, Phys. Rev. 73 (1–5) (2006) 046113.

[17] A.-L. Barabási, E. Ravasz, T. Vicsek, Deterministic scale-free networks, Physica A 299 (2001) 559–564.

[18] M. Chen, B. Yu, P. Xu, J. Chen, A new deterministic complex network model with hierarchical structure, Physica A 385 (2007) 707–717.

[19] Y. Gua, J. Suna, A tree-like complex network model, Physica A 389 (2010) 171–178.

[20] M. Catanzaro, M. Boguñá, R. Pastor-Satorras, Generation of uncorrelated random scale-free networks, Phys. Rev. E 71 (1–4) (2005) 027103.

[21] P. Allegrini, D. Menicucci, R. Bedini, L. Fronzoni, A. Gemignani, P. Grigolini, B.J. West, P. Paradisi, Spontaneous brain activity as a source of ideal 1/f

noise, Phys. Rev. E 80 (1–13) (2009) 061914.

[22] P. Allegrini, D. Menicucci, R. Bedini, A. Gemignani, P. Paradisi, Complex intermittency blurred by noise: theory and application to neural dynamics,

Phys. Rev. E 82 (1–4) (2010) 015103 (R).

[23] I. Couzin, Collective minds, Nature 445 (2007) 715.

[24] A. Cavagna, A. Cimarelli, I. Giardina, G. Parisi, R. Santagati, F. Stefanini, M. Viale, Scale-free correlations in starling flocks, Proc. Natl. Acad. Sci. 107

(2010) 11865–11870.

[25] D.R. Chialvo, Emergent complexity: what uphill analysis or downhill invention cannot do, New Ideas in Psychology 26 (2008) 158–173.

[26] D.R. Chialvo, Complex emergent neural dynamics, Nature Physics 6 (2010) 744–750.
20

Physica A 391 (2012) 6246

Contents lists available at SciVerse ScienceDirect

Physica A

journal homepage: www.elsevier.com/locate/physa

Erratum

Erratum to ‘‘A new measure of network efficiency’’ [Physica A 391 (4)

(2012) 1894–1899]

Nicholas W. Hollingshad∗, Malgorzata Turalska, Paolo Allegrini, Bruce J. West, Paolo Grigolini

Center for Nonlinear Science, University of North Texas, P.O. Box 311427, Denton, TX 76203-1427, USA

Istituto di Fisiologia Clinica (IFC-CNR), Via Moruzzi 1, 56124 Pisa, Italy

Information Science Directorate, US Army Research Office, Durham, NC 27709, USA

article info

Article history:

Available online 25 July 2012

The publisher regrets to inform you that the below mentioned Acknowledgments were not included for the article ‘‘A

new measure of network efficiency’’

Please find below the Acknowledgments of the above mentioned paper in order to include a funding agency:

‘‘N. W. Hollingshad, M. Turalska, P. Grigolini warmly thank ARO and Welch for financial support through Grants

W911NF1110478 and B-1577, respectively’’.

The publisher would like to apologize for any inconvenience caused.

DOI of original article: http://dx.doi.org/10.1016/j.physa.2011.11.017. ∗ Corresponding author at: Center for Nonlinear Science, University of North Texas, P.O. Box 311427, Denton, TX 76203-1427, USA.

E-mail address: nwh0012@unt.edu (N.W. Hollingshad).

0378-4371/$ – see front matter © 2012 Elsevier B.V. All rights reserved.

doi:10.1016/j.physa.2012.07.035
Research@

21

Optimal Topology Control and Power Allocation 

for Minimum Energy Consumption in 

Consensus Networks

Stefania Sardellitti, Sergio Barbarossa, and Ananthram Swami

IEEE Transactions on Signal Processing, 60(1), 383-399 (2012)
22

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 60, NO. 1, JANUARY 2012 383

Optimal Topology Control and Power Allocation

for Minimum Energy Consumption in

Consensus Networks

Stefania Sardellitti, Sergio Barbarossa, Fellow, IEEE, and Ananthram Swami, Fellow, IEEE

Abstract—Consensus algorithms have generated a lot of interest

due to their ability to compute globally relevant statistics by only

exploiting local communications among sensors. However, when

implemented over wireless sensor networks, the inherent iterative

nature of consensus algorithms may cause a large energy consumption. Hence, to make consensus algorithms really appealing

in sensor networks, it is necessary to minimize the energy necessary to reach a consensus, within a given accuracy. We propose a

method to optimize the network topology and the power allocation

over each active link in order to minimize the energy consumption.

We consider two network models: a deterministic model, where

the nodes are located arbitrarily but their positions are known,

and a random model, where the network topology is modeled as

a random geometric graph (RGG). In the first case, we show how

to convert the topology optimization problem, which is inherently

combinatorial, into a parametric convex problem, solvable with

efficient algorithms. In the second case, we optimize the power

transmitted by each node, exploiting the asymptotic distributions

of the eigenvalues of the adjacency matrix of an RGG. We further

show that the optimal power can be found as the solution of a

convex problem. The theoretical findings are corroborated with

extensive simulation results.

Index Terms—Consensus algorithms, minimum energy consumption, random graphs, sensor networks, topology control.

I. INTRODUCTION

AVERAGE consensus algorithms have received considerable attention in recent years because of their ability to

enable globally optimal decisions using only local exchange of

information among nearby nodes [1]–[3]. The price paid for

this simplicity and the underlying decentralized philosophy is

that consensus algorithms are inherently iterative. As a consequence, the implementation of consensus algorithms over a

wireless sensor network (WSN) requires an iterated exchange

Manuscript received October 25, 2010; revised May 13, 2011; accepted

September 21, 2011. Date of publication October 13, 2011; date of current

version December 16, 2011. The associate editor coordinating the review of

this manuscript and approving it for publication was Dr. Mark Coates. This

work was supported in part by the WINSOC project, a Specific Targeted

Research Project (Contract Number 0033914) co-funded by the INFSO DG of

the European Commission within the RTD activities of the Thematic Priority

Information Society Technologies.

S. Sardellitti and S. Barbarossa are with the DIET, Sapienza University

of Rome, 00184 Rome, Italy (e-mail: Stefania.Sardellitti@uniroma1.it;

sergio@infocom.uniroma1.it).

A. Swami is with the ARL, Adelphi, MD 20783 USA (e-mail: a.swami@ieee.

org).

Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TSP.2011.2171683

of data among the nodes, which might cause an excessive energy consumption. This must be contrasted with a centralized

strategy where there is a sink node that, after collecting all the

observations from the sensors (perhaps over multiple hops), is

virtually able to compute the desired statistic in a single shot.

Hence, to make consensus algorithms practically appealing in

a sensor network context, it is necessary to minimize the energy consumption necessary to reach consensus. Clearly, the

network topology plays a fundamental role in determining the

convergence rate [4]. It is well known that, as the network connectivity increases, so does the rate of convergence. However,

having a densely connected network requires a high power consumption to guarantee reliable direct links between many nodes.

In principle, having a fully connected network is equivalent to

having as many sink nodes as sensors, so that the convergence

time of fully connected networks is minimum. However, the

power consumption necessary to maintain a fully connected network is also maximum. On the other hand, a minimally connected network entails low power consumption to maintain a

few links, but, at the same time, it requires a large convergence

time. Since what really matters in a WSN is the overall power

spent to achieve consensus, this paper addresses the problem

of finding the optimal network topology that minimizes this

overall power consumption, taking into account convergence

time and transmit power necessary to establish reliable links

jointly. The search for the optimal topology is, per se, a combinatorial problem whose solution becomes prohibitive even for

small scale networks. Nevertheless, we will present a relaxation

technique that converts the problem into a convex problem, with

minimum performance loss.

If the links among the nodes are symmetric, or, equivalently,

if the graph describing the network topology is undirected, the

convergence rate can be lower bounded through the so called

algebraic connectivity, defined as the second smallest eigenvalue of the graph Laplacian [5]. For this reason, there have

been works aimed at maximizing the algebraic connectivity of

a given undirected graph by a suitable choice of the weights associated to each edge [6], [7]. Alternatively, in [8] it was shown

how some network topologies, such as small world graphs, can

greatly increase the convergence rate. In [9], the authors show

that nonbipartite Ramanujan graphs constitute a class of topologies maximizing the convergence rate. In [10] it was shown how

to add edges from a given set to a graph in order to maximize its

algebraic connectivity. Other works, for example [11], proposed

strategies to improve the convergence rate of gossip algorithms

through geographic routing.

1053-587X/$26.00 © 2011 IEEE
23 384 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 60, NO. 1, JANUARY 2012

However, in all these works, the focus is always on convergence time and there is no cost associated with the establishment of the graph topology. Conversely, in our work, since the

graph represents a real network, we consider as the optimization metric the power consumption necessary to maintain reliable communication links among the nodes, taking into account the radio propagation model, multiplied by the number of

iterations necessary to achieve consensus. From this perspective, enforcing a small world, scale-free, or Ramanujan graph

topology may not be the best choice for a wireless network,

whose topology should depend on the application at hand as

well as on the propagation model.

In the wireless communication network context, there have

been works on optimizing the network topology in order to minimize the power consumption necessary to guarantee connectivity, e.g., [12], [13]. These works concentrate on the communication task and do not make any specific reference to the running

application. However, it is now well established, in the sensor

network context, that, whenever possible, an efficient wireless

sensor network design should take into account the specific goal

of the sensor network [14]. For this reason, we focus on the

achievement of consensus in a WSN. It is well known that one

of the most crucial parameters in WSNs is energy consumption,

because in many contexts it is hard to recharge the batteries or

scavenge energy from the environment. For this reason, in most

applications, minimizing energy consumption is more appropriate than minimizing convergence time (although, there are

important applications where the latter could be more important). In [15], the minimum energy consumption problem was

studied, assuming a common transmit power. As shown in [15],

there typically is an optimum power that minimizes the energy

necessary to achieve consensus within a prescribed accuracy.

In this work, we generalize the initial idea suggested in

[15] and we propose a method for optimizing the network

topology and the power allocation across every link in order

to minimize the energy necessary to achieve consensus. We

consider two classes of networks: a) deterministic topologies,

with arbitrary, but known, node locations, and b) random

geometries, with unknown node locations, modeled as random

variables. In the deterministic case, we optimize both topology

and power allocation. Differently from [6], we do not assume

any prior topology, as the topology comes out as a result of the

optimization. Topology optimization is, in general, a combinatorial problem and hence an NP-hard problem (recall that an

undirected graph composed of nodes may assume

topologies). To tackle this issue, we propose a relaxation technique that allows us to formulate topology optimization as a

convex parametric problem. Then, we show that the effect of

this relaxation on the performance is negligible.

In the random topology case, where the internode distances

are unknown, we show how to optimize the single (common)

transmit power, modeling the network topology as a random

geometric graph, a model suitable for wireless networks. We

provide both theoretical and simulation results, exploiting the

theory of the eigenvalues of random geometric graphs.

The paper is organized as follows. In Section II, we briefly review the consensus algorithm. In Section III, we introduce our

communication model and formulate the optimization problem.

Section IV is devoted to topology optimization for arbitrary

networks. In Section V, we start by providing a closed form

expression, albeit approximate, for the algebraic connectivity

of a random geometric graph. Then we use this expression to

proceed with the topology optimization for random geometric

graphs. The analytical findings are corroborated with extensive

simulation results.

II. BRIEF REVIEW OF CONSENSUS ALGORITHMS

Let us consider a wireless network composed of sensors.

The network topology can be represented as an undirected graph

where denotes the set of vertices (nodes)

and is the set of bidirectional edges (links)

connecting and . Furthermore, let be the

-dimensional symmetric adjacency matrix of the graph ,

with elements if and otherwise. According to this notation and assuming no self-loops, i.e.,

, the degree of node is defined as

. The degree matrix is defined as the diagonal matrix whose diagonal entry is . Let

denote the set of neighbors of node , so that 1.

The Laplacian matrix of the graph is the symmetric

matrix , whose entries are

(1)

Given a set of measurements , collected by node , for

, the goal of the consensus algorithm is to allow

every node to compute a globally optimal function of the measurements, say , through a decentralized mechanism that does

not require the presence of a sink node, but builds on the interaction among nearby nodes only. Let us consider, for simplicity,

the case where is the average of the measurements. In this

case, reaching an average consensus can be seen as the minimization of the disagreement between the states of the interacting nodes. One of the nice properties of the Laplacian is that

the disagreement can be expressed as a quadratic form built on

the Laplacian [16]

(2)

An important property of the Laplacian is that it has, by construction, a zero eigenvalue, whose multiplicity is equal to the

number of connected components of a graph. Hence, a graph is

connected if its zero eigenvalue has multiplicity one [5]. Furthermore, if the graph is connected, the eigenvector associated

with the null eigenvalue is the vector , composed of all ones.

The minimization of the quadratic form in (2) can be achieved

using a simple steepest descent technique. In continuous time,

the minimum of (2) can be reached through the following updating rule [2]:

(3)

1By we denote the cardinality of the set.
24SARDELLITTI et al.: MINIMUM ENERGY CONSUMPTION IN CONSENSUS NETWORKS 385

initialized with 2. By construction, the eigenvalues

of are all nonnegative. Hence, the convergence of (3) is guaranteed. In particular, if the graph is connected, the state vector

in (3) converges to the projection of the initial state onto

the nullspace of , spanned by the vector , i.e., [2]

(4)

This corresponds to having all the nodes converging

to a consensus on the average of the initial observations . Moreover, the convergence

rate is lower bounded by the slowest decaying mode,

i.e., by the second smallest eigenvalue of ,

also known as the algebraic connectivity of the graph.

More specifically, if the graph is connected, the dynamic

system (3) converges to consensus exponentially [2], i.e.,

, with .

Defining the disagreement vector as , we

can write . As a consequence,

the convergence time can be defined (see also [2]) as the

time necessary for the slowest mode of the dynamical system

(3) to be reduced by a factor , i.e., the time for which

. Hence, we can set

(5)

III. OPTIMIZATION CRITERION

It is well known that one of the most critical issues in WSN is

energy consumption. Neglecting for simplicity the power spent

for processing operations with respect to the power to be used

to enable wireless communications, the overall power spent to

reach consensus is the product between the sum of the power

necessary to establish the communication links among the

nodes and the number of iterations necessary to achieve

consensus. The exchange of information among the nodes is

supposed to take place in the presence of a slotted system, with a

medium access control (MAC) mechanism that prevents packet

collisions. Denoting by the duration of a time slot unit, the

number of iterations is then approximately3 .

Introducing the power coefficients , denoting the

power used by node to transmit to node with , and

the binary coefficients assessing the presence of a

link between nodes and or not , the power spent by

the whole network in each iteration is then .

Using (5), our goal is to minimize the following performance

metric:

(6)

2The discrete-time counterpart of (3) is [2]

where is a parameter chosen so as to ensure that no eigenvalue of is greater

than one in modulus.

3We neglect rounding errors, which tend to vanish if the duration of the updating time slot is small with respect to the convergence time.

where incorporates all irrelevant constants. This metric is

proportional to the integral of power consumption with respect

to time and then it represents an energy consumption. In (6),

we have made explicit the dependence of the Laplacian on

the vector containing all binary coefficients , since

finding these coefficients is the goal of our optimization. More

specifically, our goal is to find the set of active links, i.e., the

nonzero coefficients , and the powers that minimize (6),

under the constraint of maintaining network connectivity, which

entails . The problem can then be formulated as

follows:

(7)

where is an arbitrarily small positive constant used to prevent

the algebraic connectivity from going to zero, which would correspond to a disconnected network and is the vector with entries .

Since the topology coefficients are binary variables, is

a combinatorial problem, with complexity increasing with the

size of the network as . Hence, its solution, for

medium/large scale networks is prohibitive. Our objective is to

modify in order to turn it into a convex problem, with negligible performance losses.

A first simplification comes from observing that the coefficients and are not independent of each other. Their

dependence is indeed a consequence of the radio propagation

model. In this work, given the complexity of the topology optimization problem, we assume a fairly simple communication

model. We state that there is a link between nodes and , and

then , if the signal–to-noise ratio at the receiver

node , when node transmits, exceeds a minimum value ,

i.e., . If we denote by the power received by

node when node transmits, and by the noise power, assumed for simplicity to be the same at each receiving node, we

have , if , or otherwise .

Assuming flat fading channel modeling, we use the following

propagation model:

(8)

where is the distance between nodes and , and is the

path loss exponent. The parameter plays the role of a scaling

factor or reference distance, and typically corresponds to the so

called Fraunhofer distance, such that, if , the receiver

is in the transmit antenna far-field, where the received power

is inversely proportional to ; conversely, if , the

receiver is in the transmit antenna near-field, where the received

power is approximately equal to the transmitted one. The unity

term in the denominator of (8) is used to avoid the unrealistic

situation in which the received power could be greater than the

transmitted one. Given the propagation model (8), the relation
25 386 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 60, NO. 1, JANUARY 2012

between the power coefficients and the topology coefficients

is then

(9)

In the following sections we will show how to relax this relation in order to simplify the solution of the optimal topology

control problem. We will consider two scenarios: a) a deterministic topology, with arbitrary node geometry, where the distances between the nodes are known, and b) a random topology,

where the positions of the nodes are unknown and modeled

as random variables. In case a), since the distances among the

nodes are known, we optimize the power allocated to each link.

This strategy is also instrumental in determining the topology,

which is equivalent to finding the entries of

the adjacency matrix. Conversely, in the random topology case,

since the distances are not known, we assume broadcast communications, and look for the optimization of the coverage radius of each node. The deterministic topology case will be the

subject of Section IV, whereas the random case will be studied

in Section V.

IV. OPTIMAL TOPOLOGY AND POWER ALLOCATION FOR

ARBITRARY NETWORKS

In the case where the distances between the nodes are known,

the optimization criterion amounts to solving problem ,

which involves a combinatorial strategy that makes the problem

numerically very hard to solve, especially for medium/large

scale networks. The relation (9) reduces the set of unknowns

to the set , but the problem still retains most of its difficulties. To make problem to be numerically tractable, we

introduce a first relaxation so that, instead of requiring to

be binary, we assume to be a real variable belonging to

the interval . Under this assumption, problem is

not combinatorial anymore, but it is still a nontrivial nonlinear

constrained problem. The first important contribution of this

paper is to propose a relaxation technique that transforms the

previous problem into a convex problem that can be solved with

well established and efficient numerical tools. We achieve this

goal by first introducing the following relationship between the

coefficients and the distances :

(10)

where is a positive coefficient and is the coverage radius,

which depends on the transmit power. According to (10),

is close to one when node is within the coverage radius of

node , i.e., , whereas is close to zero, when

. The switching from zero to one can be made steeper

by increasing the value of .

Given the propagation model (8), the coverage radius is

related to the power and the minimum power required for

reliable communication , as follows:

(11)

Plugging (11) in (10), the coefficients can be written explicitly in terms of the power coefficients as follows:

(12)

This relation can be also inverted to find the coefficients as

a function of , as follows:

(13)

with . Expression (12) becomes our relaxed version of (9) and it allows us to reduce the set of variables to the

only power vector . Consequently, problem can be relaxed into the following problem:

(14)

where, thanks to (12), the Laplacian is now written explicitly

in terms of the power coefficients . In principle, the last inequality in (14) makes any link feasible. But this does not imply

that the final network will be fully connected, because some coefficients might turn out to be equal zero, implying that the

link between node and is not active. The first important result, related to the solution of (14), is the following.

Theorem 1: Given the propagation model in (8), using the

relations (12) between the topology coefficients and the

power terms , problem is a convex-concave fractional

problem if .

Proof: Let us consider the objective function in .

The numerator of is clearly a convex function of . We

only have to prove the concavity of the algebraic connectivity

with respect to the transmit powers. As a first step,

we prove that is a concave function of . Then, we

use this to show that is a concave function of .

(i) is a concave function of : We compute the

second-order derivative of the function in (12) with respect to

(15)

where the constants and are given by

Note that ; since , we also have

. From (15), we see that, if , the second-order

derivative is always nonpositive and then is a concave

function of .

(ii) is concave in : Exploiting the properties of the

Laplacian [16], we can write the quadratic form associated to

as in (2). From (12), we also note that each coefficient
26SARDELLITTI et al.: MINIMUM ENERGY CONSUMPTION IN CONSENSUS NETWORKS 387

depends only on the corresponding link power , and not on

the other link powers. Given any pair of power vectors and

, let , with . Hence, we

have

(16)

where the inequality follows from the concavity of .

We further recall that the algebraic connectivity is the

solution of the following positive semidefinite program (SDP)

[5]:

(17)

Hence, using (16), we obtain

where the last equality follows from a majorization theorem for

eigenvalues4. Thus, the algebraic connectivity is a concave function of the vector of transmit powers. Hence, (14) is a convexconcave fractional problem since it is the ratio of a convex and

a concave function of .

Since (14) is a convex-concave fractional problem, we can use

one of the methods that solve quasi-convex optimization problems, see, e.g., [18], [19]. For example, we can use the nonlinear

parametric formulation proposed in [19]. To do so, we introduce

the following function:

where is a real positive parameter and

. In order to find the solution of this

problem we could use the following result, proved in [19] and

[20].

Theorem 2: Let and be continuous real-valued

functions , where is a nonempty compact subset of

and . Then

4Let be Hermitian matrices, and let . Let their eigenvalues be sorted in nondecreasing order. Then the vector of eigenvalues

majorizes the vector . Since for the graph Laplacian,

it follows that . See [17, Theorem 4.3.27].

with , if, and only if

where means that for the minimum of

is taken on at .

Before applying Theorem 2, it is useful to further convert the

convex-concave optimization problem into the following

parametric problem:

(18)

By Theorem 1, is a concave function of . Hence, the

objective function in , as sum of convex functions, is a

convex function. The constraint sets are convex. Then, problem

is a convex parametric problem, whose solution is a function of the parameter that controls the tradeoff between the

global transmit power and the convergence time. Later on, we

will show how to find the optimal .

Since problem is convex, it can be solved using numerically efficient convex programming tools. However, before

applying any convex tool, it is worth noticing that the feasible

set in is not compact. Hence, even if a solution exists, in

principle, it could be unreachable in finite time. To overcome

this potential drawback, we propose next an alternative formulation of : Instead of looking for the set of power coefficients , and then for the , using (13), we can reformulate

so as to look directly for the variables . Then the optimization problem in (18) can be rewritten in terms of vector ,

as follows:

(19)

where . We verify next that this

problem is still convex. To study the behavior of , we compute the second-order derivative of , obtaining

We infer that

We note that, if , then for ,

so that , as a sum of convex functions, is convex5. Finally,

the algebraic connectivity is a concave function of

, as can be proved following the same steps as in Theorem

5Note that is the sum of functions of the single variables . Hence the

convexity of can be studied looking at the convexity of the single functions

.
27 388 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 60, NO. 1, JANUARY 2012

1. Then, the optimization problem is also a convex parametric problem, perfectly equivalent to the original problem in

(18), since the change of variables in (13) ensures

a one-to-one mapping , for , with

image covering the problem domain in (18) (see [21, p. 130]).

Hence, assuming , with an infinitesimal positive constant so that the feasible set of (19)

is a compact convex set in , we

ensure that the set of minima of (19) is nonempty and by the convexity of the problem we can deduce that all local minima are

also global. Note that since the optimization problem in (19) is

convex an optimal solution can be found via efficient numerical

tools. Furthermore, using Dinkelbach’s algorithm [19], based

on Theorem 2, we are also able to find the optimal parameter

in . More specifically, the Dinkelbach’s algorithm, applied

to our problem, proceeds through the following steps:

1. Set and let be a feasible point of , with

;

2. Set and find that solves the minimization

problem in ;

3. If , with

an arbitrarily small positive constant, stop and take as

the optimal link coefficient vector; otherwise, set

and go to step 2.

Since the topology coefficients obtained in this way are

real variables belonging to the interval , to obtain the network topology, it is necessary to quantize them to convert them

into binary values, 1 or 0, indicating the presence or absence

of a link. This quantization is achieved by comparing each

with a threshold . Of course, the final topology will depend

on the threshold value. Moreover, the thresholding operation

will also affect the final result in terms of convergence time

and energy consumption. It is then of interest to check how

sensitive the final topology, as well as convergence time and

energy consumption, are to the choice of . In the ensuing

section, we present some numerical results to shed light on the

resulting topologies and their dependence on the propagation

model parameters.

A. Numerical Examples

Since our optimization procedure is based on a relaxation

technique, the first important step is to evaluate the impact of

the relaxation on the final topology and performance.

Example 1: Comparison Between Exhaustive Search and Relaxed Technique: We compare now the topology obtained as a

solution of the relaxed problem with the optimal graph

obtained by solving directly problem using an exhaustive

search over all possible topologies. For complexity reasons, of

course we can only perform this comparison for small scale networks. We consider networks of and 6 nodes.

To provide results not conditioned to a specific geographic

node deployment, we averaged the results over 100 statistically

independent realizations of the nodes locations. In each iteration

we compute the minimum energy reached using the optimal

exhaustive search over all possible topologies, and the energy

corresponding to the network topology whose coefficients

are obtained by solving problem and thresholding the

Fig. 1. Optimal average energy obtained with exhaustive search and average

energy obtained by solving the relaxed problem versus , for different

values.

Fig. 2. Ratio between the optimal energy obtained with exhaustive search and

the energy obtained by solving the proposed relaxed problem , versus ,

for different values of .

result with the threshold . In Fig. 1, we report the

average energies and versus . We can note from Fig. 1

that the loss in terms of optimal energy due to the relaxation of

the original problem is negligible (the energy loss is less than

1.7 dB).

Example 2: Impact of Thresholding Operation on Final

Topology: Clearly, the selection of the threshold plays a

role in the identification of the final topology. To evaluate the

impact of on the final topology, in Fig. 2 we report the

ratio versus for different numbers of nodes and for

. The energies are averaged over 100 independent node

realizations. We can observe from Fig. 2 that there is a wide

range of values of such that the energy loss is practically

independent of . This shows that our proposed procedure is
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SARDELLITTI et al.: MINIMUM ENERGY CONSUMPTION IN CONSENSUS NETWORKS 389

Fig. 3. Optimal topologies, for different threshold values and fixed : (a) . (b) . (c) . (d) .

rather robust with respect to the choice of . It is now interesting to check the effect of the threshold on the topology,

considering larger scale networks (in this case, we can only run

our algorithm, as the exhaustive search is not feasible). As an

example, in Fig. 3 we show the topologies obtained by solving

problem , for a network composed of nodes, using

different values of , for . Comparing the four cases

reported in Fig. 3, we notice that, only for very low values of

the threshold [i.e., case (d)], we appreciate a sensitive change

of topology, whereas for a large range of values of , the final

topology is practically the same. This means that the proposed

algorithm, in spite of the relaxation step and the subsequent

quantization, yields rather stable solutions.

The previous results pertain to a specific realization of the

node locations. To provide results of more general validity, in

Fig. 4, we report the average value of: a) fraction of active links

; b) ; and c) the average energy , as a function

of the threshold . The averages are carried out over 100 independent realizations of the nodes location. From Fig. 4, we observe that there is an interval of values of (roughly, between

to ) for which we obtain a strong reduction in the

fraction of active links, with respect to the situation where there

is no threshold, still achieving nearly the same performance, in

terms of algebraic connectivity and energy consumption. This

is indeed an important result, as it shows that the relaxed algorithm is weakly sensitive to the choice of .

Fig. 4. Average value of (a) fraction of active links. (b) . (c) energy

versus threshold value for .

Example 3: Impact of Propagation Parameters on Final

Topology: It is also interesting to look at the change in

topology as a function of the radio communication model. To

this end, in Fig. 5, we plot the optimal topologies achieved for

the same node locations as in Fig. 3, but pertaining to different

path loss exponents , for a given threshold. Interestingly, we
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390 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 60, NO. 1, JANUARY 2012

Fig. 5. Optimal topologies, for different path loss exponents and fixed threshold : (a) . (b) . (c) . (d) .

notice, as expected, that, as the propagation loss increases (i.e.,

increases), the network tends to become more and more

sparse.

The results shown in Fig. 5 refer to a single realization of the

node positions. To draw conclusions of more general validity,

we averaged over 100 statistically independent realizations of

the node locations, for networks of 30 nodes. In Fig. 6, we report

the same performance metrics as in Fig. 4, but now as a function

of the path loss exponent , setting . From Fig. 6,

we observe that when the attenuation is high (i.e., is large),

reducing the number of links (making the topology sparser) is

more important than reducing convergence time. Conversely,

when the attenuation is low (i.e., is small), increasing network

connectivity is more important than reducing power consumption. This behavior sounds reasonable and in agreement with

intuition.

V. OPTIMAL TOPOLOGY FOR RANDOM GEOMETRIC NETWORKS

In this section, we remove the assumption that the node locations are known a priori and model the network as a random

geometric graph (RGG). In such a case, the graph connectivity

properties and the convergence time can only be established in

a probabilistic sense, asymptotically, as the number of nodes

tends to infinity. We refer to [22] for the first basic result about

the convergence of consensus algorithms over random graphs

and to [23] for a more recent generalization of the convergence

Fig. 6. Average value of (a) fraction of active links. (b) . (c) energy

versus path loss for .

conditions under stochastic disturbances. As shown in [22],

the rate of convergence of consensus algorithms in the random

graph case is dictated by the expected value .

In the following, exploiting the concentration properties of the

eigenvalues of RGGs [24], [25], we will show how to relate

the convergence time to the expected value of . This link
30SARDELLITTI et al.: MINIMUM ENERGY CONSUMPTION IN CONSENSUS NETWORKS 391

will be fundamental to derive the optimal coverage radius, and

then transmit power, that minimize the energy consumption

necessary to achieve consensus over RGGs. To achieve this

goal, it is fundamental to recall and extend some results about

the spectrum of RGGs.

A. RGGs

A random graph is obtained by distributing points randomly

over the -dimensional space and connecting the nodes according to a given rule. Let be a set

of -dimensional vectors , belonging to a compact set

of , whose entries denote the coordinates of the

nodes in , and let be the Euclidean norm on . The

graph topology is captured by the adjacency matrix which,

in this case, is a random matrix. An important class of random

matrices, encompassing the adjacency matrix of our problem,

is the so-called Euclidean Random Matrix (ERM) class, introduced in [26]. Given a set of points located at positions

an adjacency matrix is an ERM if its generic

entry depends only on the difference , i.e.,

, where is a measurable mapping from to .

An important subclass of ERM is given by the adjacency matrices of the so-called RGGs. In such a case, the entries of

the adjacency matrix are either zero or one depending only on

the distance between nodes and , i.e.

(20)

where is the coverage radius. This is a particular case of an

ERM, corresponding to having

(21)

We will use the symbol to indicate an RGG composed

of points, with coverage radius .

The RGG model is the most appropriate to capture the

topology of a wireless network, as it basically states that there

is a link between two nodes only if they are within the coverage

radius of each other6.

Next, we recall some of the most important properties of

RGG’s, in terms of connectivity and spectrum, as they are relevant to our optimization problem.

1) Connectivity: Some interesting results on the asymptotic

connectivity of random geometric graphs have been derived in a

seminal work by Gupta and Kumar [27] who proved that, given

a set of points uniformly distributed within a unit square (i.e.,

), the graph is connected almost surely if the coverage

radius behaves as

with , as . Conversely, if , the graph

is disconnected almost surely. This means that the expression

represents a threshold distance. In the following, we

6In practice, a wireless channel is also affected by fading, multipath, and shadowing. Hence, the presence of a link between two nodes depends on two sources

of randomness: distance and fading. In this work, we concentrate on the single

source of randomness, given by the distance between the nodes, but considering

both sources of randomness will be an interesting extension of this paper.

will often use the shorthand notation to indicate the law

, with , as . Hence, a coverage

radius behaving as represents a law that ensures

connectivity with high probability, as . We will use the

notation to indicate such a behavior.

In [4], [28], it has been shown that the degree of a RGG

of points uniformly distributed over a two-dimensional

unit torus7 is equal to

(22)

with high probability, i.e., with probability , if the radius

behaves as . This implies that an RGG tends to

behave, asymptotically, as a regular graph, if the coverage radius

is chosen so as to guarantee connectivity with high probability.

We are primarily interested in the second eigenvalue of the

Laplacian, , where is the degree matrix and is

the adjacency matrix [see (1)]. From (22), , so that

we only need to investigate the second largest eigenvalue of .

Hence, in the ensuing section, we study the spectrum of .

2) Spectrum of a Random Geometric Graph: In [24], [25],

it is shown that the eigenvalues of the adjacency matrix, or of

the transition probability matrix 8, tend to be concentrated, as

the number of nodes tend to infinity. In particular, in [24] it is

shown that the eigenvalues of the normalized adjacency matrix

of an RGG , composed of points uniformly

distributed over a unit bidimensional torus, tend to the Fourier

series coefficients of the function defined in (21)

(23)

almost surely, for all , where

. Using polar coordinates, i.e.,

and , with and ,

we obtain

This integral can be computed in closed form. Setting

and , we have

with . Furthermore, using the integral expression

for the Bessel function of the first kind of order ,

, we get

Finally, using the identity , we can make

explicit the dependence of on the index pair

(24)

7A torus geometry is typically used to get rid of border effects.

8The transition probability matrix is the adjacency matrix, normalized with

respect to the node degree, so that the th row of the adjacency matrix is divided

by the degree of node .
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392 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 60, NO. 1, JANUARY 2012

Fig. 7. Verification of the inequalities in (26).

This formula allows us to rank the eigenvalues of .

In particular, we are interested in the second largest eigenvalue of . Considering that the minimum coverage radius

ensuring connectivity behaves as , i.e., it

is a vanishing function of , we can use the Taylor series

expansion of , for small . Recalling that, for small

, , we can approximate the

eigenvalues as

(25)

This expansion shows that, at least for small , the largest eigenvalue equals and occurs at , whereas the second

largest eigenvalue corresponds to the cases

and . More generally, we can check numerically that, for and , the following inequalities

hold true:

(26)

The validity of these inequalities can be verified from Fig. 7,

which shows the three terms in (26) as a function of and .

In summary, denoting the spectral radius of as

, where is the set of eigenvalues

of , it follows that

(27)

while the second largest eigenvalue of , converges

to

(28)

We are now able to derive the asymptotic expression for the

second largest eigenvalue of the normalized Laplacian

, where is the normalized degree matrix.

Because of the asymptotic property of the degree of an RGG,

Fig. 8. Asymptotic analysis and simulation results for some eigenvalues of

and for versus the transmission radius.

shown in (22), the second largest eigenvalue of tends asymptotically to

(29)

Thus, the algebraic connectivity of the graph can be approximated, asymptotically, as

(30)

Since the previous expressions have been derived in the

asymptotic regime, in Fig. 8 we compare the analytic formulas

for the first and the second largest eigenvalues of , as given

in (27) and (28), with the numerical results obtained by averaging over 100 independent realizations of RGG’s composed

of nodes. We can notice the good agreement in

Fig. 8 between the theoretical expression for the algebraic

connectivity , given in (29), and the simulation results.

B. Minimization of the Energy Consumption: An Analytic

Approach

We can now exploit the previous analytic expressions to study

the energy minimization problem for RGG’s. In the random

topology case, since the distances are unknown, we cannot optimize the power associated with each link. However, we can

seek the common transmit power that minimizes energy consumption. Thus, in the random setting we assume a broadcast

communication model, where each node broadcasts the value

to be shared with its neighbors. In the lack of any information

about distances among the nodes, we assume that each node uses

the same transmit power. In this case, the network topology can

be modeled as a random graph model. It can be shown [22],

[23] that the system in (3) converges to consensus almost surely,

i.e., assuming that each node has a

coverage radius so that the network is asymptotically connected
32SARDELLITTI et al.: MINIMUM ENERGY CONSUMPTION IN CONSENSUS NETWORKS 393

with probability one [14]. Then the rate of convergence of the

dynamical system in (3) is given [22], [23] by .

Note that defining and using the Taylor

series expansion of 9 at the point , we

can write

(31)

Consequently, taking the expected value, we get

(32)

denoting with the variance of . But

since all central moments of order greater than one of the eigenvalues tend to zero, because of the concentration property, we

can use the approximation

(33)

As a consequence, the energy spent to achieve consensus can

now be approximated as

(34)

This is the performance metric we wish to minimize in the

random scenario, with respect to the single unknown .

In particular, using the asymptotic expression (30) for the algebraic connectivity, we can introduce the following metric

(35)

We now check, numerically, that the function given in (35)

is a convex function of , for , where

, to ensure connectivity.

Let us rewrite (35) as

with and as in (30). The first–

and second–order derivatives of are, respectively

9For simplicity we drop in the dependence on .

so that is a convex increasing positive function of . Let us

now study the behavior of . Using

we obtain

with for . Moreover, since

for

we have

(36)

Observe that the first term on the right-hand side (RHS) of (36)

is always positive. Furthermore, if , the second term is

also positive, since for . Hence,

and we can conclude that the algebraic connectivity is an increasing and convex function of for , where

.

We can now compute the first- and second-order derivatives

of the energy function with respect to . We get

and, substituting the corresponding expressions, the extremal

points can be obtained by solving the following nonlinear

equation:

with and

. Furthermore, the second derivative of is

given by the equation shown at the bottom of the page. In Fig. 9,

we report as a function of , for different values of .

From Fig. 9, we can check that the second derivative is always

positive in the range of interest. This verifies that is indeed a convex function of defined on the compact convex set

. As a consequence, we can state that there

is always at least a radius that globally minimizes the energy

consumption in a RGG.

Numerical examples. In Fig. 10, we compare the value of

obtained by our theoretical approach and by simulation, for various values of the path loss exponent, . The results

are averaged over 100 independent realizations of random geometric graphs composed of nodes. For each , the pair

of radius and energy providing minimum energy consumption is

indicated by a circle (simulation) or a star (theory). We observe
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394 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 60, NO. 1, JANUARY 2012

Fig. 9. Second derivative of the energy versus the transmission radius for several values of .

Fig. 10. Global energy consumption versus transmission radius for an RGG;

theoretical values (solid) and simulation (dashed).

that the theoretical derivations provide a very good prediction

of the performance achieved by simulation. Furthermore, for

each , there is a coverage radius value that minimizes energy

consumption.

The optimal values of the coverage radius as a function of ,

as predicted by our theoretical derivations or by simulation, are

reported in Fig. 11. From this figure, we observe that there is

a clear transition from the low power attenuation regime (i.e.,

), where the optimal radius tends to make the network

fully connected, as opposed to the strong attenuation situation

, where the network is minimally connected. This

means that in the low attenuation case, minimizing convergence

time is more important than minimizing power consumption.

Fig. 11. Optimal transmission radius versus .

Fig. 12. Global energy consumption versus per node transmit power for an

RGG; theoretical values (solid) and simulation (dashed).

Conversely, in the large attenuation case, it is more important to

minimize power consumption, by limiting the number of links

as much as possible, compatibly with the constraint of ensuring

connectivity.

Fig. 12 shows the average energy consumption versus the per

node transmit power , under the same settings of Fig. 10. The

circles (simulation) and the stars (theory) represent, again, the

values of that minimize energy consumption, for each . It

can verified the existence of an optimal transmit power value

minimizing the energy consumption.

Finally, Fig. 13 shows the optimal per node power versus .

For low values of , the transmit power increases with because

it must cope with higher attenuations to guarantee connectivity.

Conversely, for large values of , the optimal power decreases

with because, in such a case, it is more beneficial to limit
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SARDELLITTI et al.: MINIMUM ENERGY CONSUMPTION IN CONSENSUS NETWORKS 395

Fig. 13. Optimal per node transmit power versus .

the number of links, until the minimum power guaranteeing the

connectivity constraint is reached.

C. Random Topologies versus Uniformly Spaced Grids

Finally, it is interesting to compare the energy consumption

achievable with a random geometric graph and a deterministic

grid. In the deterministic case, we may also distinguish between

a graph with nodes scattered arbitrarily in a given area, but with

positions known, and a regular uniform grid, where the points

are located over a rectangular grid.

1) Eigenvalues of a Planar Uniform Grid: We start by deriving the algebraic connectivity of a square grid whose nodes

are uniformly spaced within a unit square, at a distance

. To avoid undesired border effects, we consider

the wrapping of the unit square in order to form a toroidal surface. Each node is assumed to have a link with the neighboring

nodes only if they are at a distance less than a coverage radius

, that depends on the transmit power , as with geometric

graphs. In Appendix A, we derive a closed form expression for

the network degree and for the algebraic connectivity [see, e.g.,

(45)–(48)].

A numerical check of our derivations is reported in Fig. 14,

where we show the numerical value of , obtained through

the eigendecomposition of , and the value given in (48), for different values of . From Fig. 14, we can see a perfect agreement

between our closed form expression and the numerical results.

Notice, in particular, the sharp transition behavior: the eigenvalues tend to or 0 depending on whether the transmit power

is above or below a threshold; further, the threshold increases

with .

In Fig. 15, we compare the algebraic connectivity of the rectangular grid, given by (48), with the theoretical value obtained

for the random geometric graph, given in (30), assuming the

same node density, over the same toroidal surface. As shown in

previous works, see, e.g., [4], RGGs tend to behave asymptotically as a regular graph. The result shown in Fig. 15 is a further

Fig. 14. Network algebraic connectivity versus per node transmit power for

several values of the path loss coefficient.

Fig. 15. Algebraic connectivity of random geometric graph and rectangular

grid versus transmission radius.

confirmation of this property. In the same figure, we also report simulation results for the algebraic connectivity, obtained

by averaging over 100 independent realizations of a RGG. We

can check, once again, the good agreement between theory and

simulation.

Finally, in Fig. 16 we compare the energy consumption obtained assuming full a priori knowledge of the nodes’ locations

or no knowledge at all. In the first case, the topology and the

power allocation over each link are optimized according to the

method illustrated in Section IV. The optimal values, for each

, are indicated by colored dots. In the second case, we report

(solid line) the energy consumption versus the average power,

assuming that all nodes transmit with the same power (since they
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Fig. 16. Average network energy consumption versus average network power.

do not have any prior information about the other nodes positions). To make a fair comparison between the two cases, we

consider in both cases the sum of the powers associated with

each active link. This means that, in the second case, the energy

consumption is measured, for every common transmit power ,

as , where denotes the number of neighbors of node . The results shown in Fig. 16 have been obtained

by averaging the results obtained in the two settings over the

same set of 100 independent random deployments of

nodes, uniformly distributed over the unit square.

From Fig. 16 we can see that, as expected, the method assuming perfect a priori knowledge of the node positions (in the

figure denoted as ) is indeed able to achieve better performance than the other method , as it is able to take advantage of the a priori knowledge. The optimal average power

is the result of a tradeoff between transmit power and convergence time, and always depends on the path loss exponent.

VI. CONCLUSION

In this paper we have addressed the problem of finding the

network topology that minimizes the energy consumption necessary to achieve consensus in a WSN. Assuming a simple flat

fading propagation model, we have studied two main network

models: 1) arbitrary networks, where the nodes are arbitrarily located, but with known positions, and 2) a random model, where

the positions are not known a priori, but are modeled as random

variables. In the first case, we have shown how to optimize the

network topology and the power allocated to each active link

in order to minimize the total energy necessary to achieve consensus within a prescribed accuracy. Topology optimization is,

in general, a combinatorial problem and hence computationally

demanding. To simplify the solution, we introduced a relaxation

step that enabled us to reformulate the energy minimization

problem as a convex-concave fractional program. This alternative formulation can be cast as an equivalent parametric convex

problem, which enables efficient solutions. The link weights resulting from the solution of the convex problem must then be

thresholded to find out the network topology. We have shown

through numerical results that the thresholding operation may

yield a considerable reduction of the number of active links, yet

with very limited effect on performance.

In our deterministic setting, we assumed point-to-point links

and we optimized the power over each link. In practice, a consensus algorithm running over a wireless network could benefit from the broadcast channel. Deciding between broadcast or

one-to-one links entails a proper choice of the medium access

strategy, to establish in which time slot each node has to listen

to which broadcaster.

Conversely, in the random network case, lacking any information about the internode distances, we assumed a broadcast

communication strategy. In such a case, the network topology

is modeled as a random geometric graph. We have derived

closed form expressions, albeit valid only asymptotically, for

the algebraic connectivity, assuming a common transmit power.

Then, building on these expressions, we have shown that the

energy consumption is a convex function of the coverage radius.

We have also shown that a random geometric graph performs,

asymptotically, as a regular graph built over a rectangular grid.

This confirms previous results, although now in the context of

energy minimization over consensus networks.

Finally, we have compared the performance achievable with

arbitrary and random graphs. Clearly, the knowledge of the

node locations allows better power allocation, that translates

into lower energy consumption to achieve consensus. However,

in practice there is a price associated with the knowledge of

node location. This knowledge requires the acquisition of

the node positions first and then a centralized optimization.

Conversely, the random approach can be followed also in a

decentralized fashion, with only minimal information about

some global parameters like number of nodes and area covered

by the network, and it does not need any extra hardware or

computation to acquire the nodes’ locations.

In this paper, we have assumed a simple flat fading channel

model, whose effect is only to introduce attenuation and superimpose noise. The simple model captures the essence of the

problem and keeps the overall problem complexity under control. However, looking at potential applications, it would be

interesting to generalize the approach to the case where the

channel model is more complicated. Furthermore, the network

topology has been assumed to be static. However, a dynamic

topology, possibly adapted to the consensus state, might provide better performance.

APPENDIX A

In this section, we derive an analytical expression for the

eigenvalues of the adjacency matrix of a rectangular grid over

a unit torus, as a function of the coverage radius . We assume, for simplicity that the number of nodes is a square

number, i.e., , with integer. We number the rows of
36SARDELLITTI et al.: MINIMUM ENERGY CONSUMPTION IN CONSENSUS NETWORKS 397

the grid, 1 to , going from bottom to top. The matrix

can be expressed in a cyclic block form, as

.

.

. .

.

. .

.

. .

.

. .

.

. .

.

.

where each block , for , is an

circulant matrix. In particular, the block is the adjacency

matrix between the nodes lying on the same row of the grid,

starting from the leftmost node and proceeding along the row.

represents the adjacency matrix between the nodes of row

and the nodes lying on the th row, for ,

and on the th row for .

The toroidal structure, used to avoid border effects, reflects into

the cyclic structure of , so that we have if

We also note that is a block circulant matrix with circulant

blocks (BCCB). Since each block for is

an -dimensional circulant matrix, the eigenvalues of

admit the following decomposition

with

.

.

. .

.

. .

.

. .

.

. .

.

.

where is a diagonal matrix whose entries are the eigenvalues of , i.e., the discrete Fourier transform (DFT) of the first row of . Hence, the matrix admits

the following diagonalization:

where denotes the Kronecker product and

with . The entries of the

matrix can be also expressed, after a few algebraic manipulations, as

(37)

for and the coefficient is th entry

of the first row of for . Equivalently, (37)

can be seen as the two-dimensional DFT of a matrix whose

rows are given by for

, applying the 1-DFT to the rows and then to the columns of .

In general, for a regular grid where each node has degree ,

the algebraic connectivity can be expressed as

(38)

where the coefficients can be expressed as follows. Let

denote the step function, i.e., for and zero

otherwise. For odd, we have

(39)

and

(40)

and

(41)

Finally for .

Similarly, in the case of even, we have

(42)

(43)

(44)

with if .

Finally, we can derive analytical expressions of the network

degree and of the algebraic connectivity for odd as

(45)

and

(46)
37

398 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 60, NO. 1, JANUARY 2012

and for even

(47)

and

(48)

For example, in the case where , i.e., assuming a

regular grid of degree , we obtain

(49)

while is a circulant matrix whose first row is the -dimensional vector . Then, the eigenvalues

are given by and applying the

formula in (37) the eigenvalues of are

for .

The second largest eigenvalue of is obtained for and

and is given by . Thus

the eigenvalues of the Laplacian matrix can be expressed as

and the algebraic connectivity of the

network is given by .

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2005.

Stefania Sardellitti received the Dr. Eng. degree in

electronic engineering from the University of Rome

“La Sapienza,” Italy, in 1998 and the Ph.D. degree in

electrical and information engineering from the University of Cassino, Italy, in 2005.

Since 2005, she has been an Appointed Professor

of digital communications at the University of

Cassino. She is currently a Research Assistant with

the DIET Department, Sapienza, University of

Rome. She has participated in the European project

WINSOC, on wireless sensor networks. Her research

interests are in the area of statistical signal processing, in particular on multiple

antenna and multiple access systems. Currently, her primary research activity
38SARDELLITTI et al.: MINIMUM ENERGY CONSUMPTION IN CONSENSUS NETWORKS 399

is in the field of cognitive radios, femtocell networks, and wireless sensor

networks, with emphasis on distributed decision.

Sergio Barbarossa (F’11) received the M.Sc.

degree in 1984 and the Ph.D. degree in electrical

engineering in 1988, both from the University of

Rome “La Sapienza,” Rome, Italy, where he is now

a full professor.

He has held positions as a Research Engineer at

Selenia SpA (1984–1986), at the Environmental Institute of Michigan (1988) and Visiting Professor at

the University of Virginia (1995 and 1997), the University of Minnesota (1999), and he has taught short

graduated courses at the Polytechnic University of

Catalonia (2001 and 2009). He has been principal investigator in several international projects. He has been the scientific lead of the European project WINSOC,

on wireless sensor networks. He is currently the scientific lead of the European

Project FREEDOM, on femtocell networks and is a principal investigator on the

European Project SIMTISYS, on the monitoring of maritime traffic from constellations of satellites. He is also involved in the design of the radar system to

be used for the landing module of ExoMars, a spacecraft to be launched by the

European Space Agency aimed to land on Mars. He is the author of a research

monograph titled “Multiantenna Wireless Communication Systems.” His current research interests lie in the areas of cognitive radios, femtocell networks,

sensor networks, cooperative communications, spaceborne radars, distributed

detection, and estimation.

Dr. Barbarossa has been a member of the IEEE Signal Processing for Communications Technical Committee (1998–2004) and served as an Associate

Editor of the IEEE TRANSACTIONS ON SIGNAL PROCESSING (1998–2001 and

2004–2006). He has been a plenary speaker at SAM 2008 and SPAWC 2009.

He has co-edited a Special Issue on Optimization of MIMO Transceivers for

Realistic Communication Networks: Challenges and Opportunities of the IEEE

JOURNAL ON SELECTED AREAS IN COMMUNICATIONS and a Special Issue of

the EURASIP Journal of Applied Signal Processing on MIMO Communications and Signal Processing. He received the 2000 IEEE Best Paper Award

from the IEEE Signal Processing Society and the 2010 EURASIP Technical

Achievements Award.

Ananthram Swami (F’08) received the B.Tech.

degree from Indian Institute of Technology (IIT),

Bombay; the M.S. degree from Rice University,

Houston, TX, and the Ph.D. degree from the University of Southern California (USC), Los Angeles, all

in electrical engineering.

He has held positions with Unocal Corporation,

USC, CS-3, and Malgudi Systems. He was a Statistical Consultant to the California Lottery, developed

a Matlab-based toolbox for non-Gaussian signal processing, and has held visiting faculty positions at INP,

Toulouse, France. He is with the U.S. Army Research Laboratory (ARL) where

he is the ST for Network Science. His work is in the broad area of network science, with emphasis on wireless communication networks. He was the co-editor

of “Wireless Sensor Networks: Signal Processing & Communications Perspectives” (New York: Wiley, 2007).

Dr. Swami is a member of the IEEE SPS Technical Committee on Sensor

Array and Multi-channel systems, and serves on the Senior Editorial Board

of the IEEE JOURNAL ON SELECTED TOPICS IN SIGNAL PROCESSING. He is an

ARL Fellow. He has served as an Associate Editor for the IEEE TRANSACTIONS

ON SIGNAL PROCESSING, SIGNAL PROCESSING LETTERS, Signal Processing

Magazine, the IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II, the IEEE

TRANSACTIONS ON WIRELESS COMMUNICATIONS, and as Guest Editor for the

IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS. He was a tutorial

speaker on “Networking Cognitive Radios for Dynamic Spectrum Access” at

ICASSP 2008, DySpan 2008, MILCOM 2008, and ICC 2010. He received the

Best Conference Paper Award at IEEE Trustcom 2008, and was Co-Organizer

and Co-Chair of three IEEE workshops related to signal processing and

communications, including IEEE SPAWC’10.
Research@

39

Efficient Extraction of Drainage Networks 

from Massive, Radar-Based Elevation Models 

with Least Cost Path Search

M. Metz, H. Mitasova, and R. S. Harmon

Hydrology and Earth System Sciences, 15, 667–678 (2011)
40

Hydrol. Earth Syst. Sci., 15, 667–678, 2011

www.hydrol-earth-syst-sci.net/15/667/2011/

doi:10.5194/hess-15-667-2011

© Author(s) 2011. CC Attribution 3.0 License.

Hydrology and

Earth System

Sciences

Efficient extraction of drainage networks from massive, radar-based

elevation models with least cost path search

M. Metz1, H. Mitasova2, and R. S. Harmon3

1Institute of Experimental Ecology, University of Ulm, Ulm, Germany

2Department of Marine, Earth and Atmospheric Sciences, North Carolina State University, Raleigh, North Carolina, USA

3Environmental Sciences Division, Army Research Office, US Army Research Laboratory, Durham, North Carolina, USA

Received: 22 April 2010 – Published in Hydrol. Earth Syst. Sci. Discuss.: 27 May 2010

Revised: 1 December 2010 – Accepted: 11 February 2011 – Published: 25 February 2011

Abstract. The availability of both global and regional elevation datasets acquired by modern remote sensing technologies provides an opportunity to significantly improve the

accuracy of stream mapping, especially in remote, hard to

reach regions. Stream extraction from digital elevation models (DEMs) is based on computation of flow accumulation,

a summary parameter that poses performance and accuracy

challenges when applied to large, noisy DEMs generated

by remote sensing technologies. Robust handling of DEM

depressions is essential for reliable extraction of connected

drainage networks from this type of data. The least-cost flow

routing method implemented in GRASS GIS as the module r.watershed was redesigned to significantly improve its

speed, functionality, and memory requirements and make it

an efficient tool for stream mapping and watershed analysis

from large DEMs. To evaluate its handling of large depressions, typical for remote sensing derived DEMs, three different methods were compared: traditional sink filling, impact

reduction approach, and least-cost path search. The comparison was performed using the Shuttle Radar Topographic Mission (SRTM) and Interferometric Synthetic Aperture Radar

for Elevation (IFSARE) datasets covering central Panama at

90 m and 10 m resolutions, respectively. The accuracy assessment was based on ground control points acquired by

GPS and reference points digitized from Landsat imagery

along segments of selected Panamanian rivers. The results

demonstrate that the new implementation of the least-cost

path method is significantly faster than the original version,

can cope with massive datasets, and provides the most accurate results in terms of stream locations validated against

reference points.

Correspondence to: M. Metz

(markus.metz@uni-ulm.de)

1 Introduction

Shuttle Radar Topographic Mission (SRTM; Farr et al.,

2007) and various airborne Interferometric Synthetic Aperture Radar for Elevation (IFSARE) surveys provide a new

generation of elevation data in regions that have had only

limited, often low resolution coverage. These topographic

data sets are increasingly used to improve mapping of geomorphic and hydrologic features, especially in remote, hard

to reach areas and at regional to global scales (e.g., Kinner

et al., 2005; Lehner and Doll, 2004; World Wildlife Fund,  ̈

2009).

In spite of significant advances in the development of flow

routing algorithms (e.g. Quinn et al., 1991; Costa-Cabral and

Burges, 1994; Holmgren, 1994; Quinn et al., 1995; Tarboton,

1997), accurate extraction of drainage networks from Digital Elevation Models (DEMs) generated by remote sensing

technologies such as SRTM or IFSARE remains challenging. To fully understand the problem, it is important to realize that the mapped elevation surfaces include the top of the

forest canopy, as well as anthropogenic features, rather than

the ground surface required for flow routing. Although homogeneous forest canopy generally follows the shape of the

ground surface, gaps in vegetation that can stretch over hundreds of meters create large, often nested, depressions that

pose difficulties for flow routing. In addition to these large

depressions, the surface over forested areas is noisy, creating

numerous small depressions and barriers that further complicate drainage network extraction. Therefore, one of the

important questions investigated in this paper was whether

such data are suitable for drainage network extraction at all

and if yes, how accurate are the extracted drainage networks

and which methods provide the most reliable results.

At the same time, the broad availability of elevation data

dramatically increased the extent of regions that can be analyzed at relatively high resolutions, given the computational

capabilities that can support processing of massive DEMs.

Published by Copernicus Publications on behalf of the European Geosciences Union.
668 M. Metz et al.: Extraction of drainage networks from radar-based elevation models41

Therefore, significant effort has been devoted to the development of new flow tracing and watershed analysis algorithms that support efficient processing of large DEMs and

address the issue of routing through complex nested depressions (e.g. Rivix Limited Liability Company, 2001; Arge et

al., 2003; Danner et al., 2007).

The most widespread method for handling depressions is

sink filling, up to the level of the sink spill point, combined with routing through the resulting flat area (Jenson and

Domingue, 1988; Fairfield and Leymarie, 1991; Planchon

and Darboux, 2001; Wang and Liu, 2006). Improved sink

filling methods (e.g. Garbrecht and Martz, 1997; Grimaldi

et al., 2007; Santini et al., 2009) first fill sinks, and then introduce a gradient to all flat areas to provide non-zero gradients for flow routing. The method complementary to sink

filling is carving or breaching (Rieger, 1998; Martz and Garbrecht, 1998) where a channel is carved out of each sink,

breaking through the (artificial) obstacle. Both principles can

be combined in an impact reduction approach (IRA; Lindsay

and Creed, 2005) that for each sink determines the method

that causes the least impact on the source dataset. All these

approaches alter the elevation data in order to ensure full

drainage, assuming that sinks are artifacts created either by

too low (artificial pits) or too high (artificial drainage blocks)

elevation values.

An alternative to the modification of elevation data is to

determine the least-cost drainage paths (LCP) through unaltered terrain and out of sinks. LCP search methods were

generally designed to find the shortest or fastest route from a

starting point to a given destination, used for example in car

navigation devices or to generate cumulative cost surfaces. A

particular LCP search method (A* Search; Hart et al., 1968)

was adapted for flow routing and watershed analysis and implemented as the module r.watershed (Ehlschlaeger, 1989) in

GRASS GIS (Neteler and Mitasova, 2008, see GRASS Development Team 2010 for binaries and source code). When

applied to IFSARE derived DEMs, this approach has provided more accurate flow routing through large, nested depressions with fewer artifacts than traditional sink filling

(Kinner et al., 2005). However, the module was not designed to handle the massive DEMs that are currently available (Arge et al., 2003). To render the module suitable for

analysis of very large datasets at ever higher resolutions, and

to reduce the artifacts in flow patterns due to the single flow

direction (SFD) algorithm (Quinn et al., 1991; Holmgren,

1994; Tarboton, 1997), redesign of the implementation and

addition of more flow routing options was necessary.

In this study, we present substantial improvements to the

LCP method implementation and evaluation of its efficiency

and accuracy when applied to radar-based DEMs. We compare the performance of the improved module with (1) its

original implementation, (2) a module based on a disk input/output (I/O) efficient method (Arge et al., 2003), and

(3) the IRA implementation in the Terrain Analysis System

(TAS, Lindsay and Creed, 2005). We also analyze the impact

of mapping technology (IFSARE, SRTM), resolution, and

DEM resampling on the accuracy of the extracted drainage

networks. Although the LCP method has been used since

early 90ies, the literature that describes the algorithm, its

implementation and properties is very limited (Ehlschlaeger,

1989). This paper aims to fill this gap in literature and to

highlight the value of this method and its improvements for

analysis of modern DEMs.

2 Methods

2.1 Improved least cost path search algorithm

The LCP algorithm (A* Search, Hart et al., 1968;

Ehlschlaeger, 1989) used for flow routing, flow accumulation, and watershed analysis of raster-based DEMs was redesigned to increase the processing speed and decrease memory consumption. The implementation in the GRASS module r.watershed (Fig. 1) starts with potential outlet points.

Natural ultimate outlets are e.g. river mouths opening into

oceans or lakes without outflow. On gridded elevation models, potential outlets are grid cells along the map boundaries

or cells with at least one neighbor with unknown elevation,

e.g. masked ocean. All potential outlet grid points are inserted into a list sorted by costs. Costs are measured as elevation and order of addition to the sorted list (grid cells added

earlier have higher precedence in case of equal elevation).

For the actual search, the grid cell with the lowest elevation (smallest cost) is extracted and removed from the sorted

list, marked as processed and its neighbors are investigated.

This causes the search to proceed along the least steep uphill slope. At each step during the search, only neighboring

grid cells that are not yet in the search list and not yet processed are added to the list, and drainage direction for these

neighboring grid cells is set towards the current grid cell. If

a depression is encountered, the search follows the steepest

downhill slope to the bottom of a depression and then proceeds again along the least steep uphill slope. The search

proceeds in this manner upstream along the least steep slope

and terminates when all grid points have been processed. The

LCP search provides two results: (a) flow direction for each

cell in a standard D8 manner (O’Callaghan and Mark, 1984),

and (b) the order in which cells must be processed for flow

accumulation.

The search component of the original implementation was

modified by augmenting it with a minimum heap used as priority queue (Atkinson et al., 1986; Metz and Ehlschlaeger,

2010), which led to a substantial speed improvement. In

the original implementation, the time needed to keep the

sorted list sorted increased exponentially with the number

of grid cells, while in the new implementation this time increase is logarithmic. Removal of redundant information in

intermediate data as well as reduced memory requirements

for both the search and the flow accumulation components

Hydrol. Earth Syst. Sci., 15, 667–678, 2011 www.hydrol-earth-syst-sci.net/15/667/2011/
42M. Metz et al.: Extraction of drainage networks from radar-based elevation models 669

Fig. 1. Flowchart illustrating the least cost path search algorithm.

Fig. 2. SRTM and IFSARE coverage of Panama. GPS ground control points are displayed in black.

have further enhanced the module’s capacity to process larger

DEMs. For massive datasets that can not be processed with

the amount of memory available, the module can optionally

use external memory with intermediate data stored on disk.

These changes, aimed at improving the computational performance, did not affect the quality of the results.

To improve the flow routing accuracy and to reduce artifacts in the flow accumulation pattern, a multiple flow direction (MFD) algorithm, based on Holmgren (1994), was

implemented using the order determined by the LCP search

with an option to control the strength of flow convergence.

For grid-based elevation models, an MFD approach conforms better to theoretical surface flow dispersion than single

flow direction in which only one neighboring cell can receive

surface flow from the current cell (Freeman, 1991; Quinn et

al., 1991; Holmgren, 1994; Tarboton, 1997). Initial D8 flow

directions as determined by the LCP search are used to route

flow out of depressions when all unprocessed neighboring

grid cells have higher elevation than the current grid cell.

2.2 Elevation data

Two different sources of elevation data were used to demonstrate the improvements in the LCP method (Fig. 2). Countrywide elevation coverage of Panama was available as a

3 arc second resolution SRTM DEM (Farr et al., 2007).

SRTM version 2.1 provided by the NASA Jet Propulsion Laboratory (http://www2.jpl.nasa.gov/srtm/, download

at http://dds.cr.usgs.gov/srtm/) was selected for this study as

the most reliable version in terms of accuracy and minimal

artifacts, after evaluating properties of the SRTM products

available at the time of writing (v1, v2.1, v3, v4.1). The

published absolute vertical error of the version 2.1 SRTM

DEM is 6.2 m for South America including Panama (Farr

et al., 2007), although several studies report higher accuracies (see e.g. Rodriguez et al., 2006). A recent airborne IFSARE survey has provided new, more detailed information

about the topography in central Panama at 10 m resolution,

using technology with published vertical accuracy of ∼3 m

www.hydrol-earth-syst-sci.net/15/667/2011/ Hydrol. Earth Syst. Sci., 15, 667–678, 2011
43 670 M. Metz et al.: Extraction of drainage networks from radar-based elevation models

Fig. 3. Shaded relief of a Landsat false color composite (R,G,B =

4,5,3) overlaid with GPS reference points in red and points digitized

from Landsat in grey.

(Andersen et al., 2006). IFSARE (InterFerometric Synthetic

Aperture Radar - Elevation) is derived in a way very similar to SRTM, with the main difference that IFSARE is airborne and SRTM data was recorded with sensors onboard

the Space Shuttle Endeavour. Both IFSARE and SRTM used

two antennas to collect radar data; elevation information was

derived by analyzing the data recorded by the two antennas

(the general concept of the new satellite-borne TanDEM-X

mission is similar, collecting radar data with two antennas).

Neither of the surveys penetrated the forest canopy to the

ground surface and the elevation surface was over large regions defined by a triple canopy tropical forest environment

with tree heights of more than 30 m above ground. Given

the widespread use of the SRTM data, one of the important

questions explored in this paper was whether such data are

suitable for stream extraction at all, and if yes, what is the accuracy and which methods provide the most reliable results

in this challenging environment.

SRTM v2 tiles covering all of Panama were combined and

gaps in the dataset filled using the regularized spline with tension (RST) interpolation method (GRASS module r.fillnulls;

Mitasova and Mitas, 1993). The seamless SRTM coverage

was then reprojected with bicubic interpolation from geographic to the UTM zone 17N coordinate system with 90 m

resolution to keep reprojection modifications to a minimum.

To evaluate the impact of resampling on stream extraction

accuracy (see e.g. Valeriano et al., 2006) and to generate

data for additional performance testing, the reprojected 90 m

SRTM DEM was reinterpolated to 30 m resolution using the

RST method.

The IFSARE data were provided as a 10 m resolution

DEM (Kinner et al., 2005) in the UTM zone 17N coordinate system and did not require additional processing. For

additional testing purposes, the IFSARE DEM was downsampled to 30 m resolution using bicubic interpolation in order to have DEMs from two sources (SRTM and IFSARE)

with different level of detail, but identical resolution of 30 m.

2.3 Stream location data

Two sets of data points were used to evaluate horizontal accuracy of the drainage network extraction methods: (i) stream

segments digitized from Landsat imagery (TM 5, year 2000,

scene id LT50120542000087XXX02, provided by the United

States Geological Survey – USGS) and (ii) GPS field measurements. Bands 3, 4, and 5 of the Landsat TM scene were

used at its native resolution of 30 m. The geographic projection for this study (UTM 17 North) was identical to the projection of this Landsat scene, i.e. the Landsat grid geometry

and values were not modified. A Landsat false color composite with Red, Green, and Blue assigned to the channels

4, 5, and 3 respectively, clearly separated vegetation from

waterbodies and provided the background for manual digitization of points along the river centerlines. Only streams and

rivers with a width less or equal to 4 grid cells (120 m) were

digitized in order to exclude lakes, anabranching or braided

rivers, and rivers too broad to reliably determine a stream

center using Landsat imagery (Fig. 3). The horizontal accuracy of these manually digitized reference points is given

by the imagery resolution of 30 m or better, depending on

the stream width and the location of the stream centerline in

relation to the grid cells.

GPS points were collected in the field using a Corvallis Microtechnology March v3.7 GPS unit with ∼2 m positional accuracy. Sites in the Chagres river watershed were

measured during the years 2002 to 2007 and sites at lower

reaches of most major rivers across Panama were measured

during 2005 and 2009 years. GPS points were collected

along clearly identifiable perennial rivers at locations where

a reliable GPS signal was available. The GPS measurements

included a larger proportion of points acquired in mountainous regions along smaller rivers, while most of the points

located along larger rivers in low-gradient flood plain and

coastal plain landscapes were digitized from Landsat imagery (Fig. 3).

The SRTM water body database was not used for the accuracy evaluation because this cannot be regarded as an independent data source since it was used in the creation of the

SRTM DEM v2, and because even the larger rivers visible on

Landsat imagery were still too narrow to be captured in the

SRTM water body database.

We have also considered and rejected the use of hydrography (blue lines) from topographic maps because of their

low reliability and a relatively coarse scale of 1:50 000. Several studies that have compared the blue lines with on-ground

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44

M. Metz et al.: Extraction of drainage networks from radar-based elevation models 671

Fig. 4. Extraction of a single main channel: (A) Location of areas shown in images (B), (C), and (D); (B) The starting points of the extracted

streams (red lines) were well upstream of the reference points (black crosses). Flow accumulation derived from IFSARE 10 m is used as

background. The extracted streams (red lines) are located within broader stream tubes in low-gradient areas close to the coast, for both (C)

LCP-derived streams and (D) streams derived with sink filling. IFSARE 10 m was used for both (C) and (D) representing the same area.

surveys of existing streams demonstrated that the blue lines

were the least accurate source of stream location information (see e.g. Colson, 2006 and the references therein), often based on old data digitized from aerial photographs with

large errors in locations where the streams were not visible,

due to the presence of vegetation. In fact, one of the major

motivations for deriving the drainage network from the IFSARE data in Panama was the inadequate scale of available

blue lines with many larger streams missing and with errors

in the mapped streams location. Until recently, the relative

inaccessibility of some regions (e.g. the upper Chagres river)

and an incessant cloud cover have discouraged mapping in

these areas (Kinner et al., 2005).

2.4 Flow accumulation and evaluation of extracted

drainage networks

Several tests were performed to evaluate the tested flow routing algorithms. The first test evaluated the processing speed

and capability to handle large DEMs when computing flow

accumulation maps. In this test, the performance of the new

LCP implementation was compared with the old r.watershed

module (GRASS Development Team, 2008) and with a relatively new, I/O-efficient algorithm, used in the module

r.terraflow (Arge et al., 2003). All computations were done

on the same hardware (AMD Athlon X2 3 GHz and 8 GB

RAM) but different operating systems (Linux 64 bit for the

GRASS modules r.watershed and r.terraflow, and Windows

XP 32 bit for TAS GIS which is a 32 bit application for Microsoft Windows and was not available as 64 bit application

at the time of writing).

The second test evaluated the horizontal accuracy of the

extracted drainage paths for algorithms with different handling of depressions. Three approaches were tested: (i) sink

filling as implemented in r.terraflow, (ii) the impact reduction

approach (IRA) as implemented in TAS GIS (Lindsay and

Creed, 2005), and (iii) the LCP algorithm as implemented in

the enhanced version of r.watershed in GRASS GIS 6.4. To

test the sink filling method, r.terraflow was used because of

its capability to efficiently process massive DEMs.

The third test evaluated the impact of DEM resolution

and level of detail on the horizontal accuracy of the extracted drainage paths for all tested algorithms (r.watershed,

r.terraflow, and TAS).

The tests were performed for the central Panama region

using both the IFSARE and SRTM data at the 10 m, 30 m

and 90 m resolutions by computing flow accumulation with

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45 672 M. Metz et al.: Extraction of drainage networks from radar-based elevation models

MFD dispersal and extracting drainage networks from the

computed flow accumulation.

Drainage networks were extracted from all MFD flow accumulation maps using a minimum upstream catchment area

of 100 000 m2 as a threshold. This threshold was selected

for testing purposes and to ensure that all measured or digitized points were located downstream from the extraction

starting point and that all points had a drainage path associated with it (Fig. 4b). Accurate identification of the channel

head zones was beyond the scope of this paper because it requires more than elevation data as it is often significantly influenced by local geology, groundwater level, and land cover

(North Carolina Division of Water Quality NCWQ, 2010).

According to the NCWQ (2010) study, “the stream origins

usually occur as transition zones in which the location and

length of the zone is subject to fluctuations in groundwater

levels and precipitation”. The tested 90 m and 30 m resolution SRTM DEMs measured over at least 30 m high canopy

of dense tropical forest did not provide sufficient information for accurate identification and mapping of the dynamic

channel heads zones. Therefore our focus was on mapping

streams and rivers that have sufficiently large contributing

areas that the existence of a well defined stream can be expected.

In order to extract a single main channel in broader floodplains where MFD flow accumulation generates several cells

wide stream tubes that may include cells above the given

threshold, a new stream starting point was defined only if

it was not located within such a stream tube (Fig. 4). Downstream channel tracing followed the steepest slope and maximum flow accumulation within the stream tube. To egress

from depressions using the LCP method, streams again followed the maximum flow accumulation which was in these

cases identical to the drainage direction initially determined

during the LCP search. As a result, the drainage paths extracted from MFD flow accumulation were a single grid cell

wide and were easily vectorized for further analysis. This

approach of extracting single cell wide dendritic and topologically correct drainage networks from flow accumulation

obtained with any kind of flow distribution method avoids

the problems associated with skeletonizing rasterized MFD

channels. The MFD capabilities to map flow dispersal in

floodplains can then be preserved and single flow direction

(SFD) routing is not necessary for flow accumulation and

subsequent channel extraction (although it remains an option).

Assessment of drainage network extraction accuracy

was then performed by calculating distances between the

drainage paths derived from the IFSARE and SRTM DEMs

and the points measured along the streams, consisting of 338

on-ground GPS measurements and 995 points digitized along

rivers from the Landsat imagery (Figs. 3, 4).

In order to determine which method or level of detail provided more accurate results, we determined the distance of

reference points to the nearest extracted drainage path. The

difference in the distance to reference points between methods or level of detail was then used to assess which method

or level of detail provided more accurate results. Distance

differences were tested with two-tailed Student’s t-tests for

statistical significance and α = 0.05. If distance differences

were significantly different from zero, i.e. one of the tested

methods/levels of detail was more accurate than the other, the

sign of the difference indicated which method/level of detail

was more accurate.

3 Results

3.1 Performance comparison

To demonstrate the improvement in computational performance of the new LCP method implementation, the processing times needed by the old and new version of r.watershed,

and r.terraflow were compared, with the results summarized

in Table 1. For flow accumulation computation, the new

version was 350 times faster than the old version for the

central Panama area represented by a relatively small DEM

with 2 million grid cells at 90 m resolution. The improvement was even more dramatic for the countrywide coverage

at 90 m resolution with 27 million grid cells: the new version

was about 1750 times faster. Although absolute processing

times are dependent on the specific hardware used for testing, we expect that relative time differences will be similar on other systems because the software optimizations are

hardware-independent. Processing the larger regions used

in this study with the old version was impractical because

it could easily take days (Arge et al., 2003), whereas the new

version required only minutes to execute. The computational

time needed by the I/O-efficient r.terraflow was significantly

lower than that for the old r.watershed, but the r.terraflow

module was not as fast as the new LCP implementation on

our test system (Table 1). We have also considered the case

when the data do not fit into memory and r.watershed uses

segmented processing, with intermediate data being stored

on disk. This leads to longer processing times than for the

all-in-memory mode. Processing times observed on our testing system in the segmented mode were still shorter than for

r.terraflow, which uses I/O-efficient algorithms specifically

designed for large datasets (Arge et al., 2003). The size of

intermediate data created by r.watershed in the segmented

mode was less than 20% of those created by r.terraflow. This

can be of advantage if intermediate data are just a bit too large

to fit into memory. It is important to note that, theoretically,

r.terraflow with its advanced I/O-efficient algorithms should

be faster than r.watershed for DEMs much larger than the

ones processed here and on systems with less physical memory.

TAS GIS was not included in the performance comparisons, because its memory requirements for processing the

10 m IFSARE DEM and the 30 m SRTM DEM for all of

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46M. Metz et al.: Extraction of drainage networks from radar-based elevation models 673

Table 1. Processing time for the different DEMs on a Linux 64 bit system with an AMD Athlon X2 3 GHz CPU and 8 GB RAM.

SRTM 90 m IFSARE 30 m SRTM 90 m IFSARE 10 m SRTM 30 m

Central Panama central Panama all of Panama central Panama all of Panama

1.9 million cells 17 million cells 27 million cells 156 million cells 241 million cells

old r.watershed, 23.2min 23 h 10 min 30 h 46 min > 2 days > 2 days

all in memory not measured not measured

new r.watershed, 0.07 min 0.83 min 1.05 min 9.97 min 12.3 min

all in memory

new r.watershed, 0.15 min 1.73 min 2.05 min 30.7 min 32.18 min

data on disk

r.terraflow 0.45 min 4.4 min 5.05 min 66.95 min 58.7 min

Table 2. Median distances of reference points to nearest extracted

stream in meters.

GPS Landsat

filling IRA LCP filling IRA LCP

IFSARE 10 m 62.65 N/A* 34.40 43.75 N/A* 39.25

IFSARE 30 m 77.15 71.42 70.71 48.88 48.27 47.75

SRTM 30 m 99.78 92.46 89.13 84.25 82.81 83.72

SRTM 90 m 124.84 119.69 121.00 94.30 94.69 94.92

∗ Computation cancelled by software with out of memory error.

0

5

10

15

20

25

30

IFSARE

10m

IFSARE

30m

SRTM

30m

SRTM

90m

distance difference, M ± SD [m]

*

*

*

*

*

* *

GPS points

Landsat points

Fig. 5. Mean and standard deviation of the distance differences between sink filling and least-cost path search. A positive distance difference indicates that streams extracted with least-cost path search

are closer to reference points than streams extracted with sink filling.

Panama exceeded available system memory. TAS GIS is a

32 bit application for Microsoft Windows and can only utilize as much memory as a 32 bit application can manage,

i.e. 3 GB. As opposed to r.terraflow and r.watershed, TAS

GIS keeps all intermediate data in memory, thus limiting the

size of the DEMs that can be processed. The test system provided as much memory as TAS GIS could theoretically uti-

-5

0

5

10

15

IFSARE

10m

IFSARE

30m

SRTM

30m

SRTM

90m

distance difference, M ± SD [m]

*

* GPS points

Landsat points

Fig. 6. Mean and standard deviation of the distance differences

between impact reduction approach and least-cost path. A positive distance difference indicates that streams extracted with leastcost path search are closer to reference points than streams extracted

with the impact reduction approach.

lize, hence the out-of-memory error was regarded as a limitation of this application and not as a hardware limitation.

Therefore, TAS was used only for the sink treatment methods in comparison with the 30 m and 90 m DEM covering

the central Panama subregion (Fig. 2) and results of the IRA

method were not available for 10 m IFSARE. At the time of

writing, IRA was not included in WhiteboxGAT (Lindsay,

2009) that has replaced TAS.

3.2 Comparison between different methods of

sink treatment

To evaluate the impact of different approaches in the handling of elevation surface depressions, median distances of

reference points to extracted drainage paths for each sink

treatment method and each processed DEM were computed

and the results are summarized in Table 2. As expected, the

location of extracted drainage paths improved with increasing detail available in the DEM. While the drainage paths for

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47 674 M. Metz et al.: Extraction of drainage networks from radar-based elevation models

0

5

10

15

20

25

30

35

40

GPS field points Landsat

distance difference, M ± SD [m]

*

*

* *

LCP

Sink filling

IRA

Fig. 7. Mean and standard deviation of the distance differences

between the 30 m SRTM DEM and the 30 m IFSARE DEM. A positive distance difference indicates that streams extracted from the

30 m IFSARE DEM are closer to reference points than streams extracted from the 30 m SRTM DEM. LCP: least-cost path search;

IRA: impact reduction approach.

-5

0

5

10

15

20

25

30

35

40

45

GPS field points Landsat

distance difference, M ± SD [m]

*

*

* *

LCP

Sink filling

IRA

Fig. 8. Mean and standard deviation of the distance differences between the 90 m and the 30 m SRTM DEM. A positive distance difference indicates that streams extracted from the 30 m SRTM DEM

are closer to reference points than streams extracted from the 90 m

SRTM DEM. LCP: least-cost path search; IRA: impact reduction

approach.

the 90 m SRTM DEM were about 100 m away from reference points, the median distances improved down to 34.4 m

for the IFSARE 10 m DEM.

Drainage paths extracted from the LCP flow accumulation

were closer to the GPS field points (t-tests, all t337 < −5.4, all

p < 0.0001) for all DEMs at all tested resolutions (IFSARE

10 m, 30 m, and SRTM 30 m, 90 m) when compared with the

sink filling method (Fig. 5 showing bar graphs with mean and

standard deviation of the distance differences). The drainage

paths extracted from LCP were also closer to the points digitized from Landsat for all DEMs (t-tests, all t994 < −2.3,

all p < 0.0001), except for the 90 m SRTM where the distance difference was not significant for (t-test, t994 = −1.213,

p = 0.23).

When compared to the IRA method (Fig. 6 showing bar

graphs with mean and standard deviation of the distance differences), drainage paths extracted from LCP flow accumulation were closer to the GPS field points for the 30 m IFSARE

(t-test, t337 = −2.361, p = 0.018) and 30 m SRTM (t-test,

t337 = −3.088, p = 0.002), but for the 90 m SRTM the difference was not significant (t-test, t337 = −1.745, p = 0.08).

For the points digitized from Landsat, drainage paths extracted from LCP accumulation were not closer than IRA

drainage paths for any of the tested DEMs (t-tests, all absolute t994 < 1.0, all p > 0.3).

3.3 Comparison between different levels of detail

at constant resolution

The 10 m IFSARE and 90 m SRTM were interpolated to the

same resolution of 30 m. We investigated how much the different levels of detail at identical resolution influenced the accuracy of extracted drainage paths for all three sink treatment

methods (Fig. 7 showing bar graphs with mean and standard

deviation of the distance differences). The distance between

the reference points and drainage paths extracted from the

30 m IFSARE DEM was subtracted from the distance between the reference points and drainage paths extracted from

the 30 m SRTM DEM to measure the difference in accuracy

of drainage paths extracted from these two DEMs. For larger

rivers in flat areas (Landsat points), the accuracy of drainage

path locations was considerably higher for the 30 m IFSARE

DEM than for the 30 m SRTM DEM (t-test, all t994 < −13.0,

all p < 0.001; on average 35 m closer) for all methods. For

smaller rivers not broader than 30 m in mountainous regions

(GPS points), only IRA showed a significant improvement

in accuracy from the 30 m SRTM DEM to the 30 m IFSARE

DEM (t-test, t337 = −3.315, p = 0.001, all other t337 > −1.9

and <0, p > 0.05). However, as shown in the previous section for the 30 m IFSARE DEM, IRA drainage path locations

in mountainous regions were significantly less accurate than

LCP drainage path locations.

3.4 Effects of RST resampling on the accuracy of

drainage network extraction

In order to assess the effect of SRTM resampling from 90 m

to 30 m resolution by the RST method, the drainage paths

extracted from the 90 m SRTM DEM were compared with

drainage paths extracted from the resampled 30 m SRTM

DEM by calculating the difference in drainage path distances

to reference points (Fig. 8 showing bar graphs with mean and

standard deviation of the distance differences). Corresponding median distances of reference points to drainage path locations are given in Table 2. The accuracy of drainage path

locations was improved by resampling for larger rivers in flat

landscape for all sink treatment methods (t-tests, all t994 <

−6.5, all p < 0.001), however, for GPS points in mountainous regions only the results obtained by the least-cost path

search were improved (t-test, t337 = −3.734, p < 0.001, all

other t337 > −1 and <0, p > 0.3).

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M. Metz et al.: Extraction of drainage networks from radar-based elevation models 675

Fig. 9. A typical example for the impact of sink filling along a section of a river course extracted from the 30 m SRTM DEM. Along this

section, 92% of all elevation values were modified. Parts of the river course obscured by trees are highlighted in the GoogleEarth sreenshot.

Fig. 10. Typical effects of different sink treatment methods on extracted stream courses in (A) mountainous areas and (B) costal plains. Filled

sinks are grayed. Streams extracted from sink-filling in red are overlaid by streams extracted from least-cost path search in blue. Points of

confluences (yellow cross) in mountainous areas can be shifted considerably, and in coastal plains, sink filling can lead to long straight lines.

A shaded relief of the 10 m IFSARE DEM was used as background.

4 Discussion

Digital Terrain Models, as an approximation of real terrain at

a given level of detail, include random noise and errors. For

example, an in depth analysis of SRTM accuracy and source

of errors is provided by Rodriguez et al. (2006) and Farr et

al. (2007). The errors can have significant impact on the results of DEM analysis and their treatment has been the focus

of broad research (e.g. Jenson and Domingue, 1988; Lindsay

and Creed, 2005; Hutchinson, 2000). Sink (depression) removal was designed to remove small depressions introduced

as artifacts of elevation data processing for the first generation of DEMs to facilitate continuous flow routing (Jenson

and Domingue, 1988). This procedure, referred to as hydrological conditioning, has become widespread even for next

generations of elevation data and is now a pre-processing

step required by a number of applications for hydrological

analysis and modeling. In order to alleviate the problem of

routing surface flow through the flat areas created by sink

filling, efforts have been made to add a gradient to these flat

areas (e.g. Martz and Garbrecht, 1998; Wang and Liu, 2006;

Grimaldi et al., 2007; Santini et al., 2009) Nevertheless, as

modern high resolution data such as LiDAR demonstrate,

actual terrain includes many true depressions, especially in

riffle-and-pool portions of mountainous rivers, and on floodplains and coastal plains (Notebaert et al., 2009). Therefore, sink filling is not always an appropriate approach for

hydrological conditioning of a DEM because the measured

elevation data are replaced with new elevation values based

on the rather unrealistic assumption that the terrain has no

depressions (Jenson and Domingue, 1988; Tarboton, 1997),

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49 676 M. Metz et al.: Extraction of drainage networks from radar-based elevation models

be they artificial or real. Although many hydrologic models require depression-less DEMs, removal of sinks can alter

the true hydrological state that includes standing water in depressions. Moreover, in IFSARE and SRTM DEMs that include vegetation cover, many of the observed depressions are

caused by gaps in canopy, canopy closing over river courses,

or by vegetation partially obstructing narrow and deep valleys. The filling of such features requires altering elevations

by several meters and over areas of hundreds of m2, leading

to large flats and an artificial drainage network geometry that

does not represent reality. Figure 9 presents an example of

the barriers across the river channel created by overhanging

trees and a profile that illustrates the extensive modification

in elevation that can result from such filling. Preserving most

of the drainage directions of the original DEM is preferable

to hydrological conditioning in this case.

Our results suggest that the adverse effects of sink filling

become more pronounced for higher resolution DEMs (see

Fig. 10 for drainage paths extracted from 10 m IFSARE).

By contrast, at lower resolutions such as 90 m, the coarsest

resolution tested in this study, there was often no significant

difference in drainage path locations between the compared

methods, particularly for Landsat reference points. Landsat points where predominantly located in flood plain and

coastal plain landscapes where both the horizontal and the

vertical resolution of 90 m SRTM were often not sufficient to

locate rivers and no method was able to accurately delineate

drainage paths in these areas from 90 m SRTM (Hancock et

al., 2006; Valeriano et al., 2006).

When comparing SRTM and IFSARE DEMs at the same

resolution of 30 m, the drainage networks extracted from

the IFSARE DEM were, as expected, more accurate than

drainage networks extracted from SRTM DEM, particularly

for Landsat points in relatively flat areas. As shown in previous studies (Hancock et al., 2006; Valeriano et al., 2006),

the SRTM DEM close to its native resolution of 3 arc sec

does not provide sufficient vertical detail to determine river

courses in flat areas. The applied RST resampling method

significantly improved the accuracy of extracted drainage

path locations for SRTM in regions with low topography.

Apparently the level of detail has been not just increased but

also improved because resampling to a higher horizontal (and

for SRTM also higher vertical) resolution seems to be helpful in improving the results of watershed analysis from lowresolution DEMs (see also Valeriano et al., 2006). The global

SRTM coverage at 3 arc sec horizontal resolution has been

created by averaging the original 1 arc sec coverage (Farr et

al., 2007). Depending on the method used, reinterpolating

the 3 arc sec version back to 1 arc sec can result in narrowed

valleys and ridges and increased channel sinuosity. Interestingly, only the LCP search method benefited from the reinterpolation in mountainous regions and provided here the most

accurate drainage path locations for the 30 m SRTM DEM.

It is important to note that the differences in the tested

methods were significant due to the particularly challenging

study region. Smaller differences between the methods could

be expected in topography with gentle slopes and uniformly

low vegetation cover. Although network connectivity, topology and channel morphometry has not been investigated, the

visual inspection of the results indicates that there were differences between the results obtained by different methods

and resolutions in the coastal plain areas. None of the tested

DEMs or methods produced sufficiently reliable results and

higher resolution did not always lead to correct topology.

These results are similar to those found for LiDAR-based

DEMs (Colson, 2006) and require further research.

Although the tests presented here demonstrated the performance of the LCP method for radar-based data, the new,

improved version of r.watershed has in fact been developed

using LiDAR data provided by the USGS in addition to the

1 m LiDAR DEM examples and data provided by Neteler and

Mitasova (2008). The presented method processes a DEM

without modifying its elevation values and is thus to a degree (quantitatively narrowed by this study) dependent on

the reliability of the provided DEM. The LCP search approach, despite its capability to route flow through depressions, will therefore, like other methods, fail to accurately

extract drainage networks if the input DEM deviates too far

from real topography.

5 Conclusions

We presented a method for fast computation of flow accumulation and drainage network extraction for large DEMs

with nested depressions and evaluated it against other commonly used methods for sink treatment. The accuracy assessment using ground control points and Landsat satellite imagery provided insight into the accuracy of drainage network

extraction from radar elevation data in a challenging tropical forest environment and coastal plain setting. The results

suggest that the conceptually simple sink filling approach is

not suitable for the IFSARE and SRTM DEMs in regions

with significant vegetation cover whereas both the tested impact reduction approach of Lindsay and Creed (2005) and the

LCP search provide more accurate drainage networks.

The performance testing has demonstrated that the new

implementation dramatically improves computational efficiency while preserving the high accuracy of the LCP routing capabilities. The increase in computational performance

is particularly relevant for the modern mapping technologies

that can rapidly produce massive DEMs at high resolutions

for large areas and the slow processing and analysis has become bottleneck in their application. Fast processing of massive DEMs opens their application to a new level of detail and

spatial extent, for example in rapid response operations or for

mapping in remote regions where the streams are covered by

dense vegetation and no reliable stream data are available.

Hydrol. Earth Syst. Sci., 15, 667–678, 2011 www.hydrol-earth-syst-sci.net/15/667/2011/
50M. Metz et al.: Extraction of drainage networks from radar-based elevation models 677

Acknowledgements. The support of the US Army Research Office

for this research is gratefully acknowledged. We would also like to

thank two anonymous reviewers for their valuable comments.

Edited by: R. Purves

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Research@

53

Connectivity of Heterogeneous 

Wireless Networks

Wei Ren, Qing Zhao, and Ananthram Swami

IEEE Transactions on Information Theory, 57(7), 4315-4332 (2011)
54IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 57, NO. 7, JULY 2011 4315

Connectivity of Heterogeneous Wireless Networks

Wei Ren, Student Member, IEEE, Qing Zhao, Senior Member, IEEE, and Ananthram Swami, Fellow, IEEE

Abstract—We address the percolation-based connectivity of

large-scale ad hoc heterogeneous wireless networks, where secondary users exploit channels temporarily unused by primary

users and the existence of a communication link between two secondary users depends on not only the distance between them but

also the transmitting and receiving activities of nearby primary

users. We introduce the concept of connectivity region defined as

the set of density pairs—the density of secondary users and the

density of primary transmitters — under which the secondary

network is connected. Using theories and techniques from continuum percolation, we analytically characterize the connectivity

region of the secondary network and reveal the tradeoff between proximity (the number of neighbors) and the occurrence

of spectrum opportunities. Specifically, we establish three basic

properties of the connectivity region—contiguity, monotonicity of

the boundary and uniqueness of the infinite connected component,

where the uniqueness implies the occurrence of a phase transition

phenomenon in terms of the almost sure existence of either zero

or one infinite connected component; we identify and analyze

two critical densities which jointly specify the profile as well as

an outer bound on the connectivity region; we study the impacts of secondary users’ transmission power on the connectivity

region and the conditional average degree of a secondary user

and demonstrate that matching the interference ranges of the

primary and the secondary networks maximizes the tolerance of

the secondary network to the primary traffic load. Furthermore,

we establish a necessary condition and a sufficient condition for

connectivity, which lead to an outer bound and an inner bound on

the connectivity region.

Index Terms—Cognitive radio, connectivity region, continuum

percolation, critical densities, heterogeneous wireless network,

phase transition.

I. INTRODUCTION

T HE communication infrastructure is becoming increasingly heterogeneous, with a dynamic composition of interdependent, interactive and hierarchical network components

with different priorities and service requirements. One example

is the cognitive radio technology [1] for opportunistic spectrum

access which adopts a hierarchical structure for resource sharing

Manuscript received August 10, 2009; revised October 20, 2010; accepted

January 23, 2011. Date of current version June 22, 2011. This work was supported in part by the Army Research Office under Grant W911NF-08-1-0467

and by the National Science Foundation under Grant CCF-0830685.

W. Ren and Q. Zhao are with the Department of Electrical and Computer Engineering, University of California, Davis, CA 95616 (e-mail:

qzhao@ucdavis.edu; wren@ucdavis.edu).

A. Swami is with the Army Research Laboratory, Adelphi, MD 20783 USA

(e-mail: a.swami@ieee.org).

Communicated by S. Ulukus, Associate Editor for Communication

Networks.

Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIT.2011.2145650

[2]. Specifically, a secondary network is overlaid with a primary network, where secondary users identify and exploit temporarily and locally unused channels without causing unacceptable interference to primary users [2].

A. Connectivity and Connectivity Region

While the connectivity of homogeneous ad hoc networks consisting of peer users has been well studied (see, for example,

[3]–[10]), little is known about the connectivity of heterogeneous networks. The problem is fundamentally different from

its counterpart in homogeneous networks. In particular, the connectivity of the low-priority network component depends on the

characteristics (traffic pattern/load, topology, interference tolerance, etc.) of the high-priority component, thus creating a much

more diverse and complex design space.

Due to the hierarchical structure of spectrum sharing, a communication link exists between two secondary users if the following two conditions hold: (C1) they are within each other’s

transmission range; (C2) they see a spectrum opportunity determined by the transmitting and receiving activities of nearby

primary users (see Section II-B-I). We say that a topological link

exists between two secondary users if only condition (C1) holds.

Using theories and techniques from continuum percolation,

we analytically characterize the percolation-based connectivity

of the secondary network in a large-scale ad hoc heterogeneous

network. Specifically, we consider a Poisson distributed secondary network overlaid with a Poisson distributed primary

network in an infinite two-dimensional Euclidean space1. We

define network connectivity as the existence of an infinite

connected component formed by communication links almost

surely (a.s.), i.e., the occurrence of percolation. This connectivity characterizes the accessibility of two secondary users at

a given instant and is thus also called instantaneous mutual

connectivity. Given the transmission power and the interference tolerance of both the primary and the secondary users, the

connectivity of the secondary network depends on the density

of secondary users [due to (C1)] and the traffic load of primary

users [due to (C2)].

We thus introduce the concept of connectivity region , defined as the set of density pairs under which the secondary network is connected, where denotes the density of

the secondary users and the density of primary transmitters

(representing the traffic load of the primary users). As illustrated

in Fig. 1, a secondary network with a density pair inside the shaded region is connected: the secondary network has

1This infinite network model is equivalent in distribution to the limit of a

sequence of finite networks with a fixed density as the area of the network increases to infinity, i.e., the so-called extended network [11]. It follows from the

arguments similar to the ones used in [12, Chapter 3] for homogeneous ad hoc

networks that this infinite ad hoc heterogeneous network model represents the

limiting behavior of large-scale networks.

0018-9448/$26.00 © 2011 IEEE
55 4316 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 57, NO. 7, JULY 2011

Fig. 1. Connectivity region (the upper boundary is defined as the

supremum density of the primary transmitters that ensures connectivity with a

fixed density of the secondary users; the critical density of the secondary

users is defined as the infimum density of the secondary users that ensures connectivity under a positive density of the primary transmitters; the critical density

of the primary transmitters is the supremum density of the primary transmitters that ensures connectivity with a finite density of the secondary users).

a giant connected component which includes infinite number of

secondary users. The existence of the giant connected component enables bidirectional communications between distant secondary users via multihop relaying. On the other hand, a secondary network with a density pair outside this region is not connected: the network is separated into an infinite

number of finite connected components. Consequently, any secondary user can only communicate with users within a limited

range.

It is shown in [13] and [14] that when the primary network is

static, this instantaneous connectivity is equivalent to the connectivity defined in terms of the finiteness of the minimum multihop delay (referred to as fd-connectivity); when the primary

traffic has temporal dynamics (no matter how small the range

of the dynamics is), this instantaneous connectivity is stronger

than the fd-connectivity and the secondary network can be intermittently connected as long as it is topologically connected

[13], [14].

The objective of this paper is to establish analytical characterizations of the connectivity region and to study the impact of

system design parameters (in particular, the transmission power

of the secondary users) on network connectivity. The main results are summarized in the subsequent two subsections.

B. Analytical Characterizations of the Connectivity Region

We first establish three basic properties of the connectivity region: contiguity, monotonicity of the boundary and uniqueness

of the infinite connected component. Specifically, based on a

coupling argument, we show that the connectivity region is a

contiguous area bounded below by the -axis and bounded

above by a monotonically increasing function (see

Fig. 1), where the upper boundary is defined as

(1)

with denoting the secondary network of density

overlaid with a primary network specified by the density

of the primary transmitters. The uniqueness of the infinite connected component is established based on ergodic theory and

certain combinatorial results. It shows that once the secondary

network is connected, it contains a unique infinite connected

component a.s.

Second, we identify and analyze two critical parameters of

the connectivity region: and . They jointly specify the

profile as well as an outer bound on the connectivity region. Referred to as the critical density of the secondary users, is the

infimum density of the secondary users that ensures connectivity

under a positive density of the primary transmitters

(2)

We show that equals the critical density of a homogeneous

ad hoc network (i.e., in the absence of primary users), which

has been well studied [15]. This result shows that the “takeoff”

point in the connectivity region is completely determined by the

effect of proximity—the number of neighbors (nodes within the

transmission range of a secondary user).

Referred to as the critical density of the primary transmitters,

is the supremum density of the primary transmitters that

ensures the connectivity of the secondary network with a finite

density of the secondary users:

(3)

We obtain an upper bound on , which is shown to be achievable in simulations. More importantly, this result shows that

when the density of the primary transmitters is higher than the

(finite) value given by this upper bound, the secondary network

cannot be connected no matter how dense it is. This parameter

thus characterizes the impact of opportunity occurrence

on the connectivity of the secondary network: when the density of the primary transmitters is beyond a certain level, there

are simply not enough spectrum opportunities for any secondary

network to be connected.

Since a precise characterization of the upper boundary

of the connectivity region is intractable, we establish

a necessary and a sufficient condition for connectivity to provide an outer and an inner bound on the connectivity region.

The necessary condition is expressed in the form of the conditional average degree of a secondary user and is derived by the

construction of a branching process. The sufficient condition

is obtained by the discretization of the continuum percolation

model into a dependent site percolation model.

C. Impact of Transmission Power on Connectivity: Proximity

Versus Opportunity

The study of the impact of the secondary users’ transmission power on network connectivity reveals an interesting

tradeoff between proximity and opportunity in the design of

heterogeneous networks. As illustrated in Fig. 2, we show

that increasing enlarges the connectivity region along

the -axis (i.e., better proximity leads to a smaller “takeoff”

point), but at the price of reducing along the -axis.

Specifically, with a large , few secondary users experience

spectrum opportunities due to their large interference footprint

with respect to the primary users. This leads to a poor tolerance

to the primary traffic load parameterized by .

The transmission power of the secondary network should

thus be chosen according to the operating point of the hetero-
56REN et al.: CONNECTIVITY OF HETEROGENEOUS WIRELESS NETWORKS 4317

Fig. 2. Simulated connectivity regions for two different transmission powers

( denotes the transmission power of the secondary users and the large is

times the small , where is the path-loss exponent).

geneous network given by the density of the secondary users

and the traffic load of the co-existing primary users. Using the

tolerance to the primary traffic load as the performance measure, we show that the interference range of the secondary

users should be equal to the interference range of the primary users in order to maximize the upper bound on the critical

density of the primary transmitters. Given the interference

tolerance of the primary and secondary users, we can then design the optimal transmission power of the secondary users

based on that of the primary users.

D. Related Work

To our best knowledge, the connectivity of large-scale ad hoc

heterogeneous networks has not been characterized analytically

or experimentally in the literature. There are a number of classic

results on the connectivity of homogeneous ad hoc networks.

For example, it has been shown that to ensure either 1-connectivity (there exists a path between any pair of nodes) [3], [4]

or -connectivity (there exist at least node-disjoint paths between any pair of nodes) [5], the average number of neighbors

of each node must increase with the network size. On the other

hand, to maintain a weaker connectivity—p-connectivity (i.e.,

the probability that any pair of nodes is connected is at least ),

the average number of neighbors is only required to be above a

certain “magic number’ which does not depend on the network

size [6].

The theory of continuum percolation has been used by

Dousse et al. in analyzing the connectivity of a homogeneous

ad hoc network under the worst-case mutual interference [7],

[8]. In [9], [10], the connectivity and the transmission delay in

a homogeneous ad hoc network with static or dynamic on-off

links are investigated from a percolation-based perspective.

In [13] and [14], the impact of temporal dynamics of primary

traffic on the connectivity and multihop delay of the secondary

network is studied. In particular, it is shown that in the presence

of primary traffic dynamics, the secondary network is connected

(either instantaneously or intermittently) as long as it is topologically connected. Optimal power control in heterogeneous networks has been studied in [16], which focuses on a single pair of

secondary users in a Poisson network of primary users. The impacts of secondary users’ transmission power on the occurrence

of spectrum opportunities and the reliability of opportunity detection are analytically characterized.

E. Organization and Notations

The rest of this paper is organized as follows. Section II

presents the Poisson model of the heterogeneous network. In

particular, the conditions for the existence of a communication

link in the secondary network is specified based on a rigorous

definition of spectrum opportunity. In Section III, we introduce

the concept of connectivity region and establish its three basic

properties. The two critical densities are analyzed, followed

by a necessary and a sufficient condition for connectivity. In

Section IV, we demonstrate the tradeoff between proximity

and opportunity by studying the impact of the secondary users’

transmission power on the connectivity region and on the

conditional degree of a secondary user. Further, we obtain the

optimal transmission power of the secondary users, adopting

the secondary network’s tolerance of the primary traffic load as

the performance measure. Section V contains detailed proofs

of the main results and Section VI concludes the paper.

Throughout the paper, we use capital letters for parameters

of the primary users and lowercase letters for those of the secondary users.

II. NETWORK MODEL

We consider a Poisson distributed secondary network overlaid with a Poisson distributed primary network in an infinite

two-dimensional Euclidean space. The models of the primary

and secondary networks are specified in the following two subsections.

A. The Primary Network

The primary transmitters are distributed according to a twodimensional Poisson point process with density . Each primary transmitter’s receiver is uniformly distributed within its

transmission range . Here we have assumed that all primary

transmitters use the same transmission power and the transmitted signals undergo isotropic path loss. Based on the displacement theorem [17, Chapter 5], it is easy to see that the primary receivers form a two-dimensional Poisson point process

with density . Note that the two Poisson processes formed

by the primary transmitters and receivers are correlated.

B. The Secondary Network

The secondary users are distributed according to a two-dimensional Poisson point process with density , independent

of the Poisson processes of the primary transmitters and receivers. The transmission range of the secondary users is denoted by .

Communication Links: In contrast to the case in a homogeneous network, the existence of a communication link between

two secondary users depends on not only the distance between

them but also the availability of the communication channel
57

4318 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 57, NO. 7, JULY 2011

Fig. 3. Definition of spectrum opportunity.

(i.e., the presence of a spectrum opportunity). The latter is determined by the transmitting and receiving activities in the primary

network as described below.

As illustrated in Fig. 3, there exists an opportunity from ,

the secondary transmitter, to , the secondary receiver, if the

transmission from does not interfere with nearby primary receivers in the solid circle and the reception at is not affected

by nearby primary transmitters in the dashed circle [18]. Referred to as the interference range of the secondary users, the

radius of the solid circle at depends on the transmission

power of and the interference tolerance of the primary receivers, whereas the radius of the dashed circle (the interference range of the primary users2) depends on the transmission

power of the primary users and the interference tolerance of .

It is clear from the above discussion that spectrum opportunities depend on both transmitting and receiving activities of the

primary users. Furthermore, spectrum opportunities are asymmetric. Specifically, a channel that is an opportunity when

is the transmitter and the receiver may not be an opportunity when is the transmitter and the receiver. In other

words, there exist unidirectional communication links in the

secondary network. Since unidirectional links are difficult to utilize in wireless networks [19], we only consider bidirectional

links in the secondary network when we define connectivity. As

a consequence, when we determine whether there exists a communication link between two secondary users, we need to check

the existence of spectrum opportunities in both directions.

To summarize, under the disk signal propagation and interference model, there is a (bidirectional) link between and

if and only if (C1) the distance between and is at most ;

(C2) there exists a bidirectional spectrum opportunity between

and , i.e., there are no primary transmitters within distance

of either or and no primary receivers within distance

of either or .

Connectivity: We interpret the connectivity of the secondary

network in the percolation sense: the secondary network is connected if there exists an infinite connected component a.s.

Based on the above conditions (C1, C2) for the existence of a

communication link, we can obtain an undirected random graph

corresponding to the secondary network, which

is determined by three Poisson point processes: the secondary

2Since the minimum transmission power for successful reception is, in general, higher than the maximum allowable interference power, it follows that the

transmission range of primary users is smaller than . Furthermore, under

the disk signal propagation and interference model, we have

. A similar relationship holds for and .

Fig. 4. Realization of the heterogeneous network. The random graph

consists of all the secondary nodes and all the bidirectional links

denoted by solid lines. The solid circles with radii denote the interference

regions of the primary transmitters within which secondary users cannot

successfully receive and the dashed circles with radii denote the required

protection regions for the primary receivers within which the secondary users

should refrain from transmitting.

users with density , the primary transmitters with density

and the primary receivers with density (correlated

with the process of the primary transmitters)3. See Fig. 4 for an

illustration of .

The question we aim to answer in this paper is: when is the

secondary network connected, i.e., when does percolation occur

in ?

III. ANALYTICAL CHARACTERIZATIONS OF THE

CONNECTIVITY REGION

Given the transmission power and the interference tolerance

of both the primary and the secondary users (i.e., , ,

and are fixed), the connectivity of the secondary network is

determined by the density of the secondary users and the

density of the primary transmitters. We thus introduce the

concept of connectivity region of a secondary network, which

is defined as the set of density pairs under which the

secondary network is connected (see Fig. 1)

A. Basic Properties of the Connectivity Region

We next establish three basic properties of the connectivity

region.

Theorem 1: Basic Properties of the Connectivity Region.

T1.1 The connectivity region is contiguous, that is, for

any two points , , there exists

a continuous path in connecting the two points.

T1.2 The lower boundary of the connectivity region is

the -axis. Let denote the upper boundary of

the connectivity region defined by (1), then we have that

is monotonically increasing with .

3The two Poisson point processes of the primary transmitters and receivers

are essentially a snap shot of the realizations of the primary transmitters and

receivers. In different time slots, different sets of primary users become active

transmitters/receivers. Thus, even if a secondary user is isolated at one time due

to the absence of spectrum opportunities, it may experience an opportunity at a

different time and be connected to other secondary users.
58

REN et al.: CONNECTIVITY OF HETEROGENEOUS WIRELESS NETWORKS 4319

Fig. 5. Two realizations of the Poisson heterogeneous network (black stars denote primary transmitters, green plus signs denote primary receivers, red dots denote

secondary users and blue segments denote the bidirectional links between secondary users). We have removed secondary users who do not see opportunities for

clarity. The simulation parameters are given by , , , , and the critical density in this case is

(a) percolation occurs (b) percolation does not occur .

T1.3 There exists either zero or one infinite connected component in a.s.

Proof: The proofs of T1.1 and T1.2 are based on the

coupling argument, a technique frequently used in continuum

percolation [15, Section 2.2]. The proof of T1.3 is based on

the ergodicity of the random model driven by the three Poisson

point processes of the primary transmitters, the primary receivers and the secondary users. The details of the proofs are

given in Section V-B.

T1.1 and T1.2 specify the basic structure of the connectivity

region, as illustrated in Fig. 1. T1.3 implies the occurrence of a

phase transition phenomenon, that is, there exists either a unique

infinite connected component a.s. or no infinite connected component a.s. This uniqueness of the infinite connected component

excludes the undesirable possibility of having more than one

(maybe infinite) infinite connected component in the secondary

network. We point out that such a property is not always present

in wireless networks. Two examples where more than one infinite connected component exists in a homogeneous ad hoc network can be found in [20].

B. Critical Densities

In this subsection, we study the critical density of the secondary users and the critical density of the primary transmitters defined by (2) and (3), respectively. We have the following theorem.

Theorem 2: Critical Densities.

Given , , and , we have:

T2.1 , where is the critical density for

a homogeneous ad hoc network with transmission range

(i.e., in the absence of the primary network).

T2.2

(4)

where the constant is the critical density for a homogeneous ad hoc network with a unit transmission range.

Proof: The basic idea of the proof of T2.1 is to approximate

the secondary network by a discrete edge-percolation model on the grid. This discretization technique is often

used to convert a continuum percolation model to a discrete

site/edge percolation model (see, for example, [15, Chapter 3],

[8]). The details of the proof are given in Section V-C-I.

The proof of T2.2 is based on the argument that if there is an

infinite connected component in the secondary network, then an

infinite vacant component must exist in the two Poisson Boolean

models driven by the primary transmitters and the primary receivers, respectively. The key point is to carefully choose the

radii of the two Poisson Boolean models in order to obtain a

valid upper bound on . The details of the proof can be found

in Section V-C-II.

Fig. 5(a) shows one realization of the heterogeneous network

when is slightly larger than and is small. At least

one left-to-right (L-R) crossing and at least one top-to-bottom

(T-B) crossing can be found in the square network. It is thus

expected that these L-R and T-B crossings in finite square regions can form an infinite connected component in the whole

network on . If we slightly increase , then we observe

from Fig. 5(b) that the reduction in spectrum opportunities eliminates numerous communication links in the secondary network,

creating several disjoint small components.

Fig. 6 shows a simulation example of the connectivity region,

where the upper bound on the critical density of the primary transmitters given in T2.2 appears to be achievable.

C. A Necessary Condition for Connectivity

In this subsection, we establish a necessary condition for connectivity which is given in terms of the average conditional degree of a secondary user. This condition agrees with our intuition: the secondary network cannot be connected if the degree

of every secondary user is small.

Let denote the event that there exist primary receivers/transmitters within distance of a secondary user . Let

denote the complement of . Since a
59 4320 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 57, NO. 7, JULY 2011

Fig. 6. Simulated connectivity regions when , ,

and . The blue dashed line is the upper bound (4) on

the critical density of primary transmitters given in T2.2. The area of the

simulated heterogeneous network is . For a fixed density

of the secondary users, the upper boundary is equal to the minimum

density of the primary transmitters such that over all the 1000 realizations, the

percentage of the ones in which there exists at least one L-R crossing is below

50%. The intuitive reason for choosing the existence of an L-R crossing as the

criterion for connectivity is illustrated in Fig. 5 and is discussed in Section V-C.

secondary user is isolated if it does not see a spectrum opportunity, we focus on secondary users who experience spectrum opportunities and define the conditional average degree of such

a secondary user as

(5)

where denotes the degree of , the interference range

of the secondary users and the interference range of the primary users. Notice that the degree of is the number of secondary users within the transmission range of and experiencing bidirectional opportunities with . We arrive at the following necessary condition for connectivity.

Theorem 3: A necessary condition for the connectivity of

is , where is the conditional average degree

of a secondary user defined in (5).

Proof: The basic idea is to construct a branching process,

where the conditional average degree is the average number of

offspring. This branching process provides an upper bound on

the number of secondary users in a connected component. If

, then the branching process is finite a.s. It thus follows that

there is no infinite connected component a.s. in .

Details can be found in Section V-D.

To apply the necessary condition given in Theorem 3, the

conditional average degree of a secondary user needs to

be evaluated based on the network parameters. Let be a secondary user randomly and uniformly distributed within transmission range of . Let denote the

probability of a bidirectional opportunity between and conditioned on the event that sees an opportunity. Based on the

statistical equivalence and independence of different points in

a Poisson point process, the conditional average degree of a

secondary user is given by this conditional probability of

a bidirectional opportunity between and a randomly chosen

neighbor multiplied by the average number of neighbors of ,

i.e.,

(6)

The detailed derivation of (6) and an expression for are

given in Appendix A. It is also shown in Appendix A that is

a strictly decreasing function of . Thus , the inverse

of with respect to , is well-defined.

Combining (6) with Theorem 3, we obtain an outer bound on

the connectivity region. Specifically, let denote the

conditional average degree of a secondary user in .

Then those density pairs satisfying

are outside the connectivity region.

Corollary 1: Given , , and , an outer bound on the

connectivity region is given by

where is the inverse of the conditional probability

with respect to .

D. A Sufficient Condition for Connectivity

In this subsection, we establish a sufficient condition for connectivity, which provides an inner bound on the connectivity region and a criterion for checking whether a secondary network

is connected.

Our sufficient condition for connectivity is established by

using the discretization technique. The continuum percolation

model is mapped onto a dependent site-percolation model

in the following way. As illustrated in Fig. 7, we partition

into (dashed) squares with side length and locate a site at the

center of each square. Sites whose associated dashed squares

share at least one common point are considered connected (as

illustrated by solid lines in Fig. 7). Thus, each site is connected

to eight neighbors4 (see the eight neighbors of site

in Fig. 7). Let be the associated dashed square of , then

is occupied if there exists in at least one secondary user

who sees an opportunity.

Since the largest distance between two points in two neighboring dashed squares is , it follows that if we set

, then every pair of secondary users in two neighboring

dashed squares are within transmission range of each other.

Based on the definitions of occupied site in and communication link in the secondary network, we conclude that the existence of an infinite occupied component (a connected component consisting of only occupied sites) in implies the existence

of an infinite connected component in the secondary network.

Due to the fact that spectrum opportunities are spatially dependent, the state of one site is correlated with the states of its

adjacent sites. Thus, the above site-percolation model is a dependent model. Define the dependence range as the minimum

distance such that the state of any two sites at distance larger

than are independent, where the distance between two sites

4For the commonly used square site-percolation model, each site has four

neighbors. The site-percolation model constructed here provides a better inner

bound.
60REN et al.: CONNECTIVITY OF HETEROGENEOUS WIRELESS NETWORKS 4321

Fig. 7. Illustration of the dependent site-percolation model with side length

(solid dots denote sites, solid lines denote edges connecting every two sites

and dashed lines denote the squared partition).

is the minimum number of neighboring sites that must be traversed from one site to the other. Then the dependence range of

is given by

(7)

Let denote the upper critical probability of which is defined as the minimum probability of occupancy such that if

the probability of occupancy , an infinite occupied component containing the origin exists in with a positive probability (wpp.). Since the dependence range of is finite, it

follows from [12, Theorem 2.3.1] that . Now we present

a sufficient condition for connectivity in the following theorem.

Theorem 4: Let denote the upper critical probability of the

dependent site-percolation model specified above. Define

(8)

where is the common area of two circles with radii

and and centers apart. Then the secondary network is

connected if the equation shown at the bottom of the page holds.

Proof: The proof is based on the ergodicity of the heterogeneous network model and its relation with the constructed

dependent site-percolation model . Details can be found in

Section V-E.

By applying a general upper bound on the upper critical probability for a site-percolation model with finite dependence

range [12, Theorem 2.3.1], we arrive at the following corollary.

Corollary 2: A sufficient condition for the connectivity of

is

where is defined in (8) and is the dependence

range of the site-percolation model defined in (7).

IV. IMPACT OF TRANSMISSION POWER: PROXIMITY

VERSUS OPPORTUNITY

In this section, we study the impact of the secondary users’

transmission power on the connectivity and the conditional average degree of the secondary network. As has been illustrated

in Fig. 2, there exists a tradeoff between proximity and opportunity in designing the secondary users’ transmission power for

connectivity. Specifically, increasing the transmission power of

the secondary users leads to a smaller critical density of the

secondary users, but at the same time, a lower tolerance to the

primary traffic load manifested by a smaller critical density

of the primary transmitters.

A. Impact on the Conditional Average Degree

As discussed in Section III-C, the expression for the conditional average degree can be decomposed into the product

of two terms: and . The first

term is the average number of neighbors of a secondary

user, which increases with the transmission power of the

secondary users (i.e., enhanced proximity). The other term

is the conditional probability of a bidirectional opportunity, which decreases with due to reduced

spectrum opportunities. This tension between proximity and

opportunity is illustrated in Fig. 8, where we observe that

the impact of on proximity dominates when is small

( increases with ) while its impact on the occurrence of

opportunities dominates when is large ( decreases with

).

Corollary 3: Let be the transmission power of secondary

users and the conditional average degree defined in (5), then

under the disk signal propagation and interference model we

have5

where is the path-loss exponent.

Proof: We show this corollary by deriving an upper bound

on . Details can be found in Appendix B.

5Here we use the Big O notation: as if and only if

, such that for all .
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4322 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 57, NO. 7, JULY 2011

Fig. 8. Conditional average degree of secondary users vs transmission range

of secondary users. , where is the transmission power of

secondary users and is the path-loss exponent and simulation parameters are

given by , , , ,

.

For a homogeneous network, the average degree of a user is

, which increases with at rate . In sharp contrast, this corollary tells us that for a heterogeneous network,

when is large enough, the conditional average degree of

a secondary user actually decreases with at least as fast as

.

B. Impact on the Connectivity Region

From the scaling relation of the critical density [15, Proposition 2.11], we know that in a homogeneous two-dimensional

network

where the constant is the critical density for a homogeneous ad hoc network with a unit transmission range. Thus,

if each secondary user adopts a high transmission power, then

decreases. It follows from T2.1 that the critical density of secondary users required to achieve connectivity decreases due to the enhanced proximity.

On the other hand, from the upper bound on the critical density of the primary transmitters given in T2.2, we have that

where we have assumed that for some

under the disk signal propagation and interference model. Thus,

when the transmission power of the secondary network is

large enough, the critical density of the primary transmitters decreases with at least as fast as due to reduced spectrum opportunities.

C. Optimal Design of Transmission Power

Due to the tension between proximity and opportunity, there

does not exist a transmission power of the secondary users that

Fig. 9. Example of the upper bound on as a function of (parameters

are given by , ).

leads to the “largest” connectivity region (largest in the sense

that its connectivity region contains all regions achievable with

any finite transmission power of the secondary users). Thus,

the optimal design of depends on the operating point of the

heterogeneous network. For instance, when a sparse secondary

network is overlaid with a primary network with low traffic load,

a large may be desirable to achieve connectivity. The opposite holds when a dense secondary network is overlaid with a

primary network with high traffic load.

Focusing on a secondary network whose density exceeds the

critical density of the secondary users, we address the design

of its transmission power for the maximum tolerance to the primary traffic. Due to its tractability and achievability, the upper

bound on the critical density of the primary transmitters

given in T2.2 is used as the performance measure (see Fig. 6 for

a simulation example).

Theorem 5: Let and denote the interference range of

the secondary and the primary users. For a fixed , the upper

bound on given in T2.2 is maximized when the primary and

secondary networks have matching interference ranges:

.

Proof: Since under the disk signal propagation and interference model, for some , the upper bound

on can be written as

for

for .

Then the above theorem can be readily shown by finding the

maximal point for the two cases: and .

An example of the upper bound on is plotted as a function of in Fig. 9. Notice that there is a distinct difference

in the slope on the two sides of the optimal point. As a consequence, the operating region of is preferred over that of

when the optimal point cannot be achieved.

We point out that the desired operating region of is the

typical case of a secondary network coexisting with a privileged

primary network.
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V. PROOFS

In this section, we present proofs of the main results presented

in Sections III–IV. We start with a brief overview of several

basic results in percolation theory that will be used in the proofs.

A. Percolation Theory: Some Basic Results

Poisson Boolean Model: Poisson Boolean model [15], often

referred to as , is specified by two elements: a

Poisson point process on with density and a radius ,

a random variable with a given distribution. Under this model,

each point in is the center of a ball in with a random

radius distributed according to the distribution of . Radii associated with different points are independent and they are also

independent of points in . Under a Poisson Boolean model,

the whole space is partitioned into two regions: the occupied

region, which is the region covered by at least one ball and the

vacant region, which is the complement of the occupied region.

We define occupied (vacant) components as those connected

components in the occupied (vacant) region.

Assume that nodes in a homogeneous ad hoc network form

a Poisson point process with density and their transmission

range is . It is easy to see that the connectivity of this network

can be studied through examination of the occupied connected

components in the corresponding Poisson Boolean model

.

Sharp Transition in Two Dimensions: For the Poisson

Boolean model in two dimensions, the sharp transition phenomenon is remarkable in the sense that the critical density

for the a.s. existence of infinite occupied components is equal

to that for the a.s. existence of infinite vacant components.

Let denote the critical density for the Poisson Boolean

model , then we have that:

• when , there is no infinite occupied component

a.s. and there is a unique infinite vacant component a.s.;

• when , there is a unique infinite occupied component a.s. and there is no infinite vacant component a.s.

The exact value of is not known. For a deterministic radius ,

simulation results [21] indicate that , while

rigorous bounds are provided

in [15] and [22].

Crossing Probabilities: A continuous curve in the occupied

region is called an occupied path. An occupied path is an occupied L-R crossing of the rectangle

if intersects both the left and the right boundaries

of the rectangle, i.e., ,

and the segment between

the two intersecting points is fully contained in the rectangle.

Similarly, we define an occupied T-B crossing by requiring that

intersects with the top and bottom boundaries of the rectangle.

Let

Fig. 10. Continuous path connecting the two points

and in the connectivity region . (a) .

(b) .

denote the two crossing probabilities in the rectangle

. Then for a Poisson Boolean model in two

dimensions with a.s. bounded , we have [15, Corollary 4.1]

that for any

if

if . (9)

Due to the symmetry of the Poisson Boolean model,

similar results hold for the T-B crossing probability

.

B. Proof of Theorem 1

Proof of T1.1: To prove T1.1, it suffices to show that for any

two given points and in , we can find

a path in that connects these two points. In particular, the path

we constructed is given by a horizontal segment and a vertical

segment as shown in Fig. 10, where we assume, without loss of

generality, that .

Consider case (a) in Fig. 10 where . Case

(b) can be proven similarly. First, we show every point

on the horizontal segment belongs to . Let . A Poisson point process with

density is statistically equivalent to the superposition of a

Poisson point process with density and an independent

Poisson point process with density . It follows that any

realization of the heterogeneous network with densities and

can be generated by adding more secondary nodes to a

realization of the heterogeneous network with densities and

. Thus, the existence of an infinite connected component

in implies the existence of an infinite connected

component in . We thus have that

for .

Now we know that the two end points and

of the vertical segment belong to . For a point

on the vertical segment,

let , then any realization of the heterogeneous network with densities and can be obtained

by independently removing each primary transmitter-receiver

pair with probability from a realization of the heterogeneous network with densities and . It follows

from the definition of communication link in the secondary

network (see Section II-B-I) that the existence of an infinite

connected component in implies the existence

of an infinite connected component in . Thus, we

have .
63 4324 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 57, NO. 7, JULY 2011

Proof of Theorem 1.2: Suppose that

, then by using the coupling argument for showing that the vertical segment belongs to in the above proof of T1.1, we conclude that , i.e., the -axis is the lower boundary

of .

Suppose that . In order to prove the monotonicity of with it suffices to show that ,

if then . This is a direct consequence of the coupling argument for showing that the horizontal

segment belongs to in the above proof of T1.1.

Proof of Theorem 1.3: We first establish the ergodicity6 of

the heterogeneous network model.

Lemma 1: The heterogeneous network model is ergodic.

Proof of Lemma 1: The proof of this lemma is inspired

by the proof of the ergodicity of the Poisson Boolean model

[15, Proposition 2.8]. The difficulty here is that for the heterogeneous network model, we have two correlated Poisson point

processes: the primary transmitters and the primary receivers.

The definition of the shift transformation for the primary network model is thus more complicated than the standard Poisson

Boolean model with a deterministic radius . To prove Lemma

1, we first show the ergodicity of the primary network model

and then we show the mixing property7 of the secondary network model. Since the primary network model is independent

of the secondary network model, it follows from [24, Theorem

2.6.1] that the heterogeneous network model is ergodic.

Let denote the Borel -algebra in and the set of all

simple counting measures8 on . Construct a -algebra

for generated by sets of the form

where and is an integer. A point process can now

be defined as a measurable mapping from a probability space

into [23, Chapter 7]. The measure on

induced by is defined as , for all

.

In order to define the shift transformation on , it is convenient to identify with . Let denote the

number of points in , and be the shift

according to a vector . Then induces a shift transformation through the following equation for every

:

(10)

6A measure preserving (m.p.) dynamical system consists of

a set , a -algebra of measurable subsets of , a nonnegative measure

on and an invertible m.p. transformation such that

for all . A set is said to be T-invariant if

. The m.p. dynamical system is said to be ergodic

if the -algebra of T-invariant sets is trivial, i.e., for any invariant set, either it

has measure 0 or its complement has measure 0.

7An m.p. dynamical system is said to be mixing if for all

, .

8A simple counting measure on is an integer-valued measure for which

the measures of bounded Borel sets are all finite and the measure of a point is

at most 1.

Let be the probability space of

the Poisson point process for the primary transmitters with density . Let be the product space

for the primary receivers, where

. Then we equip with

the usual product -algebra and product measure with all

marginal probability measure being , where is a uniform

probability measure on . Finally, we set

and equip with the product measure and

the usual product -algebra. It follows that the primary network

model is a measurable mapping from into

defined by , where is

specified in the definition of the point process.

The positions of the primary transmitters corresponding to

are easily known from .

For the primary receivers, the positions are obtained as follows.

Consider binary cubes

For each primary transmitter , there exists a unique smallest

integer such that it is contained in a binary cube

which contains no other primary transmitters. The relative position of ’s receiver with respect to

is then given by .

Let , denote the unit vectors in , then the translation

defined by induces a shift

transformation on through the equation

Hence also induces a shift transformation on

as follows:

where is defined in (10). By using techniques similar to the

proof of Boolean models [15, Proposition 2.8], we have that the

m.p. dynamical system is ergodic.

Since the transmission range of secondary users is fixed,

the probability space of the secondary network model is the

probability space for the Poisson point process

of secondary users with density . It follows from the

proof of Poisson point processes [15, Proposition 2.6] that the

m.p. dynamical system is mixing.

Since the primary network model is independent of the secondary network model, the sample space of the heterogeneous

network model can be written as the product of and ,

i.e., . We equip with product measure

and the usual product -algebra. Similarly, the translation induces a transformation on

, which is given by

Then it follows from [24, Theorem 2.6.1] that the product m.p.

dynamical system is ergodic. Since the -algebra invariant under the transformation group

is a subset of the -algebra invariant under the transformation
64REN et al.: CONNECTIVITY OF HETEROGENEOUS WIRELESS NETWORKS 4325

, we conclude that acts ergodically, i.e., the

heterogeneous network model is ergodic.

Based on Lemma 1, we have the following lemma.

Lemma 2: The number of infinite connected component in

is a constant a.s. and it can only take value from

.

Proof of Lemma 2: Let denote the (random) number of

infinite connected components in , then since for all

, the event is invariant under the group of shift

transformations, it follows from Lemma 1 and the definition of

ergodicity that or 1. Consequently, we have

that is a.s. constant. Then it suffices to exclude the possibility

of . This is shown by contradiction, that is, if there exist

infinite connected components, then they can be linked

together as one connected component wpp. The proof is inspired

by the proof of Proposition 3.3 in [15] and a major difference is

that here we need to consider the impact of the primary network

on the connectivity of the secondary network.

Suppose that there are infinite connected components

a.s. If we remove all the secondary nodes centered inside a box

, then the resulting secondary network should

contain at least unbounded components a.s. Let, for ,

denote the graph formed by secondary nodes in . Given

a box and , consider the event shown at the

bottom of the page.

Partition the box into squares with side length

and let denote the collection of all the

squares which are adjacent to the boundary of . Clearly, for

a box and , we can find

and such that for any point with

, there exists a square

for which we have . This means that,

if we center in each square of a secondary node and there

are neither primary transmitters nor primary receivers within a

bigger box , then

every infinite component in with is

connected to some secondary node in .

Let be the event that each square in contains at

least one secondary node and the event that there are neither primary transmitters nor primary receivers within . Since

depends on the configuration of secondary nodes inside

the box , depends on the configuration of secondary

nodes outside and the configuration of primary nodes, based

on the independence of the primary network and the secondary

network, we have

If , and all occur wpp., then there is

only one infinite connected component9 wpp. By using arguments similar to the proof for [15, Proposition 3.3], we have

that there exists a large enough box and such that

. Obviously, . Moreover, it

is easy to see that .

Now we have that the number of infinite connected components is equal to zero, one or infinity a.s. To exclude the possibility of , we can use the proof of [15, Theorem 3.6]

for Poisson Boolean models, which is based on several combinatorial results. The details are omitted.

C. Proof of Theorem 2

Proof of T2.1: To prove T2.1, it suffices to show that:

(a) for any , the secondary network is not connected for any ;

(b) for any , there exists a

such that , the secondary network is

connected.

From Section IV-A, we know that for a Poisson homogeneous

ad hoc network with density and transmission range , the necessary and sufficient condition for connectivity is .

Since the existence of an infinite connected component in the

secondary network implies the existence of an infinite connected

component in the homogeneous ad hoc network with the same

density and the same transmission range, by using a coupling

argument, we conclude that when , there does not

exist an infinite connected component a.s. in the secondary network for any . This proves part (a).

The basic idea of the proof of part (b) is to approximate the

secondary network by a discrete dependent edgepercolation model on the grid. This discrete dependent edgepercolation model is constructed in a way such that the existence of an infinite connected component in implies the existence of an infinite connected component in .

Construct the square lattice on with side length . Note

that each site in is virtual and is not related to any node either

in the secondary network or in the primary network. Next we

specify the conditions for an edge being open in , which is the

key to the mapping from to .

For each edge in , let denote the middle point of .

Then we introduce three random fields , and , all associated with the edge in , where is the indicator

of the edge being open, represents the condition (C1) of

the distance between two users for the existence of a communication link in the secondary network and represents the

condition (C2) of the spectrum opportunity. Specifically, consider the Poisson Boolean model where

9Since , every secondary node in a square of is connected to

those secondary nodes in the neighboring squares.
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4326 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 57, NO. 7, JULY 2011

Fig. 11. Realization where for the edge (hollow points are sites in

and solid segments are edges in ).

is the Poisson point process generated by secondary users, then

for a horizontal edge , if the following two events (illustrated in Fig. 11) are true:

(i) there is an occupied L-R crossing of the rectangle

in

;

(ii) there are two occupied T-B crossings of the square

and the square

in

; and otherwise. For a vertical

edge , the definition of is similar, where the horizontal

and vertical coordinates are switched in the above two

events.

Next, we define the random field . For an edge in ,

if and the following two events are true:

(i) there is no primary transmitter within distance of any

secondary node of the three crossings in the definition of

;

(ii) there is no primary receiver within distance of any secondary node of the three crossings in the definition of ;

and otherwise. It follows from the definition of communication link in the secondary network (see Section II-B-I)

that if and , then the three crossings in

are also three crossings in .

Let , then we claim that the edge is open if

and is closed if . We observe from Fig. 11

that whether the edge is open is correlated with the states of

the other edges. This model thus is a dependent edge-percolation model. Furthermore, as shown in Fig. 12, if there exists

an infinite open connected component in , then those crossings associated with the edges in the infinite component in

comprise an infinite connected component in . As a

consequence, by considering the uniqueness of the infinite connected component in , we only need to prove the

following lemma in order to show T2.1.

Lemma 3: Let denote the open connected component

containing the origin in . Then given ,

, such that for and any , we have

where is the number of edges in .

Fig. 12. Percolation in (thick segments are open edges in and thin segments are closed edges in and blue curves are those crossings associated with

the open edges).

Proof of Lemma 3: For an arbitrary edge in , let

, then we have

From the result on the crossing probabilities given in (9), we

know that when ,

Thus, when , , such that

for .

Given , let be the area of the region covered by

the circles with radii centered at those secondary nodes in

the three crossings and be the area of the region covered by

the circles with radii centered at those secondary nodes in the

three crossings. Then we have

Since and

, it follows from

the basic property of Poisson point processes that
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Obviously, for fixed .

Thus if we choose , then , such that

It implies that when , for all

Thus, for and all , we have

(11)

From Fig. 11, we can see that if

, then the state of edge is only correlated with its six adjacent edges and it is independent of other edges. In this case,

by using the “Peierls argument” [25, Chapter 1], we can show

that if the probability of an edge being closed ,

then

(12)

The proof of the above statement follows the one of [7, Theorem

3] except that the upper bound on the probability of edges all

being closed is given by the following fact.

Fact 1: [8, Proposition 1]: For any collection of

distinct edges in , we have

where is the indicator of being closed, and is the probability of an edge being closed.

Thus, by combining (12) with (11), we conclude that given

, , such that for fixed

and any

Notice that depends on which is chosen according to the

crossing probability and is determined by . As a consequence,

is a function of , i.e., .

Proof of T2.2: From the conditions for the existence of

a communication link in the secondary network specified in

Section II-B-I, we know that for every secondary node in an infinite connected component, there can exist neither any primary

transmitter within distance of it nor any primary receiver

within distance of it. In other words, every secondary node

in an infinite connected component must be located outside

all the circles centered at the primary transmitters and the

primary receivers with radii and , respectively. Thus,

if there is an infinite connected component in the secondary

network, then an infinite vacant component must exist in

the two Poisson Boolean models and

driven by the primary transmitters and

the primary receivers, respectively. Here and are

Fig. 13. Counterexample for choosing . All the secondary nodes

in the infinite path are located outside those circles centered at the primary

transmitters with radii , which form a series of rings surrounding the origin

and there is no infinite vacant component in the Poisson Boolean model

driven by the primary transmitters.

Fig. 14. Infinite path in the secondary network. The dashed segments form an

inner bound on the infinite vacant component in the Poisson Boolean model

driven by the primary receivers.

some appropriate radii which will be specified later. A natural

choice for is , but if we consider the counterexample

given in Fig. 13, then we can clearly see that even if there is an

infinite path in the secondary network, no infinite vacant component exists in the Poisson Boolean model

driven by the primary transmitters. Similarly, counterexamples

can be easily constructed for choosing .

Suppose there is an infinite connected component in the secondary network. Then we can find a sequence of secondary

users such that they comprise an infinite path

starting from (see Fig. 14).

Assume that and are two adjacent secondary

nodes in the above infinite path. Notice that the distance

between and satisfies , where the

second inequality follows from the fact that the minimum transmission power for successful reception is in general

higher than the maximum allowable interference power.

As we know, all the primary receivers must be outside the

two circles with radii centered at and , respectively,

as shown in Fig. 15. Given , consider the rectangle

between and . By a

simple computation in geometry, we have that the minimum

distance from all the primary receivers to the rectangle is

. As illustrated in Fig. 14, it implies that there

exists an infinite vacant component in the Poisson Boolean

model driven by the primary

receivers10. By recalling the known results in Section V-A-II,

we thus conclude that for all

10This technique used here can also be applied to the case when ,

where only the minimum distance from all the primary receiver to the bar between and needs to be recomputed.
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4328 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 57, NO. 7, JULY 2011

Fig. 15. One edge in the infinite path.

Let , then it yields

The other term in the upper bound is obtained

by applying the same argument to the Poisson Boolean model

driven by the primary transmitters.

D. Proof of Theorem 3

Consider the connected component containing

an arbitrarily chosen secondary user . Assuming that

, we construct a branching process as follows.

Notice that if , then must see an opportunity, i.e.,

is true. Call the initial point (or

0-th generation) of the branching process. Then the children

of (i.e., the first generation of the branching process) are

secondary users which satisfy the following two conditions:

i) it is within distance of , where is the transmission

range of secondary users;

ii) there exist neither any primary receiver within distance

of the secondary user nor any primary transmitter within

distance of the secondary user.

The -th generation of the branching process is obtained similarly and these children are connected to their parents in the -th generation of the branching process via

bidirectional links. Obviously, all the secondary users in are

counted in the constructed branching process model. But some

of them may probably be counted more than once, since we do

not exclude the previous generations (including generation

0) when we consider the -th generation. Thus, this branching

process gives us an upper bound on the number of secondary

users in . It follows that if the branching process does not

grow to infinity wpp., then there does not exist an infinite connected component a.s. in , due to the stationarity

of the heterogeneous network model. Since the conditional average degree is the average number of offspring for every generation, the necessary condition follows immediately from the

classic theorem for branching processes11 [1, Theorem 2.1.1].

11The lower bound on the conditional average degree can be improved by

constructing a multi-type branching process. Specifically, when we consider the

children of a secondary user, we categorize them into different types based on

their distances from the secondary user. A similar construction has been used

to obtain the lower bound on the critical density of Poisson Boolean models [1,

Sec. 3.9]. This multi-type branching process is, however, difficult to analyze and

requires substantial further investigation.

E. Proof of Theorem 4

From the construction of the dependent site-percolation

model , we know that the existence of an infinite occupied

component in implies the existence of an infinite connected

component in . Then in order to obtain a sufficient

condition for the connectivity of the secondary network, it

suffices to find a sufficient condition for the existence of an

infinite occupied component in .

Let be the probability that one site is occupied. Then based

on the definition of the upper critical probability of , we

have that if , an infinite occupied component containing

the origin exists in wpp. It implies that if , there exists

an infinite connected component in the secondary network wpp.

Since the event that there exists an infinite connected component

in the secondary network is invariant under the group of shift

transformations, it follows from the ergodicity of the heterogeneous network model (see Lemma 1) that if , there exists

an infinite connected component in the secondary network a.s.

Let denote the event that there exist exactly secondary users in the associated dashed square of a site and

let denote the event that the secondary

user sees an opportunity. Then based on the definition of occupied site in , we obtain a lower bound on as follows:

In the last step, has been

obtained by setting the distance in the expression for the

probability of an unidirectional opportunity between two secondary users with distance apart given in [16, Proposition 1].

VI. CONCLUSION AND FUTURE DIRECTIONS

We have studied the connectivity of a large-scale ad hoc heterogeneous wireless network in terms of the occurrence of the

percolation phenomenon. We have introduced the concept of

connectivity region to specify the dependency of connectivity

on the density of the secondary users and the traffic load of

the primary users. We have shown several basic properties of

the connectivity region: the contiguity of the region, the monotonicity of the boundary and the uniqueness of the infinite connected component. We have analytically characterized the critical density of the secondary users and the critical density of

the primary transmitters; they jointly specify the profile of the

connectivity region. We have also established a necessary and a

sufficient condition for connectivity, which give an outer and an
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Fig. 16. Illustration of (the common area of two circles with radii

and and centers apart), (the union area of two circles with

radii and and cents apart) and (the intersection area

between one circle with radius and the union of the two identical circles

with radii ).

inner bound, respectively, on the connectivity region. Furthermore, by examining the impacts of the secondary users’ transmission power on the connectivity region and on the conditional

average degree of a secondary user, we have demonstrated the

tradeoff between proximity and spectrum opportunity. In establishing these results, we have used techniques and theories

in continuum percolation, including the coupling argument, ergodic theory, the discretization technique, and an approximation

using a branching process.

To highlight the interaction between the primary users and the

secondary users in a heterogeneous network, we have ignored

the fading effect and the mutual interference between secondary

users. If we take into account these factors, then the received

signal to interference-plus-noise ratios at two secondary users

will replace the distance between them in condition (C1) for the

existence of a communication link between them. This will result in a random connection model (RCM) with correlated links,

where the correlation between links is due to the mutual interference and condition (C2) on the presence of the bidirectional

opportunity. There will be a tradeoff between proximity and mutual interference in addition to the tradeoff between proximity

and opportunity and the combination of these two tradeoffs will

complicate the characterization of the connectivity of the secondary network. But since the RCM shares several basic properties (e.g., ergodicity and existence of the critical density) with

the Boolean model used in this paper, we expect that most of the

results established here (e.g., connectivity region, critical densities) can be extended to the RCM.

APPENDIX A

EXPRESSION FOR CONDITIONAL AVERAGE DEGREE

An expression for the conditional average degree of a secondary user is presented in the following proposition.

Proposition 1: Let and be the density of secondary

users and primary transmitters. Let and be the interference range of the secondary and primary users and and

the transmission range of the secondary and primary users. Let

denote the probability of a bidirectional

opportunity between and conditioned on the event that

sees an opportunity. Then the conditional average degree of a

secondary user is given by

(A1)

where

the common area of two circles with radii and

and centers apart [see Fig. 16(a)] and is the

union of two circles with radii and and centers apart [see

Fig. 16(b)]. is the intersection area between

one circle with radius and the union of the two circles with

both radii [see Fig. 16(c)]. For , the two

identical circles are centered apart and the other circle is centered at , where the middle point of the centers of the two

identical circles is chosen to be the origin .

Expressions for and can be obtained in explicit form, which can be found in [1, Appendix A].

The expression for depends on the expression for the common area of three circles which is tedious and

is given in [26]. By applying the basic property of the exponential function to (A1), we can easily show that is a strictly

decreasing function of .

Proof: Let denote the event that there exist exactly

neighbors of a secondary user . We thus have the first equation

shown at the bottom of the next page.

When , it is obvious that . When , let

be a neighbor of and an indicator function for such

that if occurs and

otherwise. Then by considering the statistical independence

and equivalence of the secondary users, we have the second

equation shown at the bottom of the next page. It follows that

we get (A2), as shown at the bottom of the next page.

According to the definition of spectrum opportunity,

can be obtained by setting

the distance in the expression for the probability of a

unidirectional opportunity between two secondary users with

distance d apart given in Proposition 1 in [16].

(A3)
69

4330 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 57, NO. 7, JULY 2011

Next, we derive the expression for the probability

of a bidirectional opportunity, i.e.,

, which depends

on the location of only through its distance to . Since

is uniformly distributed within distance of , the density

function of the distance between and is given by

for . In this case, the probability of a bidirectional

opportunity can be written as shown in (A4), also shown at

the bottom of the page, where the integrand can be written as

shown in (A5), at the bottom of the page.

Next, we compute the two probabilities in (A5) one by one.

Since the primary receivers admit a Poisson point process with

density , we have

(A6)

where is the common area of two circles with both

radii and centers apart [see Fig. 16(a)].

(A2)

(A4)

(A5)
70

REN et al.: CONNECTIVITY OF HETEROGENEOUS WIRELESS NETWORKS 4331

(A7)

(A8)

Let denote the Poisson point process formed by

primary transmitters. If we remove from primary transmitters whose receivers are within distance of or ,

then it follows from the Coloring Theorem [17, Chapter 5] that

all the remaining primary transmitters form another Poisson

point process with density , where

is the area of the circle with radius and

centered at intersecting the two circles both with radii

and centers apart [see Fig. 16(c)]. We thus have (A7), at

the top of the page. where is the union of two

circles with both radii and centers apart [see Fig. 16(b)].

Substitute (A6), (A7) into (A5), we have (A8), as shown at

the top of the page. The expression for the conditional average

degree thus follows by plugging (A8) into (A4) and then (A3),

(A4) into (A2).

APPENDIX B

PROOF OF COROLLARY 3

From [1, Appendix A] and Figs. 16(b) and (c), we know that

when

(B1)

(B2)

Substitute (B1) and (B2) into (A1), we have

(B3)
71 4332 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 57, NO. 7, JULY 2011

Plugging the expression for [1, Appendix A] into

(B3) yields the equation shown at the top of the previous page.

By applying the inequality for ,

we have the equation at the top of the previous page, where we

have assumed that under the disk signal

propagation and interference model. Since , we

arrive at Corollary 3.

ACKNOWLEDGMENT

The authors would like to thank D. Cheowtirakul for his help

in generating several simulation results.

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[12] M. Franceschetti and R. Meester, Random Networks for Communication: From Statistical Physics to Information Systems. New York:

Cambridge University Press, 2007.

[13] W. Ren, Q. Zhao, and A. Swami, “On the connectivity and multihop

delay of ad hoc cognitive radio networks,” IEEE J. Select. Areas

Commun. (Special Issue on Cognitive Radio Networking and Communications), vol. 29, no. 4, pp. 805–818, Apr. 2011.

[14] W. Ren, Q. Zhao, and A. Swami, Temporal Traffic Dynamics

Improve the Connectivity of Ad Hoc Cognitive Radio Networks

Tech. Rep. (TR-10-02) 2010 [Online]. Available: http://www.ece.ucdavis.edu/~qzhao/TR-10-02.pdf

[15] R. Meester and R. Roy, Continuum Percolation. New York: Cambridge Univerisity Press, 1996.

[16] W. Ren, Q. Zhao, and A. Swami, “Power control in cognitive radio

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ADA463920

Wei Ren (S’07) received the B.Sc. and M.Sc. degrees in electrical engineering

from Peking University, Peking, China, in 2003 and 2006, respectively, and the

Ph.D. degree in electrical and computer engineering from the University of California (UC), Davis, in 2011.

His current research interests are in cognitive radio systems and wireless networks. He is also interested in algorithmic theory and optimization techniques

for communications.

Qing Zhao (S’97–M’02–SM’08) received the Ph.D. degree in electrical engineering in 2001 from Cornell University, Ithaca, NY.

In August 2004, she joined the Department of Electrical and Computer Engineering at the University of California (UC), Davis, where she is currently

a Professor. Prior to that, she was a Communications System Engineer with

Aware, Inc., Bedford, MA, and a Postdoctoral Research Associate with Cornell

University. Her research interests are in the general area of stochastic optimization and decision theory in dynamic systems and communication networks.

Dr. Zhao received the 2010 IEEE Signal Processing Magazine Best Paper

Award and the 2000 Young Author Best Paper Award from IEEE Signal Processing Society. She holds the title of UC Davis Chancellor’s Fellow and received the 2008 Outstanding Junior Faculty Award from the UC Davis College

of Engineering. She is also a coauthor of two papers that received student paper

awards at IEEE ICASSP 2006 and IEEE Asilomar Conference 2006. She served

as an Associate Editor of the IEEE TRANSACTIONS ON SIGNAL PROCESSING from

2006 to 2009 and is currently an elected member of the IEEE Signal Processing

Society SP-COM Technical Committee.

Ananthram Swami (F’08) received the B.Tech. degree from the Indian Institute

of Technology (IIT), Bombay, the M.S. degree from Rice University, Houston,

TX, and the Ph.D. degree from the University of Southern California (USC),

Los Angeles, all in electrical engineering.

He has held positions with Unocal Corporation, USC, CS-3 and Malgudi

Systems. He was a Statistical Consultant to the California Lottery, developed

HOSAT, a MATLAB-based toolbox for non-Gaussian signal processing and has

held visiting faculty positions at INP, Toulouse, France. He is the ST for Network Science at the U.S. Army Research Laboratory. He is co-editor of the book

Wireless Sensor Networks: Signal Processing & Communications Perspectives

(New York: Wiley, 2007).

Dr. Swami is a member of the IEEE Signal Processing Society’s Technical

Committee on Sensor Array and Multichannel systems and serves on the Senior Editorial Board of the IEEE JOURNAL ON SELECTED TOPICS IN SIGNAL

PROCESSING. He was a tutorial speaker on Networking Cognitive Radios for

Dynamic Spectrum Access at IEEE ICC 2010, co-chair of IEEE SPAWC’10,

and has served on the IEEE SPS BoG.

Research@

73

Epileptic Seizures from Abnormal Networks: 

Why Some Seizures Defy Predictability

William S. Anderson, Feraz Azhar, Pawel Kudela, 

Gregory K. Bergey, and Piotr J. Franaszczuk

Epilepsy Research, 99(3), 202–213 (2012)
74

Epilepsy Research (2012) 99, 202—213

journal h om epage: www.elsevier.com/locate/epilepsyres

Epileptic seizures from abnormal networks: Why

some seizures defy predictability

William S. Andersona,∗, Feraz Azhar b,1, Pawel Kudelac,2,

Gregory K. Bergeyc,3, Piotr J. Franaszczuk d,e,4

a The Johns Hopkins University School of Medicine, Department of Neurosurgery, Meyer 5-109E, 600 North Wolfe Street,

Baltimore, MD 21287, USA b The Johns Hopkins University School of Medicine, Department of Neurosurgery, Meyer 5-157, 600 North Wolfe Street,

Baltimore, MD 21287, USA c The Johns Hopkins University School of Medicine, Department of Neurology, Meyer 2-147, 600 North Wolfe Street, Baltimore,

MD 21287, USA d The Johns Hopkins University School of Medicine, Department of Neurology, USA e U.S. Army Research Laboratory, Human Research and Engineering Directorate, Aberdeen Proving Ground, 2800 Powder Mill

Road, Adelphi, MD 20783, USA

Received 15 June 2011; received in revised form 19 October 2011; accepted 18 November 2011

Available online 12 December 2011

KEYWORDS

Computational

simulation;

Neural network

model;

Seizure prediction;

Seizure generation

Summary Seizure prediction has proven to be difficult in clinically realistic environments. Is it

possible that fluctuations in cortical firing could influence the onset of seizures in an ictal zone?

To test this, we have now used neural network simulations in a computational model of cortex

having a total of 65,536 neurons with intercellular wiring patterned after histological data. A

spatially distributed Poisson driven background input representing the activity of neighboring

cortex affected 1% of the neurons. Gamma distributions were fit to the interbursting phase

intervals, a non-parametric test for randomness was applied, and a dynamical systems analysis

was performed to search for period-1 orbits in the intervals. The non-parametric analysis suggests that intervals are being drawn at random from their underlying joint distribution and the

dynamical systems analysis is consistent with a nondeterministic dynamical interpretation of

the generation of bursting phases. These results imply that in a region of cortex with abnormal

∗ Corresponding author. Tel.: +1 443 287 4561; fax: +1 443 287 6423.

E-mail addresses: wanders5@jhmi.edu, wanderso68@gmail.com (W.S. Anderson), fazhar1@jhmi.edu (F. Azhar), pkudela@jhmi.edu

(P. Kudela), gbergey@jhmi.edu (G.K. Bergey), pfranasz@gmail.com (P.J. Franaszczuk). 1 Tel.: +1 443 287 4561; fax: +1 443 287 6423. 2 Tel.: +1 443 287 8295; fax: +1 410 955 0751. 3 Tel.: +1 410 955 7338; fax: +1 410 502 2507. 4 Tel.: +1 410 278 8003; fax: +1 410 278 8828.

0920-1211/$ — see front matter © 2011 Elsevier B.V. All rights reserved.

doi:10.1016/j.eplepsyres.2011.11.006
75 Epileptic seizures from abnormal networks 203

connectivity analogous to a seizure focus, it is possible to initiate seizure activity with fluctuations of input from the surrounding cortical regions. These findings suggest one possibility for

ictal generation from abnormal focal epileptic networks. This mechanism additionally could help

explain the difficulty in predicting partial seizures in some patients.

© 2011 Elsevier B.V. All rights reserved.

Introduction

Epileptic seizures are brief, episodic phenomena. Partial

seizures, the most common seizure type, arise from focal

brain regions (e.g. temporal, parietal) (Niedermeyer, 2005).

While in some instances there may be an identifiable

cause for the seizures (e.g. tumor, cavernoma, hippocampal sclerosis), in other instances no clear pathology is

determined. The hallmark of an epileptic seizure is the

involvement of local or regional neural networks; repetitive firing of a single neuron does not produce symptoms

without this network involvement. What causes the interictal to ictal transition? A typical partial seizure lasts less than

2 min plus any postictal state (Afra et al., 2008). Therefore,

even if a patient has very frequent seizures, the majority

of time is spent in the interictal state. While some seizures

can be provoked or are more likely to occur under certain

situations (e.g. sleep deprivation, photic stimulation), the

majority of seizures appear to occur spontaneously without

known association with definable influences.

There has been considerable interest in seizure prediction in recent years. Obviously if seizures could be reliably

predicted, then the option for targeted therapy exists (e.g.

stimulation), or at least the patient could remove themselves from potentially dangerous situations. The underlying

hypothesis for seizure prediction is that there are changes

in cerebral dynamics that may precede the clinical seizure

by minutes to hours (reviewed in Sackellares, 2008). These

changes may be local (i.e. near the seizure focus) or remote.

These changes are not apparent with visual analysis of the

EEG, even with intracranial recording arrays. Some groups

have identified high frequency activity that may signal the

onset of neocortical partial seizures, but this is an example of improved seizure detection, not prediction (Worrell

et al., 2004, 2008; Bragin et al., 2010). Reliable seizure predication has been challenging and even the most enthusiastic

proponents of the prediction hypothesis acknowledge the

difficulties with current algorithms (Lehnertz et al., 2007;

Mormann et al., 2007; Andrzejak et al., 2009).

Seizure prediction may be difficult due to rapid bistable

state changes at the time of ictal onset in the neocortex

(Suffczynski et al., 2006; Lopes da Silva et al., 2003).

The mechanisms underlying a bistable state change

may be quite different between primary generalized (e.g.

absence) and partial epileptic seizures. A bistable state

change may be more applicable to these primary generalized seizures which have abrupt bilateral cerebral onset.

In this paper, a different possible mechanism is presented

under which seizure prediction would be difficult in some

patients with focal seizure onset.

Knowing, as we do, that partial seizures are a reflection of transient abnormal regional network activity, it

is reasonable to postulate that these seizures in at least

some (perhaps many) patients result from abnormal neural

networks (e.g. the epileptogenic zone) (Jacobs et al., 2000).

We describe here a model of the epileptogenic zone where

the epileptic focus is represented by an abnormal neural network that has very slightly altered connectivity so

that, while seizures only occur infrequently, they can be

triggered by normal background activity originating from

outside the epileptogenic zone. This background activity

could be influenced by various physiologic factors (e.g.

sleep), but nevertheless this background activity would not

result in seizure activity in the non-epileptic brain. This

does not discount the possibility that some changes in neural

network synchrony may occur in the ‘‘normal’’ brain since

the cumulative lifetime incidence of unprovoked seizures

approaches 4% (Hauser et al., 1993). Often these seizures

are provoked (e.g. medications and alcohol) and less than

half of these patients have recurrent seizures. The lifetime cumulative risk of developing epilepsy only ranges

from 1.4% to 3.3% (Krumholz et al., 2007; Berg and Shinnar,

1991). In this model, however, where normal background

activity, occasionally or rarely produces a seizure in abnormal regional networks, seizure prediction would be difficult

since detectable preictal changes would not be present; the

first changes would in fact be seizure initiation.

Epileptic networks in neocortex or the hippocampus

show anatomical changes compared to normal tissue (Jacobs

et al., 2000; Sallin et al., 1995). These changes can progress

with time (Sallin et al., 1995; Arellano et al., 2004). This

could result in neuronal networks more amenable to seizure

generation (electrical or clinical) over large regional areas.

There is a complex interrelationship, much of it not well

understood, between neurons which are dysfunctional and

the neural networks which can promote seizures (Leussis

and Heinrichs, 2007; Kumar et al., 2007; Swann et al.,

2007). Even in the non-epileptic brain, excitatory connections predominate with 80—90% of synapses being excitatory

(Braitenberg and Schüz, 1998).

With neuronal network simulations it is possible to control, study, and quickly change the various influences on

network behavior. Recently, we presented the results of

computational simulation studies examining the role of

external field stimulation on ongoing bursting activity in a

neural network (Anderson et al., 2007, 2009). The cortical model used in these studies consists of discrete single

compartment Hodgkin—Huxley type cells which are spatially arranged in a realistic fashion having both a layered

and columnar structure. Since neural network behavior

reflects the aggregate output of the component neurons,

single compartment neurons allow greater computational

efficiency and the ability to model larger networks in studies

of network behavior. Arrangements of connected simulated neurons in this manner can demonstrate spontaneous

bursting phases and have spatial characteristics similar to

seizures recorded from humans (Anderson et al., 2007, 2009;

Kudela et al., 1997, 2003a,b, 2005; Franaszczuk et al.,
76 204 W.S. Anderson et al.

2003). We now present the results of a similar neuronal

network model with random surrounding background inputs.

The goal of this study was to investigate the statistical structure of the resulting bursting network activity to seek the

presence or absence of predictable patterns in the behavior.

Materials and methods

Computational model format

The individual neurons in this neocortical model were represented

by single compartment neurons bearing synaptic connections from

the rest of the network, and embedded with a fixed set of ionic

conductances. The membrane potential varies as:

Cm

dV

dt = Isyn − INa − ICa − IK − IK(Ca) − IA − IL.

The individual currents include the input synaptic current Isyn,

inward sodium and calcium currents INa and ICa, outward potassium currents including a delayed rectifier current IK, a calcium

dependent potassium current IK(Ca), a transient potassium current

IA, and a leakage current IL (Av-Ron, 1994). The minicolumns used

in the simulation consist of 16 cells with intrinsic intracolumnar

wiring, adapted from the neocortical work of Douglas and Martin

(2004), as described more fully in previous studies (Anderson et al.,

2007, 2009). This is both for its ease of implementation computationally and for its experimental support in somatosensory and

visual cortex (Douglas and Martin, 2004). The geometry imposed on

a computational model becomes relevant when studying any spatially dependent effects on the resultant spreading activity. The

minicolumns in this simulation have a 25 m center-to-center spacing in a square lattice repeating structure. The total number of

cells examined was 65,536, representing a simulated cortical surface area of 1.6 mm × 1.6 mm. Fig. 1 demonstrates a schematic of

the intracolumnar excitatory cell connections and the organization

of the minicolumns in planar space as well as snapshots of the

resultant activity in the layer II/III pyramidal cell component during model bursting activity. The model connectivity and synaptic

currents are described further in the Supplementary material.

The base connection pattern studied in this report is representative of one that can produce robust bursting as previously studied

(Anderson et al., 2007). The numbers of extra-columnar connections formed by each cell class are presented in Supplementary

Table 1. There are seven cell classes modeled: four classes of excitatory cells including layer II/III pyramidal cells, layer IV stellate

cells, layer V pyramidal cells, and layer VI pyramidal cells, and

three classes of interneurons including basket cells, double bouquet

cells, and chandelier cells. Most of the model changes described

in the described studies involve alterations in connection numbers between layer II/III pyramidal cells, one of the known robust

horizontal connections systems in the cortex supporting epileptic

propagation (Telfeian and Connors, 1998). The base connection for

this system, N2/3:2/3 = 178, is defined as the number of layer II/III

pyramidal cells a given layer II/III pyramidal cell contacts in its

axonal distribution.

The model in general illustrates consistent bursting behavior,

with epochs of spontaneous bursting onset and cessation given a

random background input of Poisson based charge injection to 1%

of the cells in the model. This is an effort to treat the underlying

cortical activity as input from neighboring cortex, with the model

itself treated as the epileptic focus given its ability to produce network bursting epochs. The synaptic input used for the background

was not periodic in nature. Average rates for these Poisson distributions are described in ‘‘Activity changes with mean background

frequency’’ section and Fig. 2. The synaptic activations used for the

background inputs were the same used in the cell to cell connections, and followed the same rise and decay times appropriate for

postsynaptic potentials.

The pseudo-random number generator used for the application

of the noise pulses was a linear congruential generator implemented

with the C-function drand48, with an intrinsic period of 281 × 1012.

For a 30 s simulation and 10−5 s time-step, this function was called

1.966 × 109 times for 1% of the cells undergoing background input.

The period length for the pseudo-random generator is 143,000 times

larger than this number.

Statistics and analysis

The interbursting phase intervals in the model were fit with a

gamma distribution (Suffczynski et al., 2005, 2006). The functional

form of this distribution f(. . .) is given by

f(��) = (ˇ ̨� ( ̨))−1 � � ̨−1 exp−��

ˇ



,

where �� is the interbursting phase interval,  ̨ is the shape

parameter, ˇ is the scale parameter, and � (. . .) represents the

gamma function. Parameters were estimated using the MATLAB

function gamfit which returns maximum likelihood estimates and

95% confidence intervals for the shape and scale parameters. A nonparametric test of randomness was used to attempt to establish

whether intervals were being drawn at random from their underlying joint distribution. This was based on the circular definition of

the lag-1 serial correlation coefficient (Wald and Wolfowitz, 1943).

p-Values were computed under the assumption of asymptotic normality of the test statistic.

A method forthe detection of unstable periodic orbits (of period1) in successive interbursting phase intervals was applied (So et al.,

1996, 1997). A period-1 orbit is a fixed point of the nonlinear map

expressing the evolution of the state of a system, iterated a single

time (Guckenheimer and Holmes, 1983). Intervals were embedded

in a two dimensional state space and 104 sets of transformed intervals were obtained after randomization. Dimensional reduction was

instituted using circles of radius 9.4 ms centered along the diagonal of the state space (Le Van Quyen et al., 1997). One hundred

surrogates were produced to test the significance of peaks which

appeared along this diagonal. The surrogates were generated using

the amplitude adjusted Fourier transform algorithm (Theiler et al.,

1992). This shuffles the original sequence of interbursting phase

intervals, maintaining the original amplitude distribution of the

data while approximately matching its Fourier power spectrum.

Results

A total of 1600 s of discontinuous 20- and 30-s data segments were obtained, holding the base connectivity of the

layer II/III pyramidal cells to layer II/III pyramidal cells at

N2/3:2/3 = 178. Only the random number seed supplied to the

background input generator was varied for each of these

runs. Additionally, five continuous segments of data were

obtained with the base connectivity set at N2/3:2/3 = 172 of

lengths 320 s, 250 s and N2/3:2/3 = 178 of lengths 195 s, 140 s,

and 208 s. These data were used for the dynamical systems analysis presented below. In addition to these data,

sixteen 20-s runs were obtained with the model while varying the mean background input frequency at the base level

of connectivity. Five 20-s runs were obtained at the base

connectivity while varying the temporal pattern of the background input, and ten 20-s runs were obtained with a fixed
77

Epileptic seizures from abnormal networks 205

Figure 1 (A) Representative connectivity of the excitatory cellular component in a given modeled minicolumn, wiring after

(Douglas and Martin, 2004). (B) Three dimensional arrangement of the 16 × 16 array of minicolumns in space. (C) Representative

snapshots of evolving activity over 0.02 s in the layer II/III pyramidal cell component. Each pixel represents one cell, color coded

proportionally to the number of action potentials fired in bins of 1/100 of a second.
78 206 W.S. Anderson et al.

Figure 2 Network activity produced by sequential increases in the mean frequency of the applied background activity (background

synaptic input provided to a fixed 1% set of the modeled cells, summed layer II/III pyramidal cell action potentials in 10 ms bins).

The model exhibits a transition from episodic bursting to a very regular bursting behavior driven by the background input.

sequence of background input while varying the overall layer

II/III to layer II/III connectivity (N2/3:2/3).

Activity changes with mean background frequency

These experiments were performed with the base connectivity of the layer II/III pyramidal cell system set at the

base value of N2/3:2/3 = 178. If the mean frequency of the

applied background synaptic input is varied from 0.25/s

up to 10/s, several patterns of activity become apparent

(Fig. 2). At 0.25/s, only the low level set of activity produced with action potential production by the background

input is observed, at the cells where the input takes place.

This plot essentially demonstrates the Poisson-based random

network activity between the bursting phases. At 0.5/s, sporadic bursting activity transmitted to the network as a whole

can be observed, with long quiescent epochs. As the applied

mean background frequency is further increased, longer and

longer periods of constant bursting activity can be observed

up to 5/s. After this, a second activity transition is observed,

in which the activity changes from continuous bursting into

short periods of very large amplitude bursting (in terms of

numbers of active neurons) punctuated by brief periods of

quiescence. At 10/s it appears to dominate the activity. This

implies a saturation mechanism in this class of connected

network, which comes into play after a critical percentage

of cells are excited per time step. This saturation behavior is again an intrinsic property of the fixed network being

probed. Additionally, these studies imply that seizure onset

can be driven by neighboring cortical activity, albeit regular patterns that might not be typical of random background

input activity utilized here.

Network activity altered with input pattern

changes

Within the context of this model, it is possible to change the

random pattern of connectivity between represented cells,

and still keep the total number of connections between the

various cell classes constant. By varying the random number

seed supplied to the generator distributing the connections,

different patterns of activity can be demonstrated, even

with the same application of underlying cortical activity

applied to the same cells. Examples of the changes in activity are presented in Fig. 3A. The pattern produced ranges

from almost constant bursting throughout the 20 s examined, to brief periods of on and off bursting. Similarly, the

connectional pattern can be held constant along with the

cells in which the background activity is applied, while varying the random number seed responsible for producing the

order in which background pulses are injected into the cells.

This produces similar alterations in network activity demonstrated in Fig. 3B, and can include several time scales of

bursting epochs. These studies imply a rich dynamics of

stochastic behavior in randomly connected neural networks

receiving temporally uncorrelated background input, and

again point toward difficulties in predicting when the bursting phases might begin.

Network activity is very sensitive to numbers of

excitatory connections

Finally, changing the numbers of connections in this network model can produce substantial alterations in network

behavior. Fig. 4 presents a sequence of plots of the layer
Epileptic seizures from abnormal networks 20779

Figure 3 (A) Network activity induced by varying the random connectivity pattern between cell classes (different connectivity

seeds, Nconn

seed , for the random number generator). Numbers of action potentials in layer II/III pyramidal cell component, 10 ms time

bins. (B) Network activity induced by varying the random time sequence of background synaptic input, Ntime

seed . In these experiments,

all cellular connections remain fixed, and the identity of the cells undergoing background synaptic input remain fixed.

II/III pyramidal cell activity (time-binned action potential

numbers) for various degrees of extracolumnar connectivity

between the layer II/III pyramidal cells. The base activity explored in this manuscript is shown in the plot with

total number of connections held at N2/3:2/3 = 178 (between

extra-minicolumnar layer II/III pyramidal cells). A rapid

reduction in network bursting is shown for a connection number reduced below this, and almost continuous activity is

shown for connection numbers above this. Fig. 3A and B data

were obtained with the connectivity set at N2/3:2/3 = 178.

Only the random pattern of connectivity is varied in Fig. 3A

and the time sequence of background input in Fig. 3B. These

studies were performed in the context of a constant average level of surrounding background input, and imply the

importance of internal connectivity in the development of

uncontrolled bursting of the network.

Statistical analysis of interburst phase intervals

The statistical properties of the interburst phase intervals

for five continuous runs of the model at connectivities of

N2/3:2/3 = 178 and 172 were analyzed. This was motivated

from a dynamical systems perspective, where periodicity in

sequential intervals was sought for. Fig. 5A displays a histogram approximation to the probability density function

for interbursting phase intervals for Continuous Run 1, which

consisted of 163 intervals collected from a 320 s run of the

simulation. Fig. 5D displays the same histogram approximation for Continuous Run 4, with a total of 58 intervals

collected from a 140 s run of the simulation. Gamma distributions were used to fit these densities (Fig. 5A and D (blue

traces)) (see ‘‘Materials and methods’’ section; Suffczynski

et al., 2005, 2006). In the case of Continuous Run 1
80 208 W.S. Anderson et al.

Figure 4 Changing the absolute connectivity in the layer II/III pyramidal cell component (number of layer II/III pyramidal cells

contacted by a given layer II/III pyramidal cell, N2/3:2/3) in the model produces alterations in the network bursting behavior. At

very low absolute connectivity (N2/3:2/3 = 110) network bursting is brief and isolated, while at higher levels of absolute connectivity

periods of constant bursting can be observed.

(Fig. 5A) we found  ̨ = 6.09 (4.61—8.04) for the shape parameter of the distribution, and ˇ = 0.26 (0.20—0.35) for the

scale parameter (95% confidence intervals in parentheses).

For Continuous Run 4 (Fig. 5D), the corresponding values

were  ̨ = 7.14 (5.00—10.19), and ˇ = 0.23 (0.16—0.33) (similar results were obtained for 3 further continuous runs and

one discontinuous run — results not shown). In accord with

the interpretation of Suffczynski et al. (2005, 2006), the fact

that the shape parameter  ̨ was larger than one in all runs,

suggests the potential presence of periodicity in the generation of bursting epochs. To probe this link further, additional

statistical tests were performed as described below.

To ascertain how intervals were being drawn from their

underlying joint distribution, we applied a non-parametric

test ofrandomness to the interbursting phase intervals (Wald

and Wolfowitz, 1943). In the case of Continuous Run 1, we

found that one cannot reject the null hypothesis of randomness at the 5% significance level (p-value 0.93). For

Continuous Run 4, we found the same conclusion at the 5%

significance level (p-value 0.29). This conclusion was also

borne out for the remaining three continuous runs.

A method for the detection of unstable periodic orbits

(of period-1) was then applied to test for the presence

of deterministic dynamics in the generation of interbursting phase intervals (So et al., 1996, 1997). This method

institutes a transformation of the sequence of intervals such

that the transformed sequence is clustered around locations

of potential periodic orbits. One can compare the peaks of

these clusters to those generated by surrogate data (Theiler

et al., 1992), to compute the statistical significance of the

peaks, and thereby ascertain the potential existence of periodic orbits in the data. Fig. 5C and F shows the peaks of the

clusters were not significantly greater than those generated

by surrogate data (see caption), and so no period-1 orbits

were detected for either run (nor for the remaining three

continuous runs), at the limit of detection in the current

data set.

Discussion

Our results demonstrate that while holding the mean properties of the network stable (mean connectivity numbers,

mean background excitation rates), very rich and strikingly

different dynamics are produced by changing the model

details. Epileptogenic behavior can be created in these networks, as described above, by changes in the random pattern

of connectivity, while holding fixed the intrinsic active or

passive membrane properties in the constituent neurons.

Such changes in connectivity could be analogous to changes

in underlying connectivity that might occur following

cerebral insults, or repetitive seizures. Similar modeling
81 Epileptic seizures from abnormal networks 209

Figure 5 Statistical analysis of the interbursting phase intervals. (A and D) Interval histograms are fit with gamma distributions

revealing shape parameters  ̨ > 1 in the case of Continuous Run 1 (A) and Continuous Run 4 (D). (B and E) Two dimensional delay

embedding of sequential intervals (Spearman correlation coefficients are not significantly different from zero) for Continuous Runs

1 and 4, respectively. (C and F) Testing the significance of potential period-1 orbits detected through a dynamical analysis of the

sequence of points displayed in (B) and (E), respectively. The y-axis represents the fraction of surrogate (shuffled, see ‘‘Materials

and methods’’ section) sequences with a maximal deviation from the mean surrogate result of greater than W (So et al., 1996,

1997) (104 random matrices and 102 surrogates were used). The horizontal dotted (red) line displays the maximal deviation for the

simulation data. Since there exists a significant fraction of surrogates with deviation greater than that for the simulation data (for

both C — 20% and F — 10%), neither plot displays convincing evidence of the existence of a period-1 orbit. (For interpretation of

the references to color in this figure legend, the reader is referred to the web version of this article.)
82

210 W.S. Anderson et al.

work has been performed on changes in connectivity with

resultant epileptic effects in hippocampus (Morgan and

Soltesz, 2008; Dyhrfjeld-Johnsen et al., 2007). In our results,

local increases or decreases in connectivity in the model may

alter the network in a similar manner (a tempting comparison which must be tempered by our lack of understanding

of the functional significance of new or absent connections,

see Sallin et al., 1995; Dinocourt et al., 2003; Marco et al.,

1997; Dudek and Sutula, 2007; Maglóscky, 2010). In the

model above, the external background activity can be held

constant or changed in a variety of time-frequency manipulations. Various simulations can be created where seizures

occur rarely or very frequently.

Similarly, human epileptic seizures are episodic, transient events. Whether epileptic seizures are random events

is not clear, but times of ictal onset can behave as a random

process (Suffczynski et al., 2006). In some patients there

are no identifiable contributing factors, in other patients

such conditions such as sleep deprivation may increase the

chance of seizure occurrence and in still other patients

seizures can be provoked by specific stimuli (e.g. hyperventilation, intermittent photic stimulation; Lu et al., 2008;

Vinogradova et al., 2009; Kaplan et al., 2009). In these

examples there may be resulting alterations of background

activity within and outside the epileptogenic zone (not evident from the EEG) that make it more likely that a seizure

may arise from the existing focal epileptogenic network.

While these influences might create a state where seizures

are more likely to occur, this facilitatory state should be distinguished from the presence (or absence) of a preictal state

in unprovoked seizures.

Random behavior of network

One interesting aspect of this model is its ability to demonstrate both spontaneous periods of bursting activity as

well as self-termination of the bursting. As illustrated in

Figs. 2—5, these periods of bursting can be quite variable in

their length. Throughout the bursting and quiescent phases,

the distributed background activity is constantly active,

affecting approximately 1% of the total cells. We believe

this model may represent the type of network behavior

described by Lopes da Silva et al. (2003), in which epileptic

activity within the network cannot be predicted from the

interictal state. The non-parametric and nonlinear dynamical analyses described in the ‘‘Results’’ section support this,

however inference from this is somewhat limited given the

finite size of the data set, the small region of modeled area,

and the gamma distribution fits described earlier.

This obviously may not be true of all types of clinical and model epileptiform behavior. For example Osorio

et al. (2009, 2010) show that pharmacoresistant seizures

tend to cluster, and may have an inherent self-triggering

capacity. This might make a prediction algorithm possible

to implement in a useful fashion. Others (Suffczynski et al.,

2005, 2006), however demonstrate that seizure onset can

be described in both experimental and model data as a random walk process, with possibly a deterministic mechanism

ascribed to seizure termination. The random onset nature

would (in a bistable network with Poisson transitions) be

difficult to predict.

In the case of our model the underlying fluctuations leading to seizure onset are the random background activity we

have imposed on the network. The properties of the network

connectivity then support the bursting frequency observed

(Anderson et al., 2007). These types of grossly synchronized

bursting states in the context of a neural network have

been studied extensively by Kowalski et al. (1992). They are

truly pathological in the sense that they would block or confound any information flow through this network. It is also a

network-generated state, and can be stopped by eliminating

synaptic transmission (Kowalski et al., 1992; Keefer et al.,

2001; Rhoades and Gross, 1994).

Many limiting cases of the gamma distribution have physical interpretations that might make it easier to understand

spiking data from cortex when used for fitting (Papoulis,

1984; Suffczynski et al., 2005). When the shape parameter,

 ̨, in the gamma distribution is an integer, the distribution

is known as an Erlang distribution and represents the probability distribution of the waiting time until the  ̨th event

from a sampled Poisson process with characteristic time ˇ.

This might be comparable to the distribution of the number

of spiking or underlying synaptic events required to trigger

the network bursting behavior. One could envision trying to

extract the integer value of  ̨ from either computational

or experimental data. Similarly, the Maxwell—Boltzmann

distribution can be related to the gamma distribution

under certain restrictions on the gamma scale and shape

parameters, implying the possibility of extracting almost

thermal-like or statistical mechanical interpretations of the

network activity (Hegyi, 1996).

Limitations of the model

Our plots demonstrate gross summed numbers of timebinned action potentials in the model for given neuron

classes. The interictal activity is a random Poisson input to

1% of the cells in the model and is demonstrated in the plots,

particularly in Fig. 2, Panel 1, inset. We chose to impose

this random interictal behavior on the model to demonstrate that these fluctuations can produce very coherent

synchronous oscillations in an unpredictable fashion. However, the surrounding input might not have to be completely

random to bring about the same effect. Epilepsies involving

specific stimuli might require a coherent surrounding input

to give rise to the seizure (Lu et al., 2008; Vinogradova et al.,

2009; Kaplan et al., 2009). A more realistic technique would

be to treat the interictal background as a log-normal process

which does have some support in the literature (see Farkhooi

et al., 2009; Waters and Helmchen, 2006). Newer recording

methods from invasively monitored epilepsy patients might

help determine what patterns of background activity are

causative (Truccolo et al., 2011; van Gompel et al., 2008).

Our intent was to represent the resting interictal cellular

activity as fundamentally sparse with a random component.

This was most easily implemented as a low frequency Poisson

process.

It is possible to view the single synaptic input (driven

by the background source) as representing several weaker

but synchronized inputs. This represents a limitation to this

modeling approach, a limitation that in large degree could

be corrected with more elaborate multicompartment and
83 Epileptic seizures from abnormal networks 211

synaptic representations of the cells to make the multiple

weaker inputs more independent. Additionally in this limited data set, we are unable to say much about time epochs

larger than several hundreds of seconds (our largest continuous simulation being 320 s). This is clearly a limitation

in this technique and future computational work, including

efforts in our laboratory will explore longer time intervals

of ictal and interictal behavior, and possibly push the detection limit for predictable activity lower (or detect it more

accurately). Models such as this particular rigid crystalline

arrangement of neurons with many fixed cellular properties

in some sense have less inherent ‘‘randomness’’ than real

neocortex. This work was primarily meant to spark interest in a possible mechanism for the difficulties inherent in

seizure prediction, but by no means should it be interpreted

too literally.

Implications for seizure prediction

The purpose of the model presented here is not to judge the

effectiveness of seizure prediction, but rather to present a

plausible, alternative hypothesis for partial seizure occurrence that could explain situations where seizure prediction

may not be possible. Indeed, it is conceptually attractive to

consider that, just as partial seizures may result from various pathologies and mechanisms, that some partial seizures

may not be reliably predicted. It is beyond the scope of

this discussion to address the various methods being used

in attempts to predict epileptic seizures. It is always important to differentiate true seizure prediction from improved

seizure detection. Other commentaries and reviews address

these methods and include discussions of the challenges

and frustrations to date in routine seizure prediction even

with intracranial electrodes (Estellar et al., 2001; Litt and

Echauz, 2002; Sackellares et al., 2006; Haas et al., 2007;

Osorio et al., 2001).

This study focused on the interval to the time of the

next ‘‘seizure’’ or busting phase in the model. Our interest

was in the occurrences of the transitions from the quiescent

or background state into the pathologic state, since that

is what most seizure prediction algorithms are optimized

for. There is fairly strong evidence, certainly in the case of

complex partial seizures in temporal lobe epilepsy, that the

length of clinical seizures can be fairly uniform in a given

patient (see for instance Afra et al., 2008). The time interval durations of the seizures themselves may also prove to

have to predictive guidance as well and should be explored

in the future in modeling efforts. This may be more useful

in the case of neocortical epilepsy with its rapid spread and

possible involvement of larger regions of tissue.

Additionally, this model can incorporate incremental

changes in connectivity in the epileptogenic zone, changes

that could be a model for progressive epileptogenesis (e.g.

sprouting). This type of model also provides data that is

comparable to clinical data from epilepsy patients. The

simulated network activity is taken from a small region of

modeled cortex comparable in size to the surface area under

a typical subdural grid electrode, and makes comparisons

between modeling efforts and clinical data easy to perform

(Anderson et al., 2007, 2009; Kudela et al., 1997, 2003a,b;

Franaszczuk et al., 2005). Indeed the major advantage of

neuronal network modeling is the ability to simultaneously

monitor activity in all of the network neurons under given

experimental conditions, something not possible with biological systems, even with sophisticated recording arrays.

Conflict of interest

None of the authors has any conflict of interest to disclose.

Acknowledgements

WSA is supported by NIH-NINDS K08 (K08NS066099-01A1).

The model described in this manuscript has been

posted to the Yale SenseLab ModelDB database of

computational neuroscience models, web address:

http://senselab.med.yale.edu/modeldb/default.asp.

Appendix A. Supplementary data

Supplementary data associated with this article can be found, in the online version, at

doi:10.1016/j.eplepsyres.2011.11.006.

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Research@

87

Evolving Communicative Complexity: 

Insights From Rodents and Beyond

Kimberly A. Pollard and Daniel T. Blumstein

Philosophical Transactions of the Royal Society B, 367, 1869-1878 (2012)
88

doi: 10.1098/rstb.2011.0221

Phil. Trans. R. Soc. B 2012 367, 1869-1878

Kimberly A. Pollard and Daniel T. Blumstein

and beyond

Evolving communicative complexity: insights from rodents

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89

Review

Evolving communicative complexity:

insights from rodents and beyond

Kimberly A. Pollard1,2,* and Daniel T. Blumstein1

1

Department of Ecology and Evolutionary Biology, University of California, Los Angeles, CA 90095, USA 2

Human Research and Engineering Directorate, Perceptual Sciences Branch, US Army Research

Laboratory, Aberdeen Proving Ground, MD 21005, USA

Social living goes hand in hand with communication, but the details of this relationship are rarely

simple. Complex communication may be described by attributes as diverse as a species’ entire

repertoire, signallers’ individualistic signatures, or complex acoustic phenomena within single

calls. Similarly, attributes of social complexity are diverse and may include group size, social role

diversity, or networks of interactions and relationships. How these different attributes of social

and communicative complexity co-evolve is an active question in behavioural ecology. Sciurid

rodents (ground squirrels, prairie dogs and marmots) provide an excellent model system for studying these questions. Sciurid studies have found that demographic role complexity predicts alarm call

repertoire size, while social group size predicts alarm call individuality. Along with other taxa, sciurids reveal an important insight: different attributes of sociality are linked to different attributes of

communication. By breaking social and communicative complexity down to different attributes,

focused studies can better untangle the underlying evolutionary relationships and move us closer

to a comprehensive theory of how sociality and communication evolve.

Keywords: sociality; individuality; repertoire size; alarm call; information theory; vocal complexity

1. WHAT IS COMPLEXITY?

While most of us have an intuitive idea of what constitutes complexity, complexity is difficult to define [1].

A system is typically considered more complex if it contains more parts, more variability or types of parts, more

connections or types of connections between parts or

more layers of embedded meaning. The information

[2] required to describe a system can be thought of as

a measure of the system’s complexity (‘Kolmogorov

complexity’, after [3]). To illustrate, Dawkins [4,

p. 265] provides a simple thought experiment: when

comparing two items or systems, imagine writing a

book to describe each one. The longer book will

describe the more complex system. Information-based

definitions of complexity are most useful because they

allow complexity to be quantified and compared with

a single metric [5].

Social complexity can be defined in numerous ways

(e.g. the number of individuals [6–8], the number of

demographic or social roles [9,10], the strength

of social bonds [11–13], the complexity of group substructure and relationships [14–18] and combinations

of these [19,20]). Each of these definitions can be considered an attribute of sociality. What these definitions

have in common is they quantify social complexity via

the number or variability of the social system’s parts,

connections or layers of categorization (table 1).

One integral attribute of social complexity is perhaps the most straightforward: social group size. The

number of individual animals present in a social group

influences that group’s resource needs [21], disease

ecology [22] and predation risk [23]. Furthermore,

social group size defines boundaries for other attributes

of social complexity, such as mating systems or social

networks [24]. Group size also directly affects the communicative landscape, since more individuals may be

communicated with, followed, discriminated or ignored

[5,25–27]. Social group size is often straightforward to

quantify. If a social group’s boundaries are well defined,

then a simple head count is all that is needed. For all

these reasons, it is not surprising that social group size

is one of the most commonly studied attributes of

social complexity and has been used in a variety of

taxa [7,28–38].

Another critical attribute of social complexity is the

number or variability of social roles in a social group.

A role can be thought of as a socially expected behavioural pattern. For instance, social roles may be based

on whether an individual is a dominant or subordinate,

a producer or scrounger, or a breeder or non-breeder.

Blumstein & Armitage [39] quantified social roles by

focusing on demographic (age/sex) roles in social

groups, which can be quantified using information

theory. The key assumption was that groups with

more overlapping generations, and those with more

age-sex classes present, were more socially complex

in their demographic roles.

Complexity in communication can be defined analogously to complexity in sociality, i.e. via the number or

* Author for correspondence (kpollard@ucla.edu).

One contribution of 13 to a Theme Issue ‘The social network and

communicative complexity in animals’.

Phil. Trans. R. Soc. B (2012) 367, 1869–1878

doi:10.1098/rstb.2011.0221

1869 This journal is q 2012 The Royal Society
90

Table 1. A selection of attributes of social systems and communication systems that can vary in complexity. The categories suggested here are neither mutually exclusive nor exhaustive,and no causal relationships are implied.

number of system parts

types of systemparts connections between parts

layers of structure, categorization ormeaning openness and variability

social complexity group size

social group sizecolony, aggregation,

or community size foraging

group size

roles

socialdemographicreproductive

social bonds

may be quantified with

social networkstatistics and may beagonistic or affiliative

matriline structuressocial hierarchiesterritorial neighbour

networks

group substructure

matrilinescoalitions

group superstructure

fission – fusiongroup aggregationspost-dispersal ties

fluidity or variability of

social attributes

group membershipsocial hierarchiesreproductive roles

communicative

complexity

repertoire size (within a signal

type)entire repertoirerepertoire within an

age class, sex orsituational context

repertoire size

(across signaltypes)

syntax

connections between

signals

referential and functionally

referential communication

communication of signal

urgency or signaller’saffective state

embedded signature

information

signaller’s age, sex, condition or

individual identity

learning and modification of

signals

openness of signal repertoires

and syntax

1870 K. A. Pollard and D. T. Blumstein Review. Evolving communicative complexity

Phil. Trans. R. Soc. B (2012)
91

variability of the signal’s or system’s parts, connections,

or layers of meaning (table 1). For acoustic communication, attributes of communicative complexity can

include a species’ repertoire size (number of call types

[32,39]) or the syntactical complexity or uncertainty

[40,41] of sequences of calls. Signals may also be complex

in their association with external referents, such as functionally referential alarm calls that communicate

predator type or response strategy, in addition to alarm

[42–45]. Communication of urgency or affective state is

another form of complexity, providing additional information about the situation [46–48]. For example, many

species of marmots alter the rate of their alarm calls

with the urgency of the situation or the degree of risk

[49]. The presence of embedded signature information

(such as age, sex or individual caller identity [50–54])

also adds complexity to communicative signals.

As social complexity, however quantified, increases,

more complex communication might be needed to

manage this social complexity [55–59]. Attributes of

social complexity may thus generate a need for animals

to exhibit different types of communicative complexity.

2. SCIURIDS AS A MODEL SYSTEM

Sciurid rodents (Rodentia: Sciuridae), particularly

ground-dwelling social species (tribe Marmotini,

including ground squirrels Spermophilus spp. and related genera, prairie dogs Cynomys spp., and marmots

Marmota spp.), present an excellent model comparative

system for studying the relationship between social and

communicative complexity, for several reasons.

First, ground-dwelling sciurids constitute a speciose

clade that exhibits a wide range of social structures.

On one end of the spectrum, some species are nearly

solitary or live in simple family groups (e.g.

woodchucks, Marmota monax, live in groups of a

mother plus her young of the year). Within the same

clade, more complicated group structures are seen,

such as those in Eurasian marmots (e.g. M. caudata,

M. marmota [60]) in which young delay dispersal for

one, two or more years, adult males participate in

group life, and females may aggregate in harems or

matrilines. Social group size also varies considerably

across ground-dwelling sciurids, as does group cohesion. In addition, sciurids also vary in their tiers or

levels of social complexity [61,62], with some species

exhibiting complex relationships between segregated

social groups (e.g. some prairie dogs). Importantly,

variation in social group size is not strictly tied to variation in other attributes of social structure [9],

allowing these different attributes to be teased apart

in comparative study.

Secondly, ground-dwelling sciurids offer a communication system amenable to comparative study.

Ground-dwelling sciurids produce vocal alarm calls

in response to predatory stimuli [63]. These calls are

typically loud and perceptually salient, and they can

be elicited and recorded by researchers. Within the

sciurid alarm call system, an array of communicative

complexity is exhibited. Species vary in the acoustic

structure of their calls [39,54,64], the size of their

alarm call repertoires [39,49] (figure 1), the response

urgency or distance to predator encoded in their calls

[65–67], syntactic or ordering differences in calls

[64], and the degree to which age and sex and individual identity are encoded [54,68]. As with social

complexity, we can view these as attributes of communicative complexity and they can be independently

studied in comparative analyses.

Thirdly, published phylogenies [69–71] aid phylogenetically controlled evolutionary analyses in this taxon.

golden

M. caudata

black-capped

M. camtschatica

alpine

M. marmota

hoary

M. caligata

Olympic

M. olympus

yellow-bellied

M. flaviventris

woodchuck

M. monax 4

kHz 2

1 s

no. alarm calls

equivocal 1 2 4

Figure 1. Alarm call repertoires of seven species of marmots (Marmota spp.). Sciurid rodents differ in the acoustic structure of

their alarm calls and in the size of their alarm call repertoires.

Review. Evolving communicative complexity K. A. Pollard and D. T. Blumstein 1871

Phil. Trans. R. Soc. B (2012)
92

To date, two phylogenetically controlled studies have

examined the correlated evolution of social complexity and communicative complexity in ground-dwelling

sciurid rodents [39,54]. Importantly, comparative studies

in sciurids permit us to decouple drivers of complexity,

and, as discussed below, have revealed that social complexity, broadly defined, does not simply select for more

complex communication. Rather, different attributes

of sociality seemingly select for specific attributes of

communicative complexity.

3. DEMOGRAPHIC COMPLEXITY DRIVES

REPERTOIRE SIZE EVOLUTION

Blumstein & Armitage [39] used 22 species of grounddwelling sciurids to test for a relationship between

one attribute of social complexity (demographic role

complexity) and one attribute of communicative complexity (alarm call repertoire size). They defined a

social complexity index via information theory, using

data on dispersal patterns, the age/sex composition of

groups and variability in these traits [39]. The resulting

numbers, expressed in bits, quantify the amount of

information needed to describe the demographic role

complexity in different species.

For species with multiple alarm call types, different

call types may be used to communicate about different

types of predators [42,43,45], to communicate different

degrees of risk or response urgency [46,48], or to communicate both predator type and urgency [44,72].

Species living in more complex social groups, such as

groups with more social demographic roles, may have

greater need to signal alarm in a more complex

manner and thus may use larger alarm call repertoires.

Across ground-dwelling sciurids, alarm call repertoire

size varies from one to five (see examples in figure 1).

Blumstein & Armitage [39] used a combination of

field recordings and literature review to establish the

alarm call repertoire size for 22 species of sciurid and

regressed this against the social complexity index.

Demographic role complexity significantly explained

variation in alarm call repertoire size, both in analyses

of raw data and independent contrasts (figure 2).

More up-to-date phylogenies have since been published, so we re-ran the independent contrasts analysis

using more recent phylogeny taken from Herron et al.

[70] and Steppan et al. [71]. The relationship between

social demographic complexity and alarm call repertoire

size remained significant (r

2 1⁄4 0.228, y 1⁄4 1.049x, p 1⁄4

0.025, n 1⁄4 21 independent contrasts). An increase in

social demographic roles appears to drive the ability to

communicate via a larger, more complex repertoire of

alarm calls. The functional explanation underlying this

correlation is unknown. Future study of alarm call

type use in different contexts or between different

caller– receiver dyads may be necessary to untangle

how more demographically complex species may benefit

from a more diverse alarm call repertoire.

4. SOCIAL GROUP SIZE DRIVES THE EVOLUTION

OF INDIVIDUAL SIGNATURES

Pollard & Blumstein [54] compared sciurid vocal signature individuality (individual distinctiveness) against

(a) 4

(b)

3

2

1

0

1.5

repertoire size

(independent contrasts)

repertoire size

1.0

0.5

0

–0.5

–1.0

0.5

social demographic complexity

social demographic complexity

(independent contrasts)

1.0 1.5

0 0.1 0.2 0.3 0.4 0.5 0.6

Figure 2. Relationship between social demographic complexity and alarm call repertoire size across 22 species of sciurid

rodents, in raw (a) and independent contrasts (b) data.

Social complexity (variability in demographic roles) correlates

with alarm call repertoire size. (a) Adapted from Blumstein &

Armitage [39], with permission from Chicago Journals.

8

MAFL

CYLE

CYLU

SPBE

SPRI

MAOL

SPTR

SPBL

(a)

(b)individuality (

Hs

) 7

6

5

4

1.5

1.0

0.5

0

5

0.5 1.0 1.5 2.0 2.5

6

social group size

social group size (independent contrasts)

Hs (independent contrasts)

7 8 9 10 11

Figure 3. Relationship between social group size and alarm call

individuality across eight species of sciurid rodents. In raw (a)

and independent contrasts (b) data, social group size correlates

with vocal individuality (Hs) in sciurid alarm calls (r

2 . 0.88,

p , 0.001 in both cases). Species in (a) are Cynomys leucurus

(CYLE), C. ludovicianus (CYLU), Marmota flaviventris

(MAFL), M. olympus (MAOL), Spermophilus beecheyi

(SPBE), S. beldingi (SPBL), S. richardsonii (SPRI) and

S. tridecemlineatus (SPTR). Figure adapted from Pollard &

Blumstein [54], with permission from Elsevier.

1872 K. A. Pollard and D. T. Blumstein Review. Evolving communicative complexity

Phil. Trans. R. Soc. B (2012)
93

social group size, Blumstein & Armitage’s [39] demographic complexity index, and two other indices of

social structure complexity [61,62]. Group size was

not correlated with these other complexity indices

[54]. Because studies quantifying vocal individuality

are not common, this analysis required novel data, and

eight species of ground-dwelling sciurid were thus captured, individually marked and recorded on multiple

occasions. Individuality was calculated from acoustic

traits using an information-theory metric developed

by Beecher [5,73] and used in previous studies

[25,27,52,74]. The individuality information statistic

quantifies, in bits, the amount of individually specific

information content present in a species’ vocalizations.

This statistic was used as the quantitative metric of

one attribute of communicative complexity.

Individual signatures in alarm calls are important

for animals such as sciurids [75–77]. Individuals

vary in their alarm signal reliability, and listeners

benefit when they recognize the individual identity of

an alarm caller, as this allows them to better calibrate

their behavioural response [76–80]. Ground-dwelling

sciurids live in groups of closely related kin, such

that the signaller and receiver are often close relatives.

This may allow callers to benefit via kin selection by

aiding related receivers in individual discrimination

and reliability assessment [54,77], although they may

also benefit via reciprocal altruism [81].

Individuality is expected to evolve with group size

[5,25–27,52,54,73]. As social group size increases,

the number of individuals that must be discriminated

increases accordingly, making individual recognition

tasks more difficult. Increased individuality would be

necessary to permit successful discrimination of all

the individuals in the group. Since other attributes of

sociality affect communicative complexity [39,82],

these other attributes may also influence individuality.

Pollard & Blumstein [54] gathered typical social group

size data from the literature and regressed this against the

individuality metric. Group size explained considerable

variation in vocal individuality, both in the raw data

and in an analysis using independent contrasts

(figure 3). As predicted, social group size appears to

drive the evolution of individual signature information.

Interestingly, it is group size, not other social

complexity attributes, that seems responsible for the

evolution of individually specific vocalizations. Signature information was also regressed against three other

measures of social structure complexity, including

Blumstein & Armitage’s [39] demographic complexity

index, Michener’s [62] social grade, and Armitage’s

[61] sociality index. ‘Social grade’ and ‘sociality index’

are somewhat subjective numerical assignments

intended to quantify the complexity of each species’

social structure and mating system. Individuality was

not significantly related to any of these three other

social complexity attributes, even after controlling for

variation in group size.

5. ATTRIBUTES OF SOCIALITY, ATTRIBUTES

OF COMMUNICATION

Results from sciurid studies highlight an important

caveat about the evolution of social and communicative

complexity. Both sociality and communication have

multiple attributes (table 1), and these attributes affect

one another in complex ways (figure 4). The main

insight from our integrative overview is that different

attributes of social complexity are likely to drive different attributes of communicative complexity. Thus, it is

group size (no.

individuals)

need for

individual

recognition

individuality and

individual

signatures

dominance

hierarchies,

coalitions or

territoriality

social bond

strength

demographic

role variability

reproductive

role variability

mating system

mating

 competition

openness of

repertoire or

syntax

timing,

ordering or

syntax

repertoire

size

functional

reference

urgency or

affect

different

response

strategy to

environmental

stimuli

signaller

reliability

environmental

complexity

communicative

complexity

social

complexity

more to

‘talk about’

Figure 4. Possible functional relationships between attributes of social complexity and attributes of communicative complexity.

For specific examples of relationships that are supported in the literature, see table 2.

Review. Evolving communicative complexity K. A. Pollard and D. T. Blumstein 1873

Phil. Trans. R. Soc. B (2012)
94

Table 2. Comparative relationships among attributes of social and communicative complexity. Studies indicate that different attributes of social complexity affect different attributes ofcommunicative complexity.attribute of socialcomplexity

attribute ofcommunicativecomplexity taxon

no. speciesin study signal type

phylogeneticcontrol citation details

social group size potential for repertoire

size

anthropoid primates 12 facial expressions

(visual)

generalized

least-squares

[35] social group size predicts facio-muscular

ability to produce varied facial expressions

social group size repertoire size primates 42 vocalizations

(auditory)

independent

contrasts

[32] social group size predicts repertoire size for

adult vocalizations

social group size signal variability whales 34 vocalizations

(auditory)

independent

contrasts

[83] social group size predicts the amount of

tonal modulation in cetacean whistles

social group size potential to perceive

diverse repertoire

strepsirrhine primates 11 acoustic signals

(auditory)

none [84,85] foraging group size predicts auditory

sensitivity and high frequency hearing

social group size individuality sciurid rodents 8 vocal alarm calls

(auditory)

independent

contrasts

[54] social group size predicts individual

distinctiveness in vocal alarm calls

colony size individuality Hirundo swallows 2 vocal contact calls

(auditory)

comparison

within genus

[25] nesting colony size predicts individual

distinctiveness in chick begging calls

colony size individuality Larus gulls 2 vocal contact calls

(auditory)

comparison

within genus

[26] mobile cre`che size predicts individual

distinctiveness in adult ‘long calls’

colony size individuality microchiropteran bats 8 vocal contact calls

(auditory)

none [27] roosting colony size predicts individual

distinctiveness in infant isolation calls

diversity/variability of

demographic roles

repertoire size sciurid rodents 22 vocal alarm calls

(auditory)

independent

contrasts

[39] diversity/flexibility in demographic roles

within social groups predicts repertoiresize for vocal alarm calls

diversity/variability of

reproductive roles

individuality Polistes wasps 25 facial coloration

patterns (visual)

concentrated

changes

[82] flexibility in nest-founding strategies predicts

variability (individual distinctiveness) invisual facial markings

social structure

complexity

repertoire size Eulemur lemurs 8 glandular chemical

secretions(olfactory)

none [86] social structure predicts diversity of

odour chemicals in female perianalsecretions

social structure

complexity

repertoire size Papio and

Theropithecusmonkeys

2 acoustic signals

(auditory)

none [87] social structure predicts vocal repertoire size

mating system

complexity

repertoire size phocid seals 12 underwater

vocalizations(auditory)

none [88] mating system predicts repertoire size for

male underwater vocalizations

mating system

complexity

repertoire size lizards 41 bodily ornamentation

(visual)

model fitting [89] mating system predicts the number of bodily

ornaments and colour dichromatism

social bonding repertoire size primates 20 vocalizations

(auditory)

independent

contrasts

[32] time spent in social grooming predicts

repertoire size for adult vocalizations

social bonding potential for repertoire

size

catarrhine primates 10 facial expressions

(visual)

generalized

least-squares

[90] time spent in social grooming predicts facial

nucleus volume in the brain

1874 K. A. Pollard and D. T. Blumstein Review. Evolving communicative complexity

Phil. Trans. R. Soc. B (2012)
95

important to think clearly about the nature of the

relationship because different attributes work different

ways (see also [1]). Studies from other taxa also support

taking a nuanced look at the correlated evolution

of social and communicative complexity (table 2). For

example, the complexity of reproductive roles was

found to influence individuality in wasp facial markings

[82], while nesting or roosting colony size was found to

influence individuality in avian and chiropteran contact

calls [25–27]. In phocid seals, underwater repertoire

size increased with mating system complexity [88]. In

primates, social group size and time spent in social

grooming influenced vocalization repertoire size [32],

while group size influenced ability to produce varied

facial expressions [35]. Furthermore, social complexity

may influence not just the complexity of communicative signals themselves but also the perceptual capacity

to receive such signals, e.g. social group size predicts

auditory sensitivity in lemurs [84]. Social and communicative complexity can also covary within species or

on short time scales; for example, chickadees placed in

larger groups displayed more variability in their call

types and combinations [40]. In all these cases, a different attribute of social complexity is associated with a

different attribute of communicative complexity.

As a general rule, we would predict that the types of

communicative complexity that are most relevant to a

specific aspect of sociality would be evolving with

those aspects of sociality. For example, if individual

recognition is important, and if all members of a

group must be recognized, we would predict a specific

type of communicative complexity (individualistic signatures) to evolve with a particular attribute of social

complexity (group size). This has been found in birds,

bats and sciurids [25–27,54]. If calls are used to attract

or impress mates, we would predict specific attributes of

communicative complexity (such as syntactical complexity or repertoire size) to evolve with a specific

attribute of social complexity (mating system). Indeed,

male repertoire size seems to have evolved with mating

system in phocid seals [88]. However, these relationships are not always obvious, and attributes of sociality

may influence communication in diverse and complicated ways (figure 4). Future studies in more taxa will

be necessary to comprehensively identify the many

ways in which distinct attributes of social and communicative complexity are evolutionarily or functionally

linked. With the discovery that social network statistics

can be used by behavioural biologists [18,91,92],

there are many precisely defined attributes of sociality

that could be studied. From a comparative perspective,

available data are often a limiting factor, but over time,

there will be more data from different species available

with which to study the evolution of social and communicative attributes [93]. As these databases are

developed, sciurid rodents can continue to play a

valuable role in this pursuit, because these species vary

in several attributes of social and communicative

complexity, and systematic study can help identify

which of these attributes covary. These attributes can

be further examined with respect to different environmental conditions in sciurids’ diverse habitats, and/or

mapped onto available phylogenies to estimate when

and how complexity evolved.

6. CONCLUSIONS

The relationship between social complexity and communicative complexity is multi-faceted. Different

attributes of sociality may drive the evolution of different attributes of communication. For example, in

sciurid rodents, social demographic complexity

explains the evolution of alarm call repertoire size,

while social group size explains the evolution of

alarm call vocal individuality. Sciurids are one of

many excellent model systems in which to explore

the evolution of social and communicative complexity,

and future work in sciurid and non-sciurid taxa will

help build a comparative database in which these questions can be thoroughly examined. The ever-growing

comparative database will ultimately help us develop

a comprehensive understanding of how social

complexity and communicative complexity evolve.

We thank Todd Freeberg, Indrikis Krams and Cecilia

Kullberg for organizing the ABS/IEC 2011 symposium on

social complexity and communicative complexity and two

anonymous reviewers for comments on a previous version

of this manuscript. We thank our field collaborators and

colleagues—Ken Armitage, Abigail Benson, Dean Biggins,

Suzanne Cox-Griffin, David Eads, Chris Evans, Greg

Grether, Jim Hare, John Hoogland, Marta Manser, Jill

Mateo, Peter Nonacs, Don Owings, Dan Salkeld and Joan

Silk—for access to animals and insightful comments over

the years. For funding, we thank the American

Philosophical Society, American Society of Mammalogists,

Animal Behavior Society, Explorers Club, Mildred

Mathias/UC Reserves, National Geographic Society,

National Institute of Health, National Science Foundation,

Rocky Mountain Biological Laboratory, Sigma Xi, UCLA,

UC Davis and the University of Kansas. We also thank the

many wonderful lab and field assistants who have helped

both of us study the evolution of complex sociality and

communication.

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Research@

99

Combining the Finite Element Method with 

Structural Connectome-based Analysis 

for Modeling Neurotrauma: Connectome 

Neurotrauma Mechanics

Reuben H. Kraft, Phillip Justin Mckee, Amy M. Dagro, and Scott T. Grafton

PLoS Computational Biology, 8(8), e1002619 (2012)
100

Combining the Finite Element Method with Structural

Connectome-based Analysis for Modeling Neurotrauma:

Connectome Neurotrauma Mechanics

Reuben H. Kraft1*, Phillip Justin Mckee2, Amy M. Dagro1, Scott T. Grafton3

1 Soldier Protection Sciences Branch, Protection Division, U.S. Army Research Laboratory, Aberdeen Proving Ground, Maryland, United States of America, 2 Dynamic

Science, Inc., Aberdeen Proving Ground, Maryland, United States of America, 3 Department of Psychology, University of California, Santa Barbara, Santa Barbara, California,

United States of America

Abstract

This article presents the integration of brain injury biomechanics and graph theoretical analysis of neuronal connections, or

connectomics, to form a neurocomputational model that captures spatiotemporal characteristics of trauma. We relate

localized mechanical brain damage predicted from biofidelic finite element simulations of the human head subjected to

impact with degradation in the structural connectome for a single individual. The finite element model incorporates various

length scales into the full head simulations by including anisotropic constitutive laws informed by diffusion tensor imaging.

Coupling between the finite element analysis and network-based tools is established through experimentally-based cellular

injury thresholds for white matter regions. Once edges are degraded, graph theoretical measures are computed on the

‘‘damaged’’ network. For a frontal impact, the simulations predict that the temporal and occipital regions undergo the most

axonal strain and strain rate at short times (less than 24 hrs), which leads to cellular death initiation, which results in damage

that shows dependence on angle of impact and underlying microstructure of brain tissue. The monotonic cellular death

relationships predict a spatiotemporal change of structural damage. Interestingly, at 96 hrs post-impact, computations

predict no network nodes were completely disconnected from the network, despite significant damage to network edges.

At early times (tv24 hrs) network measures of global and local efficiency were degraded little; however, as time increased

to 96 hrs the network properties were significantly reduced. In the future, this computational framework could help inform

functional networks from physics-based structural brain biomechanics to obtain not only a biomechanics-based

understanding of injury, but also neurophysiological insight.

Citation: Kraft RH, Mckee PJ, Dagro AM, Grafton ST (2012) Combining the Finite Element Method with Structural Connectome-based Analysis for Modeling

Neurotrauma: Connectome Neurotrauma Mechanics. PLoS Comput Biol 8(8): e1002619. doi:10.1371/journal.pcbi.1002619

Editor: Olaf Sporns, Indiana University, United States of America

Received March 22, 2012; Accepted June 6, 2012; Published August 16, 2012

This is an open-access article, free of all copyright, and may be freely reproduced, distributed, transmitted, modified, built upon, or otherwise used by anyone for

any lawful purpose. The work is made available under the Creative Commons CC0 public domain dedication.

Funding: This work was funded by the U.S. Department of Defense, Department of the Army. The funders had no role in study design, data collection and

analysis, decision to publish, or preparation of the manuscript.

Competing Interests: The authors have declared that no competing interests exist.

* E-mail: reuben.kraft@gmail.com

Introduction

The finite element method is often used to study neurotrauma

[1–9] and continues to emerge as a useful tool in the field of

neuroscience [10–13]. Models continue to advance in biofidelity

by incorporating an increased level of anatomic detail [4,13,14],

improved representation of the material behavior at various

loading rates [15–20], and advanced measures and predictions of

injury [8,21,22]. Finite element models are commonly used to

understand the biomechanics of brain and skull deformation when

the head is subjected to insult, leading to improved insight into

mechanisms of acute injury. For example, modeling axonal injury

mechanisms within white matter of the brain has been the focus of

some recent efforts and provides a means to relate an insult to a

cellular injury mechanism [8]. By using diffusion tensor imaging

(DTI) fiber tractography, the structural orientation of neuronal

axonal bundles can be incorporated into the finite element model

and can be used to compute the axonal strain during brain white

matter deformation [8,23]. Then, by using an axonal injury

threshold, the occurrence of diffuse axonal injury (DAI) is

predicted [8]. Wright and Ramesh [8] show that the degree of

injury predicted is highly dependent on the incorporation of the

axonal orientation information and the inclusion of material

anisotropy into the constitutive model for white matter. By

modeling the underlying mechanism of DAI, an enhanced

understanding of the neurotrauma is attained through a spatiotemporal description of tissue deformation. In general, finite

element simulations may help to elucidate the injury mechanisms

of neurotrauma.

As finite element models advance, experimentally based models

of neurotrauma also continue to become more sophisticated,

ranging from the macroscopic [24,25] to the cellular level [26–29].

Various biomechanical and physiological injury thresholds for

neurotrauma have been proposed in the past, including intracranial pressure [21] and strain [8,30,31]. While these thresholds offer

an immediate prediction of injury, they lack a long-term

description of functional degradation. A time-evolving injury

model is attractive since the biological response occurs on a slower

time scale than an injurious stimulus as a consequence of

mechanotransduction cascades [27]. There have been significant

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Note: Reuben Kraft’s current address is The Johns Hopkins University Applied Physics Laboratory, 11100 Johns Hopkins Road, Laurel, Maryland 20723
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efforts to develop empirically based time-evolving cellular injury

thresholds, which many times use in vitro cellular and tissue

culture models. Morrison III et al. [27] suggest that for brain

biomechanics, neuronal culture models that accurately mimic

specific brain features can be used to explore tissue properties and

tolerances or thresholds to mechanical loading. Cell death has

been primarily used as a definition of injury within the neuronal

culture model community and has been applied to determine

tissue-level tolerance criteria using local values of axonal strain,

strain rate, and time from ‘‘insult’’ [32,33]. Furthermore, as

Morrison III et al. [27] point out, empirical functions for cellular

death based on culture models could be incorporated into finite

element analyses, thereby enabling biological predictions to

supplement mechanical predictions of local tissue stress and strain.

In the study presented here, this concept is further explored and

used as a bridge to a network-based analysis of the brain.

As physics-based models become more capable of predicting

tissue-level injury mechanisms from improved computational and

experimental resources for biomechanics, there remains a need to

understand how structural damage in a given location of the brain

evolves, and how it may influence functional or cognitive

performance over time. Such a goal is complex and difficult. For

example, as Kaiser et al. point out [34], in some instances the

brain can be robust to physical damage, and in other instances

physical damage can cause severe functional deficits. Nevertheless,

Kaiser et al. [34] pursue the important and unanswered question:

Are the severity and nature of the effects of localized damage

predictable? To help explore the answer to this question, tools are

being developed for the quantitative analysis of brain network

organization, based largely on graph theory [35,36]. Typically,

network nodes represent brain regions, often obtained from high

resolution magnetic resonance imaging (MRI). The network links

or edges between brain regions represent interregional pathways

that convey neuronal signals and are commonly obtained from

non-invasive DTI or diffusion spectrum imaging (DSI) [37,38].

The connection matrix of the network of the human brain forms

the so-called ‘‘human connectome’’ [35,36,39,40]. Recently, Jirsa

et al. [41] used connectomics to establish a framework for a

‘‘virtual brain’’, in which network modeling is used to understand

the intact and damaged brain. Similar to previous work,

hypothetical or random deletion of nodes or edges were used to

degrade the structural connectome [42]. More recently, structural

and diffusion images from 14 healthy subjects were used to create

spatially unbiased white matter connectivity importance maps that

quantify the amount of disruption to the overall brain network that

would be incurred if that region were compromised [43]. In this

study, we attempt to extend the capabilities of neurocomputational

models by providing a physics-based approach for predicting

degraded regions of interest. Physics-based injury predictions may

help inform structural connectome analysis.

In this study, neurotrauma is investigated by using finite element

simulations of a single individual subjected to a simulated head

impact. Tissue damage is computed using empirically based

damage models that provide a link from macroscopic biomechanical deformation to mesoscopic damage. Axonal bundle tracts are

explicitly modeled using a multiscale description of white matter

tracts obtained from diffusion tensor imaging. Then, using the

physics-based injury predictions for white matter tissue from finite

element simulations, the structural brain connectivity or connectome is degraded, and various network measures are computed. This is an important contribution because finite element

simulation predictions of tissue damage provide physics-based

reasoning for removing nodes or degrading edges to create the

‘‘damaged’’ brain connection matrix. In turn, this approach may

provide further insight into mild traumatic brain injury by

shedding light on the relationship between mechanical stimulus

to the brain and neurobiological processes that result. Furthermore, if successful, the computational framework presented herein

could supplement ongoing efforts to evaluate the use of noninvasive medical imaging tools, such as diffusion tensor and

spectrum imaging, to detect white matter disruption for neurotrauma diagnostics [44,45] by providing a time-evolving history of

tissue injury.

Methods

A suite of medical imaging and software tools are used to obtain

an individual-specific finite element model and structural connectome-based analysis. The overarching process is schematically

shown in Figure 1 and is outlined below.

Individual Specific Models

T1 and diffusion tensor magnetic resonance images are taken

from a single individual (the corresponding data can be found in

[37]). The T1 image (Figure 1a) is segmented into different head

materials (Figure 1b) using the software Amira [46] and the

Connectome Mapper Toolkit [47]. The segmented geometry is

then used to create a biofidelic three-dimensional finite element

volume mesh (Figure 1c). In order to create a corresponding

structural connectome or network, the T1 image is parcellated into

83 regions of interest (ROI) representing the location of

anatomical regions of the brain based on the Desikan-Killiany

atlas extended to include subcortical regions [48]. Since diffusion

tensor images (Figure 1e) are used to generate axonal bundle fiber

tractography based on the direction of peak water diffusion in each

voxel [49], the DTI fiber tractography (Figure 1f) represents the

approximate location of neuronal axonal bundles [50]. Fibers are

filtered for connectome creation to include only fibers that begin

and end within ROI. The structural connectome is assembled

using the Connectome Mapper Toolkit [47] and is composed of

nodes representing ROI generated at the centroid, connected by

edges that represent structural pathways for which the DTI

tractography traverses. Following the segmentation enabled

by the Desikan-Killiany brain atlas, there are 83 network nodes

and 1029 network edges. Furthermore, DTI fiber tractography is

Author Summary

According to the Centers for Disease Control and

Prevention in the United States, approximately 1.7 million

people, on average, sustain a traumatic brain injury

annually. During the last few decades, brain neurotrauma

biomechanics has been an active area of research

involving medical clinicians and a broad range of scientists

and engineers. In addition, advances and fast growth of

human connectomics continues to reveal new insights into

the damaged brain. With recent advances in computational methods and high performance computing, we see

the need and the exciting possibility to merge brain

neurotrauma biomechanics and human connectomics

science to form a new area of investigation - connectome

neurotrauma mechanics. For neurotrauma, the idea is

simple - inform human structural connectome analysis

using physics-based predictions of biomechanical brain

injury. If successful, this technique may be further used to

inform human functional connectome analysis, thus

providing a new tool to help understand the pathophysiology of mild traumatic brain injury.

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Figure 1. A flowchart of the process for creating the finite element model and connectome from T1 and diffusion MRI. (a) The T1

image is (b) segmented into different head materials. The segmented geometry is then used to create (c) a biofidelic three-dimensional finite element

volume mesh. The mesh is required for (d) an explicit dynamic finite element simulation that captures the biomechanical response from frontal

impact. (e) Diffusion tensor images are used to generate (f) axonal bundle fiber tractography which is used to inform finite element transversely

isotropic constitutive descriptions of white matter tissue behavior (c). Tractography is also used to create (g) a network model of the brain that can be

(h) degraded over time.

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incorporated into the finite element model by using a transverse

isotropic material model specifically developed for representing

white matter tissue [23,51] where each finite element within white

matter regions is assigned an orientation based on the superposition of fiber tractography [52]. Details of the numerical

implementation are published elsewhere [52], but it is important

to point out that various different cases needed to be considered

while assigning one orientation per finite element. For example, in

the case of multiple fibers that overlap the spatial bounds of a

single element, the multiple fiber orientations were averaged.

Damage and Injury Thresholds for Coupling FEM Results

to the Connectome Analysis

Herein, finite element simulations of the human head are

designed to mimic experimental conditions for cadaveric impact

tests, which are conducted to understand the dynamics of a frontal

impact and the associated compression-tension damage [53]. In

this study, the explicit dynamic finite element method [54] is used

to capture the transient response of head impact. Supplementary

information about the finite element method is included in the

supporting information, Text S1. In addition to a volume mesh for

the head, the finite element model also requires material

constitutive descriptions and properties for all its components:

skull, cortex, brain stem, cerebrospinal fluid, and soft tissue, which

represents a homogenized mixture of muscle and skin. A detailed

list of the material constitutive laws used to relate tissue strain to

stress is included in the supporting information, Text S2. Once

again, we point out that diffusion tensor imaging is used to inform

the constitutive model for white matter tissue, which has been

applied in the past for studying non-human primates [31] and

human injury [8,22,55]. The entire finite element model consisted

of 1,394,945 tetrahedral elements and 237,115 finite element

nodes. For boundary conditions, the bottom of the neck is fixed,

and a force was applied to a circular area on the forehead (about

3 mm2) in the anteroposterior direction. The input force-time

curve was a sinusoidal shape with a peak force of 7,000 N at

2:75 ms. The finite element simulation computes the time-evolving

mechanical strain and stress in the direction of axonal fiber

bundles. The strain in the direction of axonal bundles is referred to

as the axonal strain.

One limitation of the current finite element model is the

exclusion of viscoelasticity in the constitutive description of brain

matter. The authors acknowledge that to accurately model the

progression of damage, the constitutive model should be extended

to account for the time-dependent behavior of brain tissue. The

exclusion may have an effect on the outcome of our results, leading

to larger shear stresses, but smaller shear strains, thus, less

predicted damage. For example, Chafi et al. [56] shows that

viscoelasticity plays a major role in the dynamic response of the

brain under blast loading. Future efforts are focused on improving

the mechanical description of brain tissue.

In order to model damage using a physics-based approach,

either an explicit failure mechanism should be modeled or an

empirically based failure threshold is required. For this study,

measures of axonal strain and strain rate computed for white

matter regions are used as input for empirically based injury

threshold predictions that are obtained from cellular culture

experiments. Specifically, experimental results for cellular death

are described using a mathematical function for tolerance criteria

that relates strain to resultant cell death evaluated for up to four

days post-injury [27,32,33]. Experimental data exists for the CA1

and CA3 regions of the hippocampus, dentate gyrus, and cortex

for the rat. Experiments suggest that some regions of the rat brain

are sensitive to loading rate, while other brain regions are not.

That is, Elkin et al. [33] found that cortical cell death was

dependent on applied strain rate, whereas hippocampal cell death

was not. The relations used in the present model, obtained from

Morrison III et al. [27], are:

DCA1=CA3~0:0389 e 0:3663   t 2:015   ð1Þ

DDG~0:0323 e 0:3721   t 1:8209   ð2Þ

DCortex~0:094 e 1:5293   t 0:8337   e_ 0:1175   ð3Þ

where t is time from insult, e is the local strain, and e_ is the local

strain rate [27]. Similar to Morrison III et al. [27], the units of

time are days, strain is dimensionless and strain rate is inverse

seconds. The damage parameter, D, is defined as the percent area

of cell death. Using Equations 1–3, the axonal strain and strain

rate from the finite element simulations, as well as time, are used to

calculate percent cell death for a tissue region. Due to insufficient

resolution of the MRI data used in this study, segmentation of the

hippocampus into CA1, CA2 and dentate gyrus regions was not

possible, thus the more conservative equation for DDG was used to

compute cell death for the entire hippocampus. Equations 1–3 are

monotonic functions that only increase with time, thus cellular

repair mechanisms and regeneration are not currently captured.

Potential issues with monotonic cellular death predictions, as well

as using rat brain injury thresholds instead of human cellular

injury thresholds, will be discussed later. Since the explicit

dynamic finite element method is used, the transient wave

propagation for solid mechanics is resolved; however, this is

computationally costly and limits the total time of biomechanical

prediction. The explicit dynamic simulation runs to 15 ms, thus

the ‘‘long-term’’ structural mechanics of brain swelling, relaxation,

etc., are not captured in the current model, although could be

adapted in the future by using quasi-static, implicit finite element

solvers.

The reader should understand that we use the local tissue strain

and strain rates predicted from finite element simulations of short

duration, about 15 ms, as input to experimentally based cellular

death models that were developed over a 96 hour period. This

assumes that the tissue damage due to large deformations occurs

immediately and initiates an injury process that grows with time.

This assumption is based on the observation that cell death was

not immediate in response to deformation, but instead, increased

over 4 days after injury [27].

It should be noted that the cellular death estimates that

Morrison III et al. [27] developed were not based on axonal strain

but instead on nominal strain applied to the back of the substrate

on which the neuronal cells were attached. Thus, Equations 1–3

may not accurately describe the actual relationship between

axonal strain and injury. Furthermore, the neuronal cell bodies in

the experiments were not aligned in a specific orientation, so the

response is not strictly representative of axonal strain injury. There

are research efforts examining stretching of individual axons but

have not proposed empirical relationships for cellular death in

terms of applied strain, rate of loading and time from insult

[57,58]. Experimental measurements of the cellular response of

white matter, especially axonal bundles, would be interesting to

explore and could provide a more accurate representation of cell

damage in the future.

Furthermore, the experiments performed by Morrison III et al.

[27] used the rat hippocampus, which is mostly comprised of gray

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matter. Thus, it should be noted that the empirical data described

in Equations 1–3 was not intended to describe white matter injury

response so there may be limitations in applying Equations 1–3 to

predict cellular white matter injury. Cellular injury threshold data

for isolated white matter is currently limited; however there have

been some efforts to characterize mechanical damage to axons

[57–59]. We hope a computational framework described here will

help to motivate efforts to obtain different white matter and gray

matter empirical functions, which could then be used for each of

region of the brain separately. As mentioned before, the

application of this approach raises the possible need for

understanding properties of white matter fiber bundles.

In order to map the finite element results to network-based

analysis tools, output data from each finite element that represents

white matter is mapped to a corresponding voxel in the MRI data

that is used to create the DTI tractography. This mapping is

referred to as the element-to-voxel map. Multiple finite elements

within a single voxel are averaged. The element-to-voxel map

enables voxels to be assigned additional data, including axonal

strain and strain rate from the finite element simulation. Alstott et

al. [42] chose to generate brain lesions by altering the structural

connectivity matrix of the brain by deleting nodes using various

methods. In the present work, instead of deleting nodes, we

degrade the edges of the network based on the computed cellular

death at the voxel level.

To understand how the structural network is degraded, consider

the schematic shown in Figure 2. Edges in the network are

constructed from voxels that connect two different ROIs. The

edge strength is relative to the number of tracts between two ROI.

Cellular death for each voxel is computed using Equations 1–3

and may grow to reach the chosen critical value of cellular death,

Dc (discussed shortly). Voxels with a predicted cellular death

greater than Dc are shown in red in Figure 2. For this study, if a

tract traverses a voxel that is greater than the threshold, the entire

tract is considered damaged, thus the edge strength is decreased.

This procedure is similar to that used by Kuceyeski et al. [43] who

simulated lesions at each voxel and removed tracts passing through

the lesion in order to create a damaged network to enable analysis

of changes in network measures and the evaluation of the

importance of each voxel. Similar to Honey et al. [60], the

connection strengths were resampled to a Gaussian distribution

with a mean of 0:5 and a standard deviation of 0:1. As Alstott et al.

point out [42], this transformation does not alter the rank-ordering

of strong to weak pathways, but simply compresses the scale of

connection strengths. There are other possible ways to degrade the

structural network that will be discussed later.

Since Equations 1–3 predict some degree of cell death for nonzero values of axonal strain, strain rate, and time, the additional

critical value of cell death, Dc, was required as a rule in order to

degrade a voxel. The critical cellular death of Dc~3% was chosen

Figure 2. Schematic showing how structural network edges are degraded over time. Red voxels indicate that the chosen critical cellular

death threshold, Dc, is reached. For this study, DTI tracts that traverse a damaged voxel are removed, thus degrading the connection strength of the

network edge.

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because it predicted similar levels of damage as compared to other

proposed thresholds, including 18% axonal strain, which was used

previously as a threshold that indicated degradation in electrophysiological function [30]. During the calculation of cellular

death, positive values of strain and strain rate were used (negative

strain was not used to calculate cellular death). The choice of

injury threshold, as well as the micromechanics of axonal fiber

bundles, is an active area of research that would assist making the

current methodology more accurate in the future. During the

15 ms dynamic simulation, strain and strain rate data for each

voxel is output at 0:1 ms increments. Cell death is calculated at

each voxel for each of these increments. Then, the maximum cell

death value calculated in each voxel is used to predict cell death up

to 96 hrs. It is important to note that cell death calculations are the

result of the combination of variables at each increment rather

than considering each variable independently.

Results

Finite Element Analysis of Head Impact

The deformed configurations of the head, along with contours

and response curves for various locations within the brain (frontal,

parietal, occipital, temporal, corpus callosum and cerebellum) are

shown in Figures 3a–f. Output variables including pressure, axonal

strain, and effective strain rate, useful for understanding the

anisotropic biomechanical response for white matter, are shown.

Confidence in the finite element model is established by

comparing output from the computations to pre-existing experimental data on cadaveric head impact [53]. The values of axonal

strain and strain rate were taken from specific locations in each of

the six regions plotted in Figure 3. Four out of six of the locations

represented the locations of the pressure transducers for the

experiments of Nahum et al. [53]. Locations within the cerebellum

and corpus callosum were also added for the strain and strain rate

analysis. A variation of results will exist in each region, and taking

an average of values in a anatomical region could help resolve this

problem, but including simulation data points that are located

farther from the experimental sampling points could also make the

results less accurate. The head impact simulation took about

30 hours on 32 processors in order to reach 15 ms.

As seen in Figure 3b, the intracranial pressure quickly increases

to positive values in the frontal and parietal regions, while quickly

increasing to negative pressures in the occipital and posterior fossa

regions. The anteroposterior pressure gradient (seen in Figure 3a)

is commonly observed in experimental and computational studies -

giving rise to the so-called coup and contrecoup loading scenario,

for which there are an associated number of proposed injury

mechanisms. Short duration intracranial pressure gradients with

high positive pressures are observed at the coup region, with

negative pressures at the contrecoup region. The maximum

positive pressure of approximately 160 kPa is reached in about

2:5 ms in the frontal lobe, closest to the impact, while the

maximum negative pressure of approximately {80 kPa is reached

in about 2:0 ms in the posterior fossa region. The computed

pressure response is directly compared to Nahum et al.’s [53]

experimental results in Figure 3b and show similar trends. In

addition, validation of the strain response against cadaveric

experiments of Hardy et al. [61] is also described in the

Supplemental Text S3.

Axonal strain at the various brain regions responds slower than

the pressure response. The axonal strain begins to substantially

grow at 1 ms and shows a gradual rate of change of strain, with a

maximum of 33% at 12 ms over the duration of the simulation

within the temporal region. Unlike the pressure, an obvious

transcranial gradient is not apparent for the axonal strain. While

the frontal region had the highest predicted pressure, the temporal

and occipital brain locations have the largest values of axonal

strain in the regions specifically examined. Later, in the structural

network analysis, these areas are associated with the largest

amount of cellular death. Interestingly, if a threshold of injury of

18% axonal strain is chosen [8,30], our results show the onset of

injury occurs at approximately 9.1 ms within the temporal lobe,

despite the intracranial pressure reaching approximately 100 kPa

in 2 ms. The time scales at which the pressure and strain grow will

be discussed later in the context of injury cascades.

The effective strain rate, also commonly referred to as rate of

loading or loading rate, has a maximum value of approximately

85 s{1 in the temporal lobe. From the contours shown in Figure 3e

there appears to be a strain rate focusing, with lower strain rates

closer to the skull and higher values more central to the brain.

Shear strain focusing has been reported earlier [3,62] and is

attributed to the partial conversion of energy of the axial impact

into a shear mechanical stress wave as a result of the material

response of the brain, cerebrospinal fluid, and skull [62]. The

maximum strain rate is observed before maximum axonal strain

because the axonal strain accounts for magnitudes of deformation,

while the strain rate relates to the rate of change of strain.

FEM Informed Structural Connectome Analysis

The axonal strain and strain rate output from the finite element

simulations are used to compute the amount of cellular death,

according to Equations 1–3. Figure 4 shows the evolution of

damaged tractography up to 96 hrs post-impact, using a critical

cellular death of Dc~3% as a threshold for injury and the

corresponding evolution of the degraded structural network for

sagittal and transverse views. The edges in the network, which are

fully damaged, are shown in red for visualization. In reality, each

edge is weighted and has degraded values before it is fully

damaged. The network nodes are scaled by the percentage of

connections that were removed, so that larger nodes have lost

more connections compared to original values (1{1⁄2Degreenew

=Degreeoriginal�). The sagittal and transverse views are shown to

provide insight about where the damage is predicted and how it

progresses through time. At 24 hrs, the top four regions affected

include the cuneus (medial surface of left cerebral hemisphere in

the occipital lobe), fusiform gyrus (temporal lobe), lingual gyrus

(occipital lobe), and peri-calcarine. Together, damage in these

brain regions have 559 tractography fibers removed from network

edges. Note that the total number of fiber tracts prior to impact

was 497,442, so this damage corresponds to a 0.011% degradation

of tractography. The percentage of fully degraded edges for 24, 48,

72, and 96 hrs are listed in Table 1. Using an 18% strain criterion,

17.3% of the edges were fully degraded.

Figure 4b shows that structural damage occurs first in left

cuneus and the right superiotemporal regions, resulting in one fully

damaged edge at 24 hrs. The edge between the two regions had

only three tract fibers with less than one voxel or 2 mm between

them. The edge also had a high mean fiber length of 131:7 mm

(the original total mean fiber length was 81:7 mm). This suggests

that network edges associated with few tract fibers and long fiber

length are more susceptible to damage. Asymmetry of the

predicted damage occurs due to the asymmetry of the underlying

anatomy associated with the finite element mesh, as well as

potential asymmetry of the fiber tractography, which has been

studied in recent work [63]. At 48 hrs, predicted damage seems to

show a high density of damaged fibers in the anteroposterior

direction in both hemispheres of the brain. Lateral tractography

damage is also predicted within the corpus collusum. As time

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progresses, fully damaged rostrocaudal tracts are predicted and

continue to increase.

Structural changes to the tractography and resulting network

arise because of the underlying voxel condition. That is, if a tract

goes through a voxel that has reached the critical cellular death

value, Dc, the tract is removed. Therefore, it is useful to examine

how the inherent voxel properties influence results. Figure 5a

shows the distribution of voxels above the predicted critical cellular

Figure 3. Orthographic view of the local three-dimensional response measured at various locations in the brain for (a–b) pressure,

(c–d) axonal strain and (e–f) strain rate predicted using a finite element simulation for impact to the head. Prediction of the intracranial

pressure response is compared to cadaveric experiments [53] and is shown in (b).

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death of 3% as a function of angle between the axonal fiber bundle

direction and the direction of the head impact (i.e., frontal impact

on the anteroposterior axis). If the tract fiber direction within a

voxel is parallel with the impact direction, the angle is zero. This

shows the distribution of damage as a function of angle with respect

to the impact direction. Because of the high degree of mechanical

rotation observed through the shear focusing in the temporal and

occipital brain regions, tract fibers with large angles with respect to

the impact direction are damaged initially. This is seen in Figure 5a

at 24 hrs (see red bars in plot) and in Figure 4a–b.

This data can be further analyzed by normalizing the number of

damaged voxels for a given orientation by the total number of

undamaged voxels with a given orientation from the impact

direction. This measure is referred to as the angle-normalized

number of damaged voxels. The distribution of angle-normalized

number of voxels is shown in Figure 5b, and is important because

it takes into account the original distribution of the number of tract

fibers eligible to be damaged with respect to the impact direction.

For example, Figure 5a shows 18 voxels above the 3% cell death

threshold at 48 hrs; thus, one might assume that axonal bundles

oriented 0{100 from the impact direction have little importance.

Figure 4. Evolution of damaged tracts and the corresponding structural networks. Using empirically-based cellular death predictions

obtained from in vitro models of neural tissues, local strain and strain rate values computed from finite element simulations are used to specify injury.

A computed cellular death of 3% was used as a critical value for defining white matter disruption. Damage is shown in red and the node size

represents the percent change of degree. The predicted evolution of damage is shown for the sagittal and corresponding transverse views for 24 (a

and b), 48 (c and d), 72 (e and f), and 96 hours (g and h).

doi:10.1371/journal.pcbi.1002619.g004

Table 1. The percentage of fully degraded edges and

percentage of voxels above the 3% threshold for 24, 48, 72,

and 96 hrs post-injury.

Post-Injury (hrs) % Fully Damaged % Voxels Above

Edges 3% Threshold

24 0.097 0.008

48 7.19 1.1

72 14.1 2.7

96 19.7 4.2

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However, when the original number of voxels available to be

damaged is taken into consideration (Figure 5b), 11{200 angles

have the highest percentage of damage at 48 hrs. In other words,

depending on the underlying white matter microstructure, i.e., the

axonal bundle orientation, with respect to the direction of impact,

the node, and edge degradation in the structural network may be

affected differently. This is directly related to the structural

mechanics (as opposed to structural connectomics) of the

underlying constitutive or material law used within the finite

element simulation.

However, because a monotonic function is used to describe the

empirical cellular death prediction, this trend becomes more dilute

as time progresses (but should not be extrapolated past 96 hrs

since Equations 1–3 are not validated beyond that time). From

Figure 4g–h, at 96 hours the predicted damaged axonal pathways

can be seen in many directions and across all white matter brain

regions indicating diffuse structural degradation. To summarize, at

24 hrs fibers in the areas of large rotational tissue strain are

susceptible, while at greater times, neuronal tracts that align with

the direction of impact seem more susceptible to damage (using

the assumed threshold of cellular death). In the future, additional

empirical cellular death predictions that are non-monotonic (if

valid), or cellular regeneration and repair models, would be useful

to explore.

Figure 6a–e shows the evolution of the structural connectivity

strength matrices predicted as a result of head impact simulations.

The original structural connectivity strength is defined as a

Gaussian distribution, created by using a mean of 0:5 and

standard deviation of 0:14 distributed over the total number of

edges in the network. Each figure represents a snapshot in time

starting with t~0 and ending at 96 hours. Within the 96 hr period

(for which the monotonic cellular death criteria are validated), as

long as regions have non-zero strain and strain rate computed

from the finite element simulation, edges in the network become

degraded. This decline in network strength is evident in Figure 6e.

Since a 3% critical cellular death value is only one possible injury

criteria that could be chosen, an additional criterion was

examined. Results using two different damage thresholds are

shown in Figure 6e and f. Figure 6e is the connection matrix using

the 3% threshold at 96 hrs, while Figure 6f is connection matrix

using the 18% axonal strain threshold at t~15 ms, as used in

previous work [8,31]. Connection strengths show qualitatively

similar trends in magnitude and location of edge degradation.

While the two criteria offer a similar prediction of network

damage, the cellular death Equations 1–3 are, perhaps, more

useful because effects of strain, strain rate, and time have been

decomposed into separate multiplicative terms that allow a

compartmentalized study of extrinsic biomechanical conditions.

In general, a network’s global efficiency represents how wellconnected the network is compared to a perfectly connected

network [64] and captures the network’s capacity for communication along short paths [65]. Herein, values are reported as

normalized global efficiency, which is the global efficiency of the

network divided by the efficiency of an ideal network. Ideal

network efficiency is calculated as a network where all nodes are

connected at the minimum cost. The cost for each edge is defined

as unity minus Gaussian strength. While strength offers a measure

of the capacity to send information, the cost indicates the

resistance to sending information - connections with high strength

are low in cost. Local efficiency is a network measure that Latora

[65] suggests helps to reveal the fault tolerance of the network

system and shows how efficient communication between first node

neighbors is. For this study, local efficiency is calculated for the

same set of nodes used to make the local network of the

undamaged network, even if a node loses its edge to the primary

node. This method is used to provide a more direct comparison of

Figure 5. Distribution of (a) the number of voxels with predicted cellular death above 3% for each day as a function of angle

between axonal fiber bundle direction and direction of impact to the head, and (b) the number of voxels with predicted cellular

death above 3% for each day as a function of angle between axonal fiber bundle direction and direction of impact to the head

normalized by the initial distribution of fiber angle with respect to the loading condition.

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the network after damage by accounting for edges that were

removed. As a result, edges can no longer be part of a short path

between nodes and cannot contribute to the efficiency, while also

taking into account all nodes that should be a part of the local

network without damage.

The normalized global and mean local efficiencies as a function

of time are shown in Figure 7. The normalized global efficiency is

approximately 0.14 at t~0 hrs and is about the same at 24 hrs,

indicating the network remains capable to send information.

However, at 48 hrs an 8.8% reduction in normalized global

efficiency is predicted and continues to reduce with time. A similar

trend is observed in the mean local efficiency. At 96 hrs the

normalized global efficiency was reduced 24%, while the mean

local efficiency was reduced 27%. By 96 hrs, all but sixteen nodes

had greater than 20% reduction of local efficiency. Also, note that

there were no nodes completely disconnected from the network,

although there were edges completely removed. The total number

of edges at t~0 hrs was 1029; at t~96 hrs, 203 edges were

removed using the cellular death threshold. Using an injury

threshold of 18% strain, 161 edges were removed.

Watts and Strogatz [66] define the small-world network based on

the clustering coefficient of the network and the characteristic path

length of the network. The clustering coefficient, C, is the fraction of

triangles around a node and is weighted by the geometric mean of

weights associated with edges of a triangle [64]. It measures how

well the first neighbors of a node are connected to each other. The

characteristic path length, L, is the average shortest path between all

nodes of the network. To qualify as small-world, the networks

clustering coefficient should be greater than that of an equivalent

random network, whereas the characteristic path length should be

approximately equal [66]. The small-world coefficient is defined as

(C=Crand)=(L=Lrand) [64]. This value was found to be 1.87, 1.87,

2.03, 2.31, and 2.64 for t = 0, 24, 48, 72, and 96 hrs, respectively.

When using axonal strain as the threshold variable, a value of 2.38

was computed. The increase seen in the small-world coefficient is

due to the more rapid decrease in clustering coefficient of the

random network compared to the damaged network that it is based

on. In addition, the increase seen in the small-world coefficient also

shows the ability of the network to maintain its modular structure

more effectively than the random network that it was compared to.

Figure 6. Connection strength matrices showing degradation over time, t. Connection strengths were resampled to a Gaussian distribution

with a mean of 0.5 and a standard deviation of 0.1. Because of the monotonic cellular death criterion, as long as regions have non-zero axonal strain

and strain rate from the finite element simulation, edges in the network eventually become degraded. The connection strength matrices at t~0 hrs is

shown in (a). The evolution of connection strength matrices for 0 (a), 24 (b), 48 (c), 72 (d), and 96 hours (e) are shown, as well as (f) the connection

strength matrix for the case when a 18% strain threshold is used.

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The percent reduction of local efficiencies and the associated

betweenness for the top 10 regions affected by impact at 96 hrs are

listed in Table 2. Betweenness is defined as the number of times a

node is part of a shortest path between nodes, and is one indication

of a node’s network centrality that help to describe the

‘‘importance’’ of a node [64,65]. Hubs within the connectome

are identified using betweenness centrality. The maximum

betweenness of the original network at t~0 is 226, while the

minimum value is zero. Recall from Figures 4a–b and 5a, white

matter disruption is observed in the left cuneus and the right

superiotemporal regions, which has a betweenness centrality of 41

and 17, respectively. The moderate to low betweenness of the

regions first affected at 24 hrs helps to explain why the normalized

global efficiency was not significantly reduced during this time and

shows the importance of a robust brain network. At 96 hrs, the

right hemisphere lateral orbitofrontal region shows the largest

degradation in local efficiency of approximately 45.4%; however,

that region also has a zero betweenness centrality. On the other

hand, the right hemisphere medial orbitofrontal region has a

betweenness of 110 and shows a 37.4% reduction of local

efficiency, indicating it may have a more significant effect to the

brain network. At 96 hrs, there are 10 network nodes with

betweenness greater than 100 that have at least a 20% reduction of

local efficiency. Local efficiency and betweenness for all ROIs are

included in Supplemental Figures S2 and S3, respectively.

The results reported thus far are based on a critical cell death

value of Dc~3%, as well as a critical strain threshold of 18%.

However, it is also useful to evaluate the sensitivity of the results to

the choice of Dc. An analysis of network properties was performed

for multiple critical cell death values, in the range from

Dc~2{10%, and are shown in Figure 8a–b. Figure 8a shows

the percent reduction in total edge strength as a function of the

critical cell death threshold choice. Figure 8b shows the percent

reduction in global efficiency as a function of the critical cell death

threshold choice. Results show that the choice of the critical

threshold significantly affects both network measures that were

examined. For example, for a change of Dc between 3% or 4%,

the reduction in global efficiency is 23.9% and 16.4%, respectively. In other words, from only a 1% change in the choice of Dc

results in a 7.5% change in global efficiency reduction. Therefore,

Figure 7. Global and local efficiency in terms of time from impact.

doi:10.1371/journal.pcbi.1002619.g007

Table 2. Structural measures of simulated lesions for the top

ten regions ranked according to percent reduction in local

efficiency.

Brain Region % Reduction Betweenness

Local Eff. at 96 hrs at 0 hrs

Lateral Orbitofrontal (RH) 45.4 0

Parahippocampal (RH) 44.2 0

Parsorbitalis (RH) 44.1 0

Transverse Temporal (RH) 42.0 0

Pericalcarine (RH) 39.4 1

Parstriangularis (RH) 39.3 2

Temporal Pole (RH) 39.1 3

Rostral Anterior Cingulate (RH) 38.4 0

Frontal Pole (RH) 38.1 12

Superior Temporal (RH) 37.9 21

RH and LH refer to right and left hemispheres, respectively.

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the choice of Dc will be important in obtaining accurate results

and highlights the need for future experimentation to characterize

cellular injury criteria for all areas of the brain in order to improve

the accuracy of this modeling approach.

Discussion

For the first time, a physics-based model has been linked to a

network-based analysis that establishes a coupled computational

method to study the effects of localized structural damage or

lesions. In the presented work, lesions are simulated by using a

state-of-the-art finite element model of the human head, developed

for a single individual directly from MRI, impacted on the

forehead region. The local finite element variables are coupled to a

network-based analysis through an empirically based cellular

injury model. The new approach attempts to capture the

spatiotemporal structural characteristics of brain trauma. Foundations of this idea stem from previous studies that attempted to

establish relationships of lesion sites and the resulting functional

impact [42,67].

The first part of our study involved developing a new threedimensional human head finite element model for simulating the

biomechanical response from frontal impact and comparing it to

experimental data obtained from cadavers. The finite element

model is unique in that it uses diffusion tensor imaging

tractography to inform structural mechanics constitutive laws of

the underlying white matter anisotropy in an effort to help

elucidate the injury mechanisms of impact neurotrauma. Simulations of frontal impact capture the coup and contrecoup loading

profiles, including short duration intracranial pressure gradients

with high positive pressures at the coup region, and negative

pressures at the contrecoup region (Figure 3a–b). The computed

pressure compares well to past experimental results [53]. While the

frontal region had the highest predicted pressure, the temporal

and occipital brain locations had the largest values of axonal strain

in the regions specifically examined and responded slower than the

pressure response. Because of varied mechanical tissue properties

and constraints with the head, translational cranial motion causes

relative brain movements that happens after peak pressure, and

leads to large brain deformations and significant localized regions

of axonal strain (Figure 3e–f).

Empirically based cellular death thresholds were used to predict

the time-evolving damage in various brain regions based on finite

element-based predictions of local axonal strain and strain rate.

The biomechanical simulations predict that the temporal and

occipital regions undergo the most axonal strain and strain rate at

short times (less than 24 hrs), which leads to cellular death

initiation that produces damage which shows dependence on angle

of impact and underlying microstructure of brain tissue. The

cellular death model that was used in this study is based on

experimental observations that cell death was not immediate in

response to deformation, but instead increased over four days after

insult. Tissue damage becomes more dilute as time progresses

(Figure 4g–h). At 96 hours, the predicted damaged axonal

pathways can be seen in many directions and across all white

matter brain regions, indicating diffuse structural degradation.

Interestingly, when using injury criteria proposed in the past,

including thresholds of axonal strain [8,30] or intracranial pressure

[21], the finite element simulations predict injury within 10 ms of

frontal impact. However, when using a cellular death criteria,

damage takes longer to develop but offers a similar result to the

resulting network strength (Figure 6). Thus, while the model may

be limited in the most accurate description of cellular injury, it

does seem to capture the underlying pathology of diffuse brain

injury, i.e., widespread damage to axons in the brain, to some

degree. While the two criteria offer similar prediction of network

damage, cellular death, is perhaps, more useful because effects of

strain, strain rate, and time have been decomposed into separate

multiplicative terms that allow a compartmentalized study of

extrinsic biomechanical conditions. On the other hand, as

Morrison et al. [27] point out, purely mechanical definitions of

injury thresholds may be too insensitive to identify the onset of

injury for the brain, since biological tissues are alive and perform

some form of active, physiological functions. For brain tissue,

Figure 8. Changes in (a) total edge strength and (b) global efficiency with change in critical cell death threshold for 24, 48, 72, and

96 hours.

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failure can be defined in various ways and may occur far below

mechanical failure limits. Therefore, additional tolerance criterion

that capture degradation of the electrophysiological function may

be also required for brain tissue, since injury mechanisms that may

alter neuronal function without requiring cell death are observed

and may become activated more quickly than cell death [68].

This work has attempted to establish a physics-based methodology to inform structural connectome analysis. In the current

model, network edges are degraded by weight rather than simply

deleting nodes, in an attempt to include the effects of damage on

white matter ‘‘fibers of passage’’ that Alstott et al. [42] refer to. It is

assumed that tracks representing bundles of axons are able to be

disconnected when passing through regions of high cell death,

simulating reduction in the strength of a structural connection, and

are related to a decrease in localized white matter integrity. All

network damage was linked directly to predicted tissue deformation and predicted cellular death. Interestingly, at 96 hrs postimpact, the methods used here did not lead to any node

completely disconnected from the network (although there were

edges completely disconnected). At early times (tv24 hrs) network

measures of global and local efficiency were degraded little,

however, as time increased to 96 hrs the network properties were

significantly reduced (Figures 6 and 7). Alstott et al. [42] found that

random removal of nodes did not affect network integrity until

almost all of the nodes were deleted. Thus, this method may

capture some structural network features of very mild neurotrauma at early times, but would benefit from a functional network

analysis to explore the potential outcomes. As the network was

damaged in certain areas, edges were lost or degraded, raising the

cost to send information between nodes and producing different

short paths between nodes. As a result, the brain network was not

able to maintain efficiency by using alternate paths or finding

strong hub connections past 24 hrs post-impact, thus demonstrating a potential limitation of the brain to retain network robustness

in extreme conditions that cause neurotrauma.

While global efficiency is able to demonstrate widespread effects

of damage, local efficiency provided a measure to investigate

localized damage within the network related to areas of

concentrated axonal strain and strain rate in particular areas of

the brain. There was a much larger reduction in local efficiency at

areas of high cell damage compared to reduction in the

normalized global efficiency, indicating that the brain as a whole

is resistant to some degree of localized damage. Brain regions that

experienced the largest cellular death showed a larger reduction in

local efficiency compared to the global efficiency of the network

(Table 2). However, the low betweenness of many of these ROI

suggest that they were not as necessary for communication outside

their local area. This suggests that the modular nature of the

network, including its small world properties, helped to prevent

loss of efficiency on the global scale from damage at the local scale.

This is interesting because the network considered here is a rigid,

static, anatomic network in which there is no adaptability built into

the model. However, it seems that damaged structural network

hubs retain the ability for long distance communications. In

addition, the increase predicted in the small-world coefficient also

shows that modularity was not as affected by the damaged static

network, as compared to a random network.

Limitations and Future Work

There are exciting possibilities for future work, as well as

limitations to the current modeling approach. The current model

did not attempt to model the coupled effects to the functional

network; instead, we provided an example for a single individual in

order to establish a methodology to link physics-based predictions

of tissue damage with structural network analysis for frontal

impact neurotrauma. It is important to note that the empirical

relationships may only be accurate for the rat (not the human), and

most likely, there are many more regions that need to be

characterized. Our results and conclusions may be altered

according to these injury thresholds. Although human injury

thresholds are currently limited, as additional brain region injury

thresholds are experimentally characterized and improved they

can be included in the future. Due to the computational cost of the

finite element simulations, the brain was only segmented into 83

different regions. In the future, increased segmentation of regions

of interest and improved biofidelity of the finite element model

would increase the resolution of the analysis. While this study does

not address the resulting functional outcome from structural

degradation, coupled structure-function relationships as a result of

neurotrauma would be interesting to explore. For example, a

coupled analysis may enable functional stimulus that may prohibit

or enhance further cell death. For the prediction of tissue damage,

additional physics, such as electrochemical reactions, may be

useful by incorporating diffusional properties. In addition,

increased resolution of the biomechanical response may also be

improved by further developing white matter material response

descriptions that use multiple fiber tract orientations within a

single element, thereby enabling the capability to use diffusion

spectrum imaging. There is also an opportunity to use this

framework to explore additional injury mechanisms or thresholds

from empirical or experimental data, such as intracranial pressure.

Note that the current methodology degrades network edges,

instead of nodes. In the future, it may be useful to investigate

methods for degrading nodes in addition to edges because the all

of the nodes represent cortical gray matter, and the gray matter

does experience significant strains. This would require a choice of

strain measurement other than axonal strain since the gray matter

is treated as isotropic. With this type of criteria, it may be possible

to degrade a node based on a ratio of damaged voxels within a

region of interest compared to the voxel volume of the region as a

whole.

Further work should also work to validate this approach in

humans. There are at least two distinct areas associated with

connectome damage that should be explored in order to validate

the approach described herein. The first deals with how well the

location of damage within the network description is captured

using physics-based predictions. The second validation strategy

should address the nature in which edges and nodes in the network

are degraded. There are various approaches that may be useful for

addressing both areas. For example, in order to validate how well

the location of damage is captured, further understanding about

how cellular level changes effect fractional anisotropy, that results

in altered fiber tractography would be useful to develop. Some of

this information may be obtained from various ongoing studies

that are examining the ability of DTI to diagnose mTBI, which

could also be extended to create degraded structural connectomes.

One way to do this may be to use DTI studies pre- and post-injury

from typical loading conditions that cause rotation-induced diffuse

axonal injury. By providing similar loading profiles within the

simulation and comparing computed DTI tractography damage

with the clinical data set, maybe results can be compared. Perhaps

this may be accomplished using sports-related impact injury, such

as American football, where many helmets have sensors built-in to

record impact loads.

In conclusion, this work has explored ‘‘connectome neurotrauma mechanics’’ by using physics-based finite element simulations to help elucidate injury mechanisms associated with

neurotrauma by using various cellular injury thresholds to define

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tissue damage, and established a coupled computational framework to inform structural connectome analysis.

Supporting Information

Figure S1 The magnitude of peak relative displacement between

skull and brain for validation of finite element model.

(EPS)

Figure S2 Percent reduction in local efficiency at 96 hrs for all

brain regions.

(TIF)

Figure S3 Percent reduction in betweenness at 96 hrs for all

brain regions.

(TIF)

Text S1 A concise description of the finite element method.

(PDF)

Text S2 Material constitutive laws and associated parameters

used for the head finite element model.

(PDF)

Text S3 Additional finite element validation.

(PDF)

Author Contributions

Conceived and designed the experiments: RHK PJM AMD STG.

Performed the experiments: RHK PJM AMD STG. Analyzed the data:

RHK PJM AMD STG. Contributed reagents/materials/analysis tools:

RHK PJK AMD STG. Wrote the paper: RHK PJM AMD STG.

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Connectome Neurotrauma Mechanics

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115

On Allometry Relations

Damien West and Bruce J. West

International Journal of Modern Physics B, 26(18), 1230010-1-57 (2012)
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International Journal of Modern Physics B

Vol. 26, No. 18 (2012) 1230010 (57 pages)

c World Scientific Publishing Company

DOI: 10.1142/S0217979212300101

ON ALLOMETRY RELATIONS

DAMIEN WEST∗ and BRUCE J. WEST†

∗Physics Department, Renesselaer Polytechnique Institute, Troy, New York

†

Information Sciences Directorate, US Army Research Office,

Research Triangle Park, NC 27709, USA

Received 8 June 2012

Published 6 July 2012

There are a substantial number of empirical relations that began with the identification

of a pattern in data; were shown to have a terse power-law description; were interpreted

using existing theory; reached the level of “law” and given a name; only to be subsequently fade away when it proved impossible to connect the “law” with a larger body of

theory and/or data. Various forms of allometry relations (ARs) have followed this path.

The ARs in biology are nearly two hundred years old and those in ecology, geophysics,

physiology and other areas of investigation are not that much younger. In general if X

is a measure of the size of a complex host network and Y is a property of a complex

subnetwork embedded within the host network a theoretical AR exists between the two

when Y = aXb. We emphasize that the reductionistic models of AR interpret X and

Y as dynamic variables, albeit the ARs themselves are explicitly time independent even

though in some cases the parameter values change over time. On the other hand, the

phenomenological models of AR are based on the statistical analysis of data and interpret X and Y as averages to yield the empirical AR: Y  = aXb. Modern explanations

of AR begin with the application of fractal geometry and fractal statistics to scaling phenomena. The detailed application of fractal geometry to the explanation of theoretical

ARs in living networks is slightly more than a decade old and although well received it

has not been universally accepted. An alternate perspective is given by the empirical AR

that is derived using linear regression analysis of fluctuating data sets. We emphasize

that the theoretical and empirical ARs are not the same and review theories “explaining” AR from both the reductionist and statistical fractal perspectives. The probability

calculus is used to systematically incorporate both views into a single modeling strategy.

We conclude that the empirical AR is entailed by the scaling behavior of the probability

density, which is derived using the probability calculus.

Keywords: Allometry; fractals; physiology; scaling; statistical analysis; ontogenetic

growth.

Contents

1 Introduction ................................... 3

1.1 What is the equation? .......................... 4

1.2 Is there universality? ........................... 5

1.3 Some comments on fractals ........................ 6

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1. Introduction

At the turn of the nineteenth century the German polymath Gauss64 and the American mathematician Adrian1 introduced into science the law of frequency of errors,

the French physicist Laplace116 proved the central limit theorem and the French

zoologist Curvier34 determined that the brain mass of a mammal increases more

slowly than does its total body mass (TBM). Cuvier’s observations initiated the

field of Allometry that relates the size of an organ to that of the host organism with

mass serving as a measure of size. The roads of statistical analysis and observation

(experiment) converge in the twenty-first century to explain the origin of allometry

relations (ARs).

Allometry, literally meaning by a different measure, has acquired a mathematical

description through its relations along with a number of theoretical interpretations

to account for its mathematical form. However no one theory has been universally

accepted as successfully explaining ARs in their many guises so the corresponding

origins remain controversial. Consequently, in reviewing the properties of allometry

data along with their various theoretical explanations we herein provide a glimpse

as to those origins.

We use the generic term network in our narration in order to slip smoothly from

ARs in physical networks with identical particles and van de Waal’s forces, to biological networks with structured elements and chemical interactions, to geophysical

networks with complex tributaries and branching architectures. This nomenclature

also enables us to transition from arcane historical theory to a modern perspectives

of complex networks. The mathematics of renormalization group (RG) theory,103,259

fractional differential equations,126,144 fractional stochastic differential equations236

and transitioning from dynamic variables to phase space variables to express the

probability calculus in terms of fractional diffusion equations108,207,236 are herein

found to provide insight into different aspects of the origins of ARs.

Allometry has been defined as the study of body size and its consequences79,176

both within a given organism and between species in a given taxon. Gayon65 reviewed the history of the concept of allometry, defined as the study of body size

and its consequences79,176 within a given organism and between species in a given

taxon, and distinguished between four different forms: (1) ontogenetic allometry,

which refers to relative growth in individuals; (2) phylogenetic allometry, which

refers to constant differential growth ratios in lineages; (3) intraspecies allometry,

which refers to adult individuals within a species; (4) interspecies allometry, which

refers to the same kind of phenomenon among related species. The theoretical

entailment of static from dynamic allometry models has not been systematically

studied, although there has been some recent effort in that direction.67,243 Herein

we review the use of phenomenological dynamic equations from physical biology to

relate the dynamic allometry of the first two categories to their static counterparts

in the last two.

Galileo62 recognized that in order for an organism or physical structure to retain

a constant function as size increases requires its shape (architecture) and/or the

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materials with which it is constructed to change. This simple physical picture has

biological and social analogs in which the size of a network has unavoidable consequences.164,213 We focus on ARs for physiologic phenomena, but we do indicate

where insight might also be gained from other disciplines as well.

1.1. What is the equation?

Sir Julian Huxley98; grandson of the Huxley of Darwin evolution fame, brother of

the novelist Aldous (Brave New World) and half-brother of the biophysicist Andrew

(the Hodgkin–Huxley equations)209; proposed that two parts of the same organism

have proportional rates of growth. In this way if Y is a living subnetwork observable

with growth rate θ and X is a measure of the size of a living host network with

growth rate γ then the fractional increase in the two is denoted according to Huxley

by:

dX/γX = dY /θY . (1)

This equation can be directly integrated to obtain:

Y = aXb , (2)

the time-independent AR where a and b (= γ/θ) are empirically determined. The

theoretical AR given by Eq. (2) considered by Huxley is the basis of subsequent

theoretical discussions in such excellent books as Schmidt–Nielson195 and Calder.31

The intraspecies AR relates a property of an organism within a species to its

TBM. The interspecies AR relates a property across species such as the basal

metabolic rate (BMR) to TBM.31,195 These two allometry groups are distinctly

different and the models developed to determine the theoretical forms of the allometry coefficient a and exponent b in the two cases are quite varied, as shown

subsequently.

Equation (2) looks very much like the scaling relations that have become so

popular in the study of complex networks over the last decade.3,30,153,221,239 Historically the nonlinear nature of Eq. (2) has precluded the direct fitting of the

equation to data. A logarithmic transformation is traditionally made and a linear

regression to the data on the equation

ln Y = ln a + b ln X (3)

is used to estimate the parameters a and b. In Sec. 2 we review a myriad of phenomena from a number of disciplines in which ARs have been brought to light. In

Sec. 3 we discuss the fitting of ARs to data.

All complex dynamical networks manifest fluctuations, either due to intrinsic

nonlinear dynamics producing chaos124,155 or due to coupling of the network to an

infinite dimensional, albeit unknown environment,118 or both. The modeling strategies adopted to explain ARs have traditionally taken one of two roads: the statistical

approach in which residual analysis is used to understand statistical patterns and

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to identify the causes of variation in the AR;31,195 or the reductionist approach to

identify mechanisms that explain specific values of the allometry parameters.10,247

We find that neither approach separately provides a complete explanation of the

variety of phenomena described by ARs. The influence of the environment, whether

inducing fluctuations in a reductionist model, or producing a systematic change in a

statistical model, has been taken into account in multiple studies.71,73,141 In Sec. 5

we take the road of the probability calculus that systematically incorporates both

reductionistic and statistical mechanism into the phenomenological explanation of

ARs. This calculus enables modelers to associate characteristics of the measured

probability density function (pdf ) with specific deterministic mechanisms and with

structural properties of the coupling between variables and fluctuations.118,175

1.2. Is there universality?

In physics the notion of universality occurs in the context of critical phenomena,

where in the vicinity of a phase transition a network ceases to have a characteristic

scale and can be characterized by a set of critical exponents. The principle of

universality claims that networks that undergo a phase transition can be grouped

into one of a small number of universality classes. Gisiger68 discusses universality in

the context of invariance in biology; specifically addressing the role of 1/f noise. In

the context of AR universality might imply that the allometry exponent plays the

role of a critical exponent. The empirical evidence suggests that this interpretation

of the allometry exponent is not appropriate. A somewhat less stringent definition

of universality may be defensible; one in which the details of the interactions are

washed out but the form of the AR persists with the allometry exponent in a

restricted interval.

In biology the ARs associate functional variables with measures of body size,

such as the TBM X = M raised to a noninteger power Mb. The average BMR is one

such functional variable Y = B that can be expressed in terms of TBM. The most

prevalent theories of metabolic allometry argue for either b = 2/3, based on body

cooling, or b = 3/4, based on energy efficiency. Selected data sets have been used

by various investigators to support either of these two values. However, there is also

strong evidence that there is no universal value of b that is satisfied by all metabolic

data. In fact, Bokma21 presents a large amount of fish data to demonstrate that no

single universal value of the allometry exponent b exists. This argument is supported

by Glazier72 who observes that the isometric (b = 1) metabolic scaling of pelagic

animals is an evolutionarily malleable trait that responds to environmental changes.

On the other hand, West and Brown252 argue that living networks do have universal scaling laws. Their arguments rest on patterns observed in biological and

botanical data and the quantitative theory of the fractal structure, organization

and dynamics of the branching networks in living systems. The theory developed

by West et al.,247 which we review in Sec. 4, has as one of its tenets the existence of

hierarchical fractal-like branching networks for the delivery of nutrients resulting in

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b = 3/4. They attribute this origin of metabolic AR to evolution’s solution to the

grand challenge of how highly complex, self-sustaining, reproducing, living networks

service enormous numbers of localized microscopic units in an efficient and “democratic” way. Their conclusion, like that of the earlier analysis others,

230,231 was

that fractal networks have an evolutionary advantage over those that scale classically,213 independently of what the networks distribute from macroscopic reservoirs

to microscopic sites.

Scaling is a ubiquitous property of large complex networks indicating that the

observables simultaneously fluctuate over many time and/or space scales. In the

physical sciences such phenomena have historically been categorized as 1/f noise225

or 1/f variability.239 Mandelbrot131 was probably the first to recognize the wideranging significance of this 1/f variability with his introduction of fractals into the

scientist’s lexicon. The existence of ARs has been closely tied to fractal geometry

by some investigators247 and in Sec. 5 we show that the origin of AR may reside in

fractal statistics, the scaling of pdfs, and not in fractal geometry.

1.3. Some comments on fractals

Mandelbrot131,132 identified ARs masquerading under a variety of empirical “laws”

and argued that they were a consequence of complex phenomena not having characteristic scales. Subsequent interpretations of ARs often involve fractals and so

we recall some fundamental properties of fractals that are subsequently useful. To

begin let us give a qualitative definition of a fractal:133 “A fractal is a shape made

of parts similar to the whole in some way.”

The fractal concept arises in three distinct, but related guises; geometry, statistics and dynamics. Geometric fractals deal with the self-similarity of complex geometric forms. A fractal object examined with ever increasing magnification reveals

ever greater levels of detail; detail that is self-similar in character. The basic mathematical properties of geometric fractals and their myriad of applications can be

found in a number of excellent books17,55,57,131,133,143,238 and is not repeated here.

We merely record and interpret those properties that may be needed for the analyses of ARs. The number of self-similar objects N required to cover an object of

dimension D is given by N = r−D, where r is the size of the “ruler”. In this way

the fractal dimension of the object being covered can be mathematically defined as:

D = − ln N/ ln r (4)

in the limit of vanishing r. As the ruler size goes to zero the number of rulers

necessary to cover the object diverges to infinity in such a way that D remains

finite for self-similar objects and this dimension is not necessarily integer valued.

An observable Z(t) is scaling if for a positive constant c it satisfies the homogeneity relation

Z(ct) = cHZ(t). (5)

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Modifying the units of the independent variable therefore only changes the overall

observable by a multiplicative factor; this is self-affinity. Barenblatt13 remarked

that such scaling laws are not merely special cases of more general relations; they

never appear by accident and they always reveal self-similarity. Note that scaling

alone is not sufficient to prove that a function is fractal, but if a function is fractal

it does scale.

In the sequel we relax the distinction between self-affine and self-similar, since

self-similarity has been extended to encompass both meanings in the physics literature. Meakin143 asserts that in homogeneous scaling relations it is the coefficient

that embodies the “real physics” behind power-law relations. He further observed

that the allometry exponents are universal in many homogeneous scaling phenomena and the allometry coefficients provide the only means to control physical properties and behavior.

Scale invariance or scaling requires that a function Φ(X1,...,XN ) be such that

scaling each of the N variables by an appropriate choice of exponents (α1,...,αN )

always recovers the same function Φ(X1,...,XN ) up to an overall constant:

Φ(X1,...,XN ) = γβΦ(γα1X1,...,γαN XN ). (6)

We observe that Eq. (2) is possibly the simplest of such scaling relations between

two variables such that they satisfy the RG relation

X(γY ) = γb

X(Y ).

The lowest-order solution to this equation is, of course, given by Eq. (2) and we

provide the general solution subsequently. Changes in the host network X (size)

control (regulate) changes in the subnetwork Y (property) in living networks and

in some physical networks through the homogeneous scaling relation.

Inhomogeneity in space and intermittency in time are the hallmarks of fractal

statistics and it is the statistical rather than the geometrical sameness that is evident at increasing levels of magnification. In geometrical fractals the observable

scales from one level to the next. In statistical fractals where the phase space variables (z, t) replaces the dynamic variable Z(t) it is the pdf P(z, t) that satisfies a

scaling relation:

P(αz, βt) = β−μP(z, t) ; μ = ln α/ ln β , (7)

where μ = 2H and the homogeneity relation is interpreted in the sense of the pdf in

Eq. (7). Time series with such statistical properties are found in multiple disciplines

including finance,134 economics,135 neuroscience,4,226 geophysics,215 physiology240

and general complex networks.244 A complete discussion of statisical data with such

scaling behavior is given by Beran20 in terms of the long-term memory captured

by the scaling exponent. One example of a scaling pdf is given by:

P(z, t) = 1

tμ

Fz

 z

tμ



(8)

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and in a standard diffusion process Z(t) is the displacement of the diffusing particle

from its initial position at time t, μ = 1/2 and the functional form of Fz(·) is a

Gauss distribution. However, for general complex phenomena there is a broad class

of distributions for which the functional form of Fz(·) is not Gaussian and the

scaling index μ = 1/2, see Sec. 5 for additional discussion.

Dynamic fractals do not directly enter our discussion of ARs. However for completeness we mention that in a dynamic fractal the geometry of the manifold on

which the dynamics of a network unfolds is fractal, so that the associated chaotic

time series is also fractal.155

2. Empirical Allometry

In this section we review areas of investigation where size has been observed to

control the properties of a phenomenon. In so doing we catalog a number of phenomenological relations that are not often discussed from the same perspective.

There is a nontrivial number of empirical relations that began as the identification

of a pattern in data; were shown to have a terse power-law description; were interpreted using existing theory; reached the level of “law” and given a name, not

always after the discoverer; only to subsequently fade away when it proved impossible to connect the “law” with a larger body of theory and/or data. An example

drawn from the Notebooks of Leonardo da Vinci177 relates the diameter of a parent

limb d0 to two daughter limbs d1 and d2 :

dα

0 = dα

1 + dα2 . (9)

The da Vinci scaling relation supplies the phenomenological mechanism necessary

for AR to emerge in a number of disciplines, as we subsequently discuss.

Nearly five hundred years after de Vinci recorded his observations Murray150

used energy minimization to derive the same equation with the theoretical value

α = 3, which is known in the literature as Murray’s Law or the Murray–Hess Law.

In the simplest case d1 = d2 the da Vinci scaling relation reduces to scaling between

sequential generations of a bifurcating branching network having daughter branches

of equal radii:

dk+1 = 2−1/αdk (10)

resulting in an exponential reduction in branch diameter from generation to

generation.

2.1. Living networks

Living networks have static intraspecies ARs that link two distinct but interacting

parts of the same organism in terms of mass, with mass serving as a measure of

size. Smith202 maintained that concentrating on a power function as the method

for evaluating the biological consequences of size has masked the complexity ofthe

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On Allometry Relations

allometry problem. We agree with this observation, but perhaps in ways that Smith

would not have anticipated, as will become evident.

2.1.1. Biology

Cuvier34 was the first to recognize that brain mass increases more slowly than

TBM as we proceed from small to large species within a taxon. This empirical

observation was subsequently made between many other biological observables and

was first expressed mathematically as an allometric relation by Snell204:

brainweight = a(bodyweight)b , (11)

where on log–log graph paper a is the intercept with the vertical axis and b is the

slope of the line segment. Mammalian neocortical quantities Y have subsequently

been empirically determined to change as a function of neocortical grey matter

volume X as an AR. The neocortical allometry exponent was first measured by

Tower214 for neuron density to be approximately −1/3. The total surface area of

the mammalian brain was found to have an allometry exponent of approximately

8/9.94,100,167 Changizi33 points out that the neocortex undergoes a complex transformation covering the five orders of magnitude from mouse to whale depicted in

Fig. 1 but the ARs persist; those mentioned here along with many others.

Fig. 1. Mouse to elephant curve. BMR of mammals and birds are plotted versus TBM on log–log

graph paper. The solid line sement is the best linear regression of Eq. (2) to the data with a slope

very close to 3/4. [reproduced from Schmidt–Nielsen195 with permission].

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Another quantity of interest is the time; not the chronological time measured by

a clock but the intrinsic time of a biological process first called biological time by

Hill.93 Hill reasoned that since so many properties of an organism change with size

that time itself may scale with TBM. Lindstedt and Calder119 develop this concept

further and determine experimentally that biological time, such as species longevity,

satisfies an AR with Y being the biological time. Lindstedt et al.121 clarify that

biological time τ is an internal mass-dependent time scale

τ = aMb (12)

to which the duration of biological events are entrained. They present a partial list

of such events that includes breath time, time between heart beats, blood circulation

time and time to reach sexual maturity. In all these examples and many others the

allometry exponent clusters around the theoretical value b = 1/4. Note that the

total energy of an organism seen as a bioreactor is proportional to volume (M) and

the biological time is proportional to M1/4, so the metabolic rate (energy/time)

would scale as M3/4.

2.1.2. Botany

Niklas154 shows in Fig. 2 an impressive statistical trend spanning twenty orders of

magnitude in the mass of aquatic and terrestrial nonvascular and vascular plant

species. The annual growth in plant body biomass GT (net annual gain in dry

mass per individual) and MT (total dry mass per individual) are related by the

Fig. 2. Log–log bivariate plot of total annual growth rate in dry body mass per individual GT

versus TBM. Line segment denotes reduced major axis regression curves for the entire data set.

[adapted from Niklas154 with permission].

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empirical AR:

GT ∝ M3/4

T . (13)

Figure 2 shows linear regression on logarithmically transformed data

log GT = log a + 0.75 logMT (14)

and the parameter a is the intercept with the vertical axis. In the data analyses

recorded herein the terms weight, mass and volume are used almost interchangeably as measures of size. The allometry exponent is 3/4 for the data in Fig. 2 but

empirically differs from this value when the data sets are graphed individually. The

allometry coefficients of the separate data sets may vary as a function of habitat as well. The agreement between the biomass data and the AR with exponent

3/4 is very suggestive but it must be viewed critically because of methodological

limitations.

Reich et al.174 analyzed data for approximately 500 observations of 43 perennial

plant species of coupled measurements of whole-plant dry mass and GT from four

separate studies. Collectively, the observations span five of the approximately 12

orders of magnitude of size in vascular plants.53 The result of each experiment

yielded an isometric scaling of b ≈ 1 and not b = 3/4 as did the scaling of GT to

TBM for whole plants. Consequently, even when data look as appealing as they do

in Fig. 2 things are not always what they seem.

2.1.3. Clearance curves

Another allometry phenomenon is the dependence of drug-dosing range on TBM

and is referred to as clearance.97 Zenobiotic clearance is the rate at which any foreign

compound not produced by an organism’s metabolism is passed from the organism.

The application of allometry ideas to pharmacokinetics and to determining human

parameters from those in animals is fairly recent.22,127,191 The early studies did

not address questions of variability in the allometry parameters and were primarily

concerned with whether the allometry exponent more closely tracked the value 2/3

or 3/4 by doing linear regression on log-transformed data.122

Hu and Hayton97 addressed the possible impact of statistical variability in the

AR parameters on the predicted pharmacokinetic parameter values. They found

considerable uncertainty in the value of the allometry exponent, which they fit to a

Gaussian distribution with mean value 0.74. Even though they could not determine

whether the variability in the allometry exponent was due to experimental error

or to biological mechanisms they did find that there was no systematic deviation

from the AR. However it appears that whether b = 3/4 or 2/3 depends on which

of the major elimination pathways is used, metabolism for the 3/4 value and renal

excretion for the 2/3 value.

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2.1.4. Physiology

The most studied of the interspecific ARs does not concern relative growth but is

that associating the average BMR measured in watts to average TBM measured in

kilograms of multiple species such that:

B = aMb . (15)

The metabolic rate refers to the total utilization of chemical energy for the generation of heat by the body of an animal and is often measured by the oxygen intake

during respiration.

The earliest physiologic model for the value of the allometry exponent in the

intraspecies AR was given by Sarrus and Rameaux.188 Schmidt–Nielsen195 records

that this team of a mathematician and a physician reasoned that the heat generated by a warm blooded animal is proportional to the volume and the heat loss

is proportional to the animal’s free surface. Experiments on dogs by Rubner183

supported their argument and lead to the wide acceptance of the “surface law” in

which b = 2/3. In Fig. 1 the “mouse-to-elephant” curve depicts the BMR for mammals and birds plotted versus TBM on log–log graph paper spanning six orders of

magnitude. The solid line segment is the fit of the AR to the data and yields the

empirical value b ≈ 3/4.

As Niklas154 noted the expectation was that the metabolic rate would be proportional to the 2/3 power of the TBM as prescribed by the surface law. Surprisingly, this turned out not to be the case. The research of Kleiber109 and Brody24

revealed that the slope was closer to 3/4 than to 2/3. Subsequent observational

studies have reinforced the allometric pattern observed in the data predicted by

Eq. (15) including some relating the 3/4-rule to plants, see, for example, Hemmingsen.86 Consequently the phenomenological value of the allometry exponent b

remains controversial.47,71,91

Controversy also persists regarding the theoretical explanation as to why the

allometry exponent b should have a specific value. The simple geometrical argument

of Sarrus and Rameaux suggests b = 2/3 as reviewed in a number of excellent

sources.31,47,91 On the other hand, the quarter-power AR is explained by West,

Brown and Enquist (WBE) using geometric scaling arguments from fractal physics

to establish the value b = 3/4 and other quarter-power scaling laws in physiology,

see Sec. 4.

Heusner91 adopted geometric scaling arguments to obtain b = 2/3 in the AR

between BMR and TBM. He argued that the various other values experimentally

observed for the power-law index by investigators are a consequence of differing

values of the allometric coefficient a. He reasoned that two or more data sets with

b = 2/3 but with different values of a graph as parallel line segments on log–log

graph paper, but when the two or more data sets are grouped together and analyzed

as a single data set the aggregate is fit by a single line segment with net slope

b > 2/3. The same argument can be found in a number of other references.176,195

But unlike those earlier references Heusner91 concluded that it is the allometry

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coefficient that remains the central mystery of allometry and not the allometry

exponent. We investigate the implications of Heusner’s conjecture in Sec. 3 where

we numerically explore the implications of treating the allometry coefficient as a

random variable.

It is useful to list the various forms of physiological allometry given by Lindstedt

and Schaeffer122: pulmonary and cardiac allometry with b = −1/4; renal allometry

with b = −0.85; liver allometry with b = −0.85; pulmonary blood volume with

b = 1.0; cardiac output with b = 3/4 and pulmonary transit times with b = 1/4.

2.1.5. Information transfer

There are literally dozens of physiologic ARs for physiologic time τ, relative to clock

time t, that increases with increasing body size τ = aMb120 and describes chemical

processes such as the turnover time for glucose with b = 1/49 to the life span of

various animals in captivity with b = 0.20.185 Schmidt–Nielsen195 explains how a

variety of physiologic time scales such as the length of a heart beat and respiration

all scale with body size and from that deduce a number of interesting relations.

It is only recently however that Hempleman et al.88 hypothesized a mechanism to

explain how information about the size of an organism is communicated to the

organs within the organism. Their hypothesis involved matching the neural spike

code to body size to convey this information.

Hempleman et al.88 suggest that mass-dependent scaling of neural coding may

be necessary for preserving information transmission with decreasing body size.

They point out that action potential spike trains are the mechanisms for long

distance information transmission in the nervous system. They go on to say that

neural information may be “rate coded” with average spike rate over a time period

encoding stimulus intensity or “time coded” with the occurrence of a single spike

encoding the occurrence of a rapid stimulus transition. The hypothesis is that some

phasic physiological traits are sufficiently slow in large animals to be neural rate

coded, but are rapid enough in small animals to require neural time coding. These

trait include such activities as breathing rates that scale with b = −1/4.

They tested for this allometry scaling of neural coding by measuring action potential spike trains from sensory neurons that detect lung CO2 oscillations linked to

breathing rate in birds ranging in body mass from 0.045 to 5.23 kg. While it is well

known that spike rate codes occur in the sensing of low frequency signals and spike

timing codes occur in the sensing of high frequency signals, their experiment was

the first designed to test the transition between these two coding schemes in a single

sensory network due to variation in body mass. The results of their experiments

on breathing rate was an allometry exponent in the interval −0.26 ≤ b ≤ −0.23

and although taken on a small number of birds their results do suggest a preservation of information transmission rates for high frequency signals in intrapulmonary

chemoreceptors and perhaps other sensory neurons as well. The implications of

these experiments strongly suggest the need to continue such investigations.

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On the more theoretical side Moses et al.149 apply the scaling ideas of metabolic

allometry developed by West et al.247 to information networks consisting of microprocessors to form a network. Moses et al.149 use a fractal argument to construct a

two-dimensional hierarchal self-similar branching network, an H-tree that Mandelbrot131 originally used in his discussion of the space filling behavior of the human

lung. They show that this branching network of microprocessors have a striking

similarity to such networks in organisms even though the latter has evolved by

natural selection and the former are designed by engineers.

Along this same line E. F. Rent, while an IBM employee in the 1960s, wrote a

number of internal memos (unpublished) relating the number of pins at the boundaries of an integrated circuit (X) to the number of internal components (Y ), such

as logic gates, to obtain an AR with b < 1.0. This rule has historically been used by

engineers to estimate power dissipation in interconnections and for the placement

of components in very large scale integrated (VLSI) circuit design. More recently

Rent’s Rule has been used to model information processing networks in the human

brain16 where the mass of grey and white matter are shown to satisfy an AR as first

noted by Schlenska.194 Beiu and Ibrahim18 suggested that the allometry exponent

for grey and white matter between species is identical to the Rent exponent within

a species and this was supported using MRI data by Bassett et al.16

2.2. Physical networks

Some of the oldest ARs involve physical networks, or more specifically geophysical networks. The skeptic need only return to da Vinci’s scaling relation. In his

notebooks da Vinci explains the meaning of this equation not only in the context

of relating tree trunks to subsequent branches, but to the branchings of rivers as

well. Long before the conservation of energy and the continuity of fluid flow were

known to scientists, the enigmatic Italian painter, sculptor, military engineer and

anatomist understood the basics of hydrologic networks.

2.2.1. Geology and geomorphology

Horton’s law of river numbers is another empirical regularity observed in the topology of river networks.95 As observed by Scheidegger192 the number of river segments

in successive order form a geometrical sequence such that, the

number of rivers with k tributaries ∝ R1−k

b . (16)

The bifurcation parameter Rb is the constant ratio between successive numbers of

river networks, known as Horton’s law of stream numbers nk/nk+1 = Rb and has

the empirical value between 4.1 and 4.7 in natural river networks,159,160 in contrast

to the random model that predicts a value of four. Note that the system of counting

begins at the smallest tributary that are the most numerous Rb > 1 and these feed

into larger tributaries that are fewer in number.

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Dodds and Rothman46 point out that universality arises when the qualitative

character of a network is sufficient to quantify its essential features, such as the

exponents that characterize scaling laws. They go on to say that scaling and universality have found application in the geometry of river networks and the statistical

structure of topography within geomorphology. They maintain that the source of

scaling in river networks and whether or not such scaling belongs to a single universality class is not yet known. They do provide a critical analysis of Hack’s law,

see also, Rodriguez–Iturbe and Rinaldo.182

2.2.2. Hydrology

Hack’s law is a hydrologic AR having to do with the drainage basins of rivers. Hack83

developed an empirical relation between mainstream length of a river network and

the drainage basin area at the closure of the river. Hack’s law is given by,

mainstreamlength ∝ (area)h , (17)

where h is the Hack exponent with the typical empirical value h ≈ 0.57. Hack

asserted that river networks are not self-similar.

Mandelbrot131 relates Hack’s exponent to the fractal dimension of the river network and debates the interpretation of the equation. Feder57 observed that defining

a fractal dimension for river networks was obscure and required further study. A

modern version of this discussion in terms of hydrologic allometry is given by Rinaldo et al.181 who point out that optimal channel networks yield h = 0.57 ± 0.02

suggesting that feasible optimality178 implies Hack’s law. Another viable model is

given by Sagar and Tein186 that is geomorphology realistic giving rise to general

ARs in terms of river basin areas, as well as parallel and perpendicular channel

lengths.

Maritan et al.136 consider an analogy with the metabolic AR using M ∝ Bα,

that is, α = 1/b so that α =1+ h with the limiting values α = 3/2 and h = 1/2

in the case of geometric self-similarity. Geometric self-similarity is the preservation

of the river’s shape as the basin increases in area. The observed values lie in the

range 1.5 ≤ α ≤ 1.6 and the scatter of individual curves (analog of intraspecies

data) is remarkably small. These values suggest that the branching nature of rivers

is fractal, that is, α > 3/2 in most cases. The ensemble average of Hack’s exponents

from different basins extend over 11 orders of magnitude and is indistinguishable

from h = 1/2.146 Maritan et al.136 conclude that like the interspecies metabolic rate,

the slope of the intraspecies h’s are washed out in the ensemble average, resulting

in the value h = 1/2.

2.3. Natural History

Natural History embraces the study, description and classification of the growth

and development of natural phenomena. The focus of investigation includes such

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important contemporary areas as ecology and paleontology, parts of which rely

heavily on AR and scaling.

2.3.1. Ecology

Ecology is the scientific study of the distribution, abundance and relations of organisms and their interactions with the environment. Such living networks include

both plant and animal populations and communities along with the network of relations among organisms of different scales of organization. Of concern to us here

are the species traits determined by body size and how these in turn affect food

web stability. Woodward et al.261 along with many others point out that the largest

metazoans, for example, whales (108 grams) and giant sequoias, weigh over 21 orders of magnitude more than the smallest microbes (10−13 grams).102,234 They go

on to stress the considerable variation in body mass among members of the same

food web.

We do not address the traditional linking of species through such mechanisms as

the prey–predator relation within food webs76 but instead we focus on body size.

The significance of body size has been systematically studied in ecology.27,35,102

Indentifying X with species abundance and Y with TBM in Eq. (2) there is, in

fact, an AR between the species at the base of a food web and the largest predator

at the top.35 We note that species-area power functions have a vital history in

ecology166,258 even though the domain of sizes over which the power law appears

valid is controversial.25,256

Woodward et al.261 emphasize that AR has been used to explain the observed

relations between body size and each of: home range size, nutrient cycling rates,

numerical abundance and biomass production. They speculate that body size may

capture a suite of covarying species traits into a single dimension, without the

necessity of having to observe the traits directly.

Brown et al.26 discuss the universality of the documented ARs in plants, animals and microbes; to terrestrial, marine and freshwater habitats; and to humandominated as well as “natural” ecosystems. They emphasize that the observed

self-similarity is a consequence of a few basic physical, biological and mathematical

principles; one of the most fundamental being the extreme variability of the data.

The variety of distributions of allometry coefficients and exponents are discussed

both phenomenologically and theoretically in subsequent sections.

Farr’s law29 is an example of the change in ARs in the transition from organismic to environmental allometry. Farr collected data on the number of patients

committed because of their mental condition and their mortality from a variety of

asylums in 1830s England.56 From these humble beginnings he was able to summarize the “evil effect of crowding” into a relation between mortality rate R and

population density ρ96:

R = aρb . (18)

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Here we see that the size measure used in the metabolic AR, the mass, is replaced

with a measure of community structure, the population density. The ARs that capture life histories in ecology and sociology are often expressed in terms of numbers

of animals and areas in addition to body mass. Calder31 points out that size and

time seem to be the principle characteristics of life history and ecology.

The factors necessary for the formation of social groups with a restricted

geographic area are not understood, but certain ARs help clarify them. The population density ρ of herbivorous mammals has been determined to be related to their

mass M by39 ρ−1 = cM0.75 where the animals freely roam on a “home range” of

given area: Ahr = c′

M1.02. A similar relation exists for carnivores.145 Consequently,

as explained by Calder,31 herbivorous mammals above a certain size range over an

area greater than their per capita share of the local habitat. The degree of overlap

was empirically determined to be40 Ahrρ = c′′M0.34, where the empirical exponent

0.34±0.11 is statistically indistinguishable from that obtained by combining the separate exponents for the population density and area, that being, 1.02−0.75 = 0.27.

Calder conjectures that the greater the product Ahrρ the greater the intensity of

competition and the greater the desirability of social networks that contribute to

mutual tolerance within these groups. Makarieva et al.129 argue that animal home

range represents a biological footprint of the undisturbed state of an econetwork,

however the population density adapts to disturbances in the econetwork. Consequently, the deviation of the home range-population product from isometry reflects

the degree of econetwork disturbance.

The AR between maximum abundance and body size for terrestrial plants N ∝

M−b was extended by Belgrano et al.19 to the maximum population densities of

marine phytoplankton with b = 3/4. They draw the implication that maximum

plant abundance is constrained by rates of energy supply in both terrestrial and

marine networks as dictated by a common AR. Earlier investigators found b >

3/4.37,49

2.3.2. Zoology and acoustics

Mice squeak, birds chirp and elephants trumpet due to scaling. Fitch59 discusses the

relationship between an organism’s body size and acoustic characterization of its

vocalization under the rubric of acoustic allometry. Data indicate an AR between

palate length (the skeletal proxy for vocal tract length) and body mass for a variety

of mammalian species. He shows that the interspecies allometry exponent attains

the geometric value of three in the regression of skull length and body mass, whereas

the intraspecies allometry exponent varies a great deal. The significant variability

in the intraspecies allometry exponent suggest taxon-specific factors influencing the

AR.137,199

Fitch59 gives the parsimonious interpretation that the variability in the intraspecies allometry exponent could be the result of each species adopting allometric scaling during growth as postulated by Huxley, with a different proportionality

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factor for each species. On the other hand, the interspecies allometry exponent

could result from the common geometric constraints across species due to the wide

range of body sizes. He concludes that the AR between vocal tract dimensions

and body size could provide accurate information about a vocalizer’s size in many

mammals.

2.3.3. Paleontology

Pilbeam and Gould164 provide reasons as to why body size has played such a

significant role in biological macroevolution. The first is the statistical generalization

known as Cope’s Law or Rule, which states that population lineages increase in

body size over evolutionary time scales,36 that is, the body size of a species is an

indication of how long it has survived on geological time scales. A second reason is

the one mentioned earlier, Galileo’s observation that large organisms must change

shape in order to function in the same way as do their smaller prototypes.

One quantitative measure of evolution is the development of the brain in mammals at various stages of evolution. Jerison99 showed that the brain-body relation

given by Eq. (2) is satisfied by mammals with an exponent that is statistically

indistinguishable from 2/3. He suggested that a may be an appropriate measure

of brain evolution in mammals as a class, following the proposal of Dubois48 that

a quantitative measure of cephalization in contemporary mammals be based on

the ratio; a = brain weight/(body weight)b

. These hypotheses were directly tested

by Jerison99 using endocranial volumes and body volumes for fossil mammals

at early and intermediate evolutionary stages. The data did in fact support the

hypothesis.

White and Gould253 emphasize in their review of the meaning of the allometry coefficient a that it had generated a large and inconclusive literature. Reiss176

notes that if brain mass is regressed on TBM across individuals in a species the

slopes are shallower than of regressions calculated across mean values for different

species within a single family (genus). This argument had also been presented by

Gould80 who emphasized the importance of the allometry coefficient in the geometric similarity of allometric growth. This interpretation of the allometry coefficient

was at odds with the belief of the majority of the scientific community at the time

that the allometry coefficient was independent of body size. This latter view is also

contradicted by the data analysis in Sec. 3.

Allometry has been used by Alberch et al.2 as the first step in creating a unifying theory in developmental biology and evolutionary ecology in their study of

morphological evolution. They demonstrate how their proposed formalism relate

changes in size and shape during ontogeny and phylogeny.

3. Data Statistics

In this section the phenomenology of analyzing AR measurement using statistics

is discussed; collecting data, identifying patterns (laws) in the data and developing

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methods of statistical analysis. Warton et al.220 point out that fitting a line to a

bivariate data set is not a simple task and the AR literature is filled with debate

over the proper methodology. Sir Julian readily adopted the statistical approach of

linear regression to Eq. (3) on multiple data sets to determine the allometry coefficient a and exponent b. The sophisticated statistical techniques such as principle

component analysis were not available to Sir Julian and although they can be found

in the modern AR literature least-square regression still appears to be the method

of choice.73,189

Measures of complex phenomena always contain uncertainty and the data sets

invariably fluctuate from measurement to measurement. Thus, X(t) and Y (t) are

stochastic variables, and by implication X(Y ) is a random function of its argument as well. Consequently the deterministic algebraic equations relating network

size and network function never unambiguously represent what is actually being

measured. Subsequently, we examine the corresponding mathematical modeling,

shifting the focus from the dynamic stochastic variable to the associated dynamic

pdf.

We examine the phenomenology of the random data from measurements of various properties of allometry networks that determine empirical ARs. In particular

we focus on how allometry coefficients and exponents are interpreted given that the

data on which they are based fluctuate to such a large extent. Part of the reason

for taking this approach is that a significant number of scientists adopt the viewpoint that fluctuations reflect lack of control and/or ignorance about what is being

measured. Here we adopt the more sympathetic view that networks are generically

random because they are dynamic and complex in which case statistics provide

information about the fundamental nature of that complexity.

3.1. Fluctuations

All complex dynamic networks are stochastic, either due to intrinsic nonlinear dynamics producing chaos,123,124,155 or due to coupling of the network to an infinite

dimensional albeit unknown environment,118 or both; completely aside from measurement errors. Consequently, it is necessary to understand how statistical uncertainty may be included in modeling allometry data. Kaitaniemi104 pointed out

that the potential information content of the allometry coefficient has been largely

neglected, an observation also made by Glazier73 among others. Kaitaniemi examined the different ways this parameter may vary for different sources of random

fluctuations. Here we follow a similar strategy, but we use actual data rather than

computer generated random fluctuations.

The normal or Gauss pdf suggests that the statistical variations between the

variables in the AR Eq. (2) may be additive leading some scientists70,130,203 to

propose the form:

Y =  ̄aX ̄b + η , (19)

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where η depicts the random fluctuations and the overbars denote the fitted values

of the parameters. Packard and Boardman157 also investigate the regression of data

to a three-parameter power law that does not pass through the origin

Y = Y0 +  ̄aX ̄b + η . (20)

In the near isometry case where  ̄b ≈ 1 linear regression analysis is appropriate

and additive fluctuations provide a satisfactory representation of the statistical

variability. On the other hand when the allometry exponent is substantially different

from unity the determination of the nature of the fluctuations requires preliminary

statistical analysis, see, for example, Packard.156

Packard and Boardman157 emphasize that the additive form of fluctuations in

ARs is quite different from the situation involving the logarithmically transformed

data. For the transformed data introducing additive random fluctuations yields:

log Y = log  ̄a +  ̄b log X + η , (21)

and the empirical constants  ̄a and  ̄b are fit to the transformed data. In terms of the

original AR we obtain:

Y =  ̄aeηX ̄b (22)

with the fluctuations being exponentially amplified through eη. It is evident that

when the fluctuations are considered to be focused in the allometry coefficient, as

they are here, they are multiplicative. The multiplicative character of the fluctuations implies that the influence of the random variations is amplified far beyond

their additive cousins.

They emphasize that the focus of the research on log-transformed data is to

characterize patterns of variation in morphology, physiology and ecology in organisms spanning a broad range in body size in an attempt to identify underlying

principles in the design of biological networks, see, for example, Brown et al.

27 and

references therein. They go on to assert that many of the patterns identified by

this research are inaccurate and misleading and these mischaracterizations likely

contribute to the ongoing debate about ways in which animals are constructed.

We saw that fluctuations in the log-transformed data may be equivalent to multiplicative fluctuations in the original data. So the important question is whether it

is necessary to perform the logarithmic transformation at all. Packard and Boardman157 point out that the original motivation for the log-transform was to linearize

the equations thought to represent the data and therefore facilitate the implementation of graphical and statistical analysis.162,203 However, they go on to show the

biasing problems associated with log-transforms using computer generated data

sets and caution that with the present day computer software for fitting nonlinear

equations linearization is no longer a sufficient rationale for log-transforms. So are

there other reasons to transform the data?

Kerkhoff and Enquist106 strongly disagree with the conclusions of Packard

(2008) that standard methods for fitting allometry models produce “biased and

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misleading” results. They point out that most biological phenomena are inherently

multiplicative63,66 and it is the proportional rather than absolute variation that

matters. The multiplicative influence of noise seen in the log-transformed data is

often misinterpreted as bias.203,263 Kerkhoff and Enquist106 maintain that the multiplicative error model is an appropriate feature, rather than a defect, of standard

allometry analysis. Recent research suggests that geometric error resulting from

multiplicative fluctuations should be the default standard for parameter estimation

in biology66,81 and not additive error.

3.2. Phenomenological distributions

We now consider the statistics of the fluctuations in the AR using data from the

literature. The data relating the average energy expended by a given species in watts

to the average TBM of that species in kilograms for 391 species of mammal is plotted

in Fig. 3 and also in Heusner91 as well as in Dodds et al.47 A fit of Eq. (3) to these

data that minimizes the mean-square error is a straight line on double logarithmic

graph paper and was found to have slope  ̄b = 0.71 ± 0.008 so that empirically

2/3 <  ̄b < 3/4 and the allometry coefficient  ̄a = 0.02. As West and West245 reviewed

Heusner90 had somewhat earlier questioned Kleiber’s value of 3/4 and concluded

from data analysis that this value of 3/4 was a statistical artifact. Feldman and

McMahon58 agreed with Heusner’s conclusions, but suggested that there was no

compelling reason for the intraspecies and interspecies allometric exponents to be

the same, with the intraspecies exponent being 2/3 based on geometric similarity

and the interspecies exponent being 3/4 based on McMahon’s elastic similarity.

There is a great deal of variability around the line segment that gives the AR

model in Fig. 3. West and West245 interpret these fluctuations as random variations

Fig. 3. The linear regression to Eq. (3) for Heusner’s data91 is indicated by the line segment.

The slope of the dashed line segment is 0.71 ± 0.008.

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in either the allometry coefficient or the allometry exponent. If the fluctuations are

assumed to be contained in the allometry coefficient we can define

a

a ̄ = eη = B

aM ̄  ̄b (23)

so that each data point in the (X, Y )-plane yields a single value of the allometry

coefficient. In a complex network a linear response such as suggested by Fig. 3 does

not necessarily occur since there can be independent fluctuations in both X and

Y resulting in what Warton et al.220 call equation error; also known as natural

variability, natural variation and intrinsic scatter. In the present case the AR is not

predictive, but instead summaries vast amounts of data.255 This natural variability

is manifest in fluctuations in the allometry parameters (a, b).

West and West242 calculate the statistical distribution for the random allometry

coefficient determined from Eq. (23) under the assumption that  ̄b is fixed. The

variability in the allometry coefficient determined by the data is partitioned into

20 equal sized bins in the logarithm of the allometry coefficient. A histogram is

then constructed by counting the number of data points within each of the bins

as indicated by the dots in Fig. 4. The solid line segment in this figure is the best

fit to these 20 numbers with minimum mean-square error. The functional form

for the histogram is indicated by the curve in Fig. 4241,242 and the quality of the

fit to the diversity data is determined by the correlation coefficient r2 = 0.98.

The normalized histogram G(ln a′

) on the interval (0, ∞) using the transformation

Fig. 4. The histogram of the deviations from the prediction of the AR using the data depicted

in Fig. 3 partitioned into 20 equal sized bins in the logarithm of the normalized variable a′ = a/a ̄.

Here  ̄a = 0.02 and  ̄b = 0.71. The solid line segment is the best fit of Eq. (24) to the 20 histogram

numbers, which yields the power-law index α = 2.79 and the quality of the fit is measured by the

correlation coefficient r2 = 0.98. (Reproduced from West and West242 with permission).

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G(ln a′)d ln a′ = P(a′)da′ gives the empirical pdf:

P(a′

) = α

2







1

a′1−α ; a′ ≤ 1

1

a′1+α ; a′ ≥ 1

(24)

and α = 3.28 yielding a standard deviation 0.017 in essentially exact agreement

with the empirical data. Note that this coefficient differs from the best fit value

given in the caption of Fig. 4 but this results in only a 1% change in the value

of r2.

The same inverse power-law form is obtained with α = 3.89 and r = 0.96 using

the avian BMR data of McNab142 for 533 species of bird. The distribution of the

deviations from the AR for both the avian and mammalian data sets fall off as

inverse power laws on either side of a =  ̄a. Equation (24) quantifies the qualitative

argument used earlier to associate power-law pdf ’s with multiplicative fluctuations,

see also Sec. 5.

Alternatively the fluctuations can be assumed to be contained within the allometry exponent as:

η = log(B/a ̄)

log M −  ̄b . (25)

If we assume  ̄b = 0.71 and a =  ̄a then Eq. (25) provides us with the statistical

fluctuations in the allometry exponent are used to construct a histogram exactly as

we did previously. The solid line segment in Fig. 5 is the best fit to the 20 numbers

of the histogram with minimum mean-square error. The functional form for the

histogram of deviations from the allometry exponent  ̄b is determined by the curve

in Fig. 5 and the quality of the fit to the histogram is determined by the correlation

coefficient r2 = 0.97. The histogram is fit by the Laplace pdf

Ψ(b) = β

2 exp[−β|b −  ̄b|] , (26)

with the empirical value β = 12.85.

3.2.1. Co-variation of allometry coefficient and exponent

We have obtained two separate distributions for two different parametric representations of the same data. As pointed out by Glaizer73 species within a taxon represent

a cloud of different metabolic rates and not the linear metabolic level determined by

linear regression. His cloud is incorporated into the present context by abandoning

the assumption that the allometry coefficient and exponent are independent and

requiring that the probability of a given fluctuation is the same regardless of the

representation so that P(a)da = Ψ(b)db. In order to calculate a nonzero Jacobian

of the transformation between the two allometry parameters requires that they be

functionally related. West and West243 assume b =  ̄b−c ln a, so that by inserting the

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Fig. 5. The histogram of the deviations from the prediction of the AR using the allometry

exponent fluctuation data partitioned into 20 equal sized bins. The solid line segment is the best

fit of Eq. (26) with ∆b ≡ b− ̄b, to the 20 histogram numbers, and the quality of the fit is measured

by the correlation coefficient r2 = 0.97. (Reproduced from West and West242 with permission).

allometry exponent pdf Eq. (26) into that for equal probabilities and simplifying

yields:

P(a) = βc

2







aβc−1 for a ≤ a ̄

1

aβc+1 for a ≥ a ̄

(27)

and comparing Eq. (27) with Eq. (24) they identify α = βc and obtain the empirical

pdf for the allometry coefficient P(a′

)da′ = P(a)da. Using the fitted values α = 2.79

and β = 12.85 yields c = 0.217 and consequently the empirical transformation can

be written as:

b = 0.71 − 0.50 log10 a . (28)

Note that this functional dependency of the allometry exponent on the allometry coefficient is consistent with the empirical co-variation relation obtained by Glazier73

who used linear regression on large amounts of parametric data. West and West243

emphasize that this co-variation of the allometry parameters implies that there is

no universal value for the allometry exponent.

3.2.2. Other scaling distributions

The tent shaped distribution in Fig. 5 also arises using a different approach to

quantifying the variability of BMR. Labra et al.115 investigate BMR fluctuations

by considering O2 volume time series and examining the scaling of the high frequency fluctuations across species. They determined empirically that the standard

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deviation in the BMR is proportional to the average BMR by the scaling law:

SDBMR ∝ Bλ

and λ = −0.352 ± 0.072. This relation is known as the power curve in the ecology

literature85 and was discovered by Taylor210 in his determination of the number of

new species that can be found in a given plot of ground. It was expanded to the

determination of scaling in time series by Taylor and Woiwold211 and applied in

physiology as Taylor’s law in West.238

On the other hand Labra et al.115 determined the standard deviation of BMR

to be proportional to a power of the TBM:

SDBMR ∝ Mγ

and γ = −0.241±0.103. Consequently, combining the two empirical expressions for

the variance we obtain:

B ∝ Mγ/λ (29)

with γ/λ = 0.69 a value consistent with the many fits made to the allometry data.

They determine that all the species they studied show the same invariant distribution of BMR fluctuations, regardless of the difference in their phylogeny, physiology

and body size. The distribution has the form Eq. (26) with the independent variable given by the fluctuation in the BMR ∆B and β = √2/SDBMR; in terms of the

scaling variable Bsc ≡ √2∆B/SDBMR curves for 12 different species collapse onto

a single universal curve.

3.2.3. Paleobiology and scaling

Phenomena with intermittent properties described by inverse power-law statistics

are apprarently ubiquitous233 and paleobiology is no exception. Bak and Boettcher8

interpreted Charles Lyell’s125 uniformitarianism as meaning that all geologic activity should be explainable in terms of readily available processes working at all times

and all places with the same intensity. They go on to argue that the existence of

earthquakes, volcanic eruptions, floods and tsunamis all indicate that the physical

world is not in equilibrium. Moreover the intermittent nature of the paleontological

record indicates that macroevolution is also out of equilibrium and consequently

the inverse power-law statistics are possibley suitable for their description.

Eldredge and Gould52 argued that punctuated change dominates the history of

life and that relatively rapid episodes of speciation constitute biological macroevolution. The intermittency of speciation in time has been explained by one group

as punctuated equilibria51 and has been indirectly related to fractal statistics by

identifying it as a self-organized critical phenomenon.8 In the self-organized criticality model of speciation Bak and Boettcher8 associate an avalanche of activity

with exceeding a threshold and the distribution of returns to the threshold with a

“devil’s staircase” having a distribution of steps of stasis of lengths given by the

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D. West & B. J. West

inverse power-law pdf, that is, T −γ with γ = 1.75. Moreover, as explained by Sneppen et al.,

206 the number of genera N, with a lifetime T can be fitted very well

to a power law T −β with β ≈ 2.173 More recently Rikvold and Zia179 put forward

an explanation of punctuated equilibrium based on 1/f noise in macroevolutionary

dynamics that also yields an inverse power-law pdf for the life time of ecological

communities with β = 2.

Sol ́e et al.208 analyzed the statistics of the extinction fossil record (time series)

and determined that the power spectrum has the form:

S(f) ∝ 1/fμ ,

with 0 <μ< 2. They find 0.80 ≤ μ ≤ 0.90 and argue that these values support

the self-organized criticality interpretation of extinction. On the other hand, Plotnick and Sepkowski165 also find a 1/f power spectrum with a power-law index for

extinction consistent with Sol ́e et al.208 and indices for species generation of approximately half that for extinction. However the latter authors conclude that their

results are incompatible with self-organized criticality and instead are compatible

with multifractal self-similarity in both the extinction and generation records.

4. Models of Allometry

Two distinct methods dominate the many derivations of AR. One method is based

on the first-principles reductionistic approach starting from an assumed form for the

underlying mechanisms and from that deducing the necessity of Eq. (2). The other

method is phenomenological involving statistical analysis and identifying patterns

in the data analysis and from these patterns deducing the necessity of the empirical

AR. A number of the statistical methods were reviewed in the previous section. The

ARs stand out as empirical relations that have withstood the test of time, whereas

the same cannot be said for the models developed to explain how they come about.

In this section we examine attempts to formulate general principles from which the

underlying mechanisms, whether reductionist or statistical, producing the ARs can

be identified.

The search for an unifying principle parallels such historical activities as the

determination of “least action” in analytic mechanics or optimization in control

theory. More recently investigators have rediscovered Newton’s “principle of similitude”’ introduced in the Principia (II, Proposition 32). Scaling and the principle of similitude have been present in the study of complex physical phenomena

since physics became a science. In modern times it is RG theory that provides a

formalism for determining how force is transferred across multiple scale.103,111,259

Part of the reason for exploring this approach is that fractal geometry and fractal statistics are able to capture some of the regularity observed in vast amounts

of data in the life and social sciences in addition to the physical sciences. The

implementation of fractal geometry and RG theory to study the architecture of

physiological forms,17,228,247 interacting networks of chemical reactions45,78,169 and

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On Allometry Relations

the topology of ecological webs27 over the past quarter century has lead to some

remarkable insights. In particular the descriptive success of investigations using fractals15,131,135,143 suggests that branching networks with fractal architectures have an

evolutionary advantage230,231,235 as do fractal stochastic processes.236

4.1. Optimization principles

Optimization provides one of the best recipes for determining network dynamics

consistent with a given set of constraints. It is through the specification of the

constraints that insight into a phenomenon is introduced. The biological sciences

employ ideas such as energy minimization; the minimal use of materials along with

the maximization of efficiency. So whether the extremum is a minimum or maximum

depends on one’s purpose and the quantity being varied.

4.1.1. Energy minimization

Scaling relations in living networks result from the balancing of various constraints.

One such biological balancing is that of energy utilization against the energy cost of

carrying out a biological function. A technique that has been used both implicitly

and explicitly in the derivation of ARs is supplying nutrients to various parts of

an organism through the venous and capillary networks as well as through the

respiratory network. Murray150 considered a fluid with viscosity ν and laminar flow

Q within a tube of length l and radius r. The flow had to overcome the vascular

resistance that generates a pressure difference along the length of the tube given

by Poiseuille’s law ∆p = 8lνQ/πr4.

A constraint on the flow is the cost of transporting fluid of cylindrical volume

V = πlr2 along the tube so that introducing c as the cost factor the total work to

be done per unit time is given by:

E = Q∆p + cV = 8lνQ2

πr4 + cπlr2 . (30)

Consequently, minimizing this expression with respect to the radius yields the optimal flow: Q = Cr3 with the constant C = cπ2/16ν. The cubic dependence of the

flow rate on radius is known as Murray’s law223 and is indicative of the maximum

efficiency of the flow.

Murray151 subsequently extended his result to bifurcating networks such as

occurs in bronchial airways. The flow from a parent vessel of radius r0 branches

into two daughter vessels r1 and r2 such that the flow divides:

Q0 = Q1 + Q2 .

Therefore inserting Murray’s law into this expression yields da Vinci’s equation with

α = 3 and in the case of equal radii in the daughter branches r1 = r2 we obtain

the scaling relation r1 = 2−1/3r0. Thus, the maximally efficient bifurcating network

in terms of energy transport cost has radii decreasing as the cube root of two. As

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Fig. 6. Sketch of a branching structure such as a blood vessel or bronchial airway with the

parameters used in a bifrucating network model.

pointed out by Weibel223 this last result is also known as Murray’s law. However

this law was actually first formulated by Hess89 for blood vessels and subsequently

is more properly called the Hess–Murray law.

4.1.2. Optimal design

Rashevsky172 introduced the principle of optimal design in which the material used

and the energy expended to achieve a prescribed function is minimal. He applied the

principle to the basic problem of how the arterial network could branch in space

in order to supply blood to every element of tissue. To address this problem he

used the model of a bifurcating branching network supplying blood to a restricted

volume and reducing the total resistance to the flow of blood. His purpose was to

determine the condition imposed by the requirement that the total resistance is

minimum.

Here we assume the branching network is composed of N generations from the

point of entry (0) to the terminal branches (N). A typical tube at some intermediate

generation k has length lk, radius rk and pressure drop across the length of the

branch ∆pk as sketched in Fig. 6. The volume flow rate Qk is expressed in terms

of the flow velocity averaged over the cross sectional area uk : Qk = πr2

kuk. Each

tube branches into n smaller tubes with the branching of the vessel occurring over

some distance that is substantially smaller than the lengths of the tubes of either

generation. Consequently, the total number of branches generated up to generation

k is Nk = nk. The pressure difference at generation k between the ends of a tube

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On Allometry Relations

is given by a suitably indexed version of Poiseuille’s law and the total resistance to

the flow is given by the ratio of the pressure to flow rate

Ωk = ∆pk

Qk

= 8νlk

πr4

k

. (31)

The total resistance for a network branch with m identical tubes in parallel is

1/m the resistance of each individual tube. Thus, in this oversimplified case we can

write the total network resistance as:

ΩT = 8νl1

πr4

0

+

8ν

π



N

j=1

1

Nj

lk

r4

j

. (32)

In order to minimize the resistance for a given mass Rachevsky first expressed the

initial radius r0 in terms of the total mass of the network. The optimum radii

for the different branches of the bifurcation network having the total mass M are

then determined such that the total resistance is a minimum ∂ΩT /∂rj = 0 yielding

the equality rk = N −1/3

k r0. The ratio of the radii between successive generations

is rk+1/rk = (Nk/Nk+1)1/3so that inserting the number of branches at the kth

generation Nk = nk yields:

rk+1/rk = n−1/3 , (33)

yielding an exponential reduction in the branch radii across generations. Note the

formal similarity of this ratio to Horton’s law in which the ratio of numbers of river

branches is independent of generation number.

Rashevsky considered the bifurcating case n = 2 where the ratio of radii reduces

to rk+1/rk = 2−1/3 = 0.794. This is the classic “cube law” branching of Thompson213 in which he used the “principle of similitude”. The value 2−1/3 was obtained

by Weibel and Gomez222 for the reduction in the diameter of bronchial airways for

the first ten generations of the bronchial tree. However they noted a sharp deviation away from this constant fractional reduction after the tenth generation. The

value 2−1/3 was also obtained by Wilson260 who explained the proposed exponential decrease in the average radius of a bronchial tube with generation number by

showing that this is the functional form for which a gas of given composition can

be provided to the alveoli with minimum metabolism or entropy production in the

respiratory musculature. He proposed minimum entropy production69 as the design

principle for biological networks to carry out a given function.

The deviation from classical (exponential) scaling above generation ten shown in

Fig. 7 was eventually explained using an alternative model of the bronchial airways

in terms of fractal statistics228 as we subsequently discuss.

4.2. Metabolic allometry

Barenblatt and Monin14 proposed that metabolic scaling might be a consequence

of the fractal nature of biology, but they did not provide a mechanistic model for its

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Fig. 7. As the bronchial tree branches out, its tubes on average decrease in size. A theory consistent with the Principle of Similitude213 predicts that their diameters should decrease by about

the same ratio from one generation to the next; exponential decline. This semilog graph shows

measurements from Weibel and Gomez222 for 23 generations of bronchial tubes in the human

lung. The prediction is a straight line that fits the anatomic data (dots) until about the tenth

generation, after which they deviate systematically from an exponential decline. [adapted from

West and Goldberger.229]

description. This shortcoming has been overcome by a number of investigators who

have devised numerous fractal models to describe AR in a variety of contexts.228,247

4.2.1. Elastic similarity model

The first model to analytically predict the allometry exponent 3/4 was constructed

by McMahon.139,140 His argument rests on the observation that the weight of a column increases more rapidly with size than does its strength. Moreover as discussed

by Schmidt–Nielsen195 if the column is tall and slender it can fail due to elastic

buckling in which small lateral displacements exceed the elastic restoring forces.

For a sufficiently slender column with Young’s elastic modulus E and density ρ the

critical length of a column of diameter d is:

lcr = k(E/ρ)

1/3d2/3 (34)

and k is a constant. The elastic criteria of McMahon is therefore given by l

3 ∝ d2.

The weight of the column is given by the product of the density, length and crosssectional area,

Mg = ρd2/3πd2/4 ,

where the length has been replaced using the elastic criteria yielding d ∝ M3/8.

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The implications for allometry of the scaling between cylinder diameter and

mass was discussed by Calder31 using the symmorphosis hypothesis of Taylor and

Weibel:212 “... is regulated to satisfy but not exceed the requirements of the functional system”.237 First off we recognize that locomotion requires the contraction

of muscles. During contraction, muscles exert a force that increases with the crosssectional area of the muscle. The power output of the muscle is the work done per

unit time and may be equated with the metabolic rate, the product of the force

generated by the muscle and the velocity u of the shortening of the muscle B ∝ d2u.

The velocity of the shortening of the muscle appears to be a size-independent constant from species to species92 so that using these two equations for the diameter

yields:

B ∝ M3/4 .

Consequently, the allometry exponent 3/4 in McMahon’s elastic similarity model

is required to maintain the flow of energy to the working muscles and is consistent

with the principle of symmorphosis.

Versions of the above argument given by both Calder31 and Schmidt–Nielsen195

seem to explain the value of the allometry exponent for warm blooded animals.

Dodds et al.

47 critique McMahon’s model by noting that there is no compelling

reason why the power output of muscles should be the dominant factor in the scaling

of BMR. Moreover, Savage et al.

189 point out that while the elastic similarity model

might apply to the bones of mammals or the trunks of trees that have adapted to

gravitational forces, it is doubtful that it is applicable to aquatic or unicellular

organisms that also display an allometry exponent of 3/4.87

4.2.2. WBE model

West, Brown and Enquist247 (WBE) published a quantitative model of metabolic

AR that has had significant impact on how a significant fraction of today’s biology/ecology community understands metabolic ARs. WBE model nutrient distribution within a hierarchal network in which vessels become narrower, shorter

and more numerous between successive levels proceeding from the initial to the

terminal level reminiscent of representations of river branchings. The scaling in

the transport network is a consequence of the constraints imposed by three assumptions: (1) The entire volume of the organism is crammed with a space-filling

branching network. (2) The tube properties at the terminus of the network are

size-invariant. (3) The energy required to distribute resources using this network

is minimal, that is, the hydrodynamic resistance of the network is minimized. We

note their claim that this model is the origin of universal scaling laws in biology248,249 with b = 3/4. However we recall at the start that the existence of an

empirical exponent b = 3/4 for metabolic AR has been questioned by numerous investigators47,91,113,114,128 and we address these concerns and others in due

course.

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WBE introduce two parameters to characterize the network branching process,

βk = rk+1/rk the ratio of successive branch radii, as was done in the energy minimization arguments172 and the other is γk = lk+1/lk, the ratio of successive branch

lengths. They conclude that the total number of terminal branches scales with TBM

as NT = (M/M0)b. The total number of branches at level k is Nk = nk. Consequently at the network terminus the self-similarity of the network yields NT = nN

and consequently the total number of branches is:

N = b ln(M/M0)/ ln n . (35)

Rashevsky’s energy minimization argument indicates that the transport of nutrients

in complex networks is maximally efficient when βk is independent of k. This is the

“fractal” scaling assumption made by WBE.

The estimates of the ratio parameters required two separate assumptions. To

estimate the ratio of lengths WBE assume that the volume of a tube at generation k

can be replaced by a spherical volume of diameter lk and in this way implement the

space-filling assumption. The conservation of volume between generations therefore

leads to the space filling condition:

(lk+1/lk)

3 ≈ Nk/Nk+1 = n−1

and consequently to the k-independent parameter

γ = γk = n−1/3 .

They maintain that this level-independent scaling of the lengths is a generic property of all the space-filling networks they consider. This condition for an n-branching

network with a reduction in size of n−1/3 between successive generations yields using the definition for the fractal dimension D = log n/ log n1/3 = 3.

A separate and distinct assumption is made to estimate β using the classic

rigid-pipe model to equate the cross-sectional areas between successive generations:

πr2

j = nπr2j+1, so that we have

β = βk = n−1/2 .

Note that this scaling of β differs from the ratio parameter obtained using energy

minimization. Inserting these values of the scaling parameters into the expression

for b yields the sought after exponent b = 3/4. This value of the exponent in turn

determines a number of other quarter-power scaling laws.

WBE point out that this is strictly a geometrical argument applying only to

those networks that exhibit area-preserving branching. Moreover the fluid velocity

is constant throughout the network and it is independent of size. They go on to

say that these features are a natural consequence of the idealized vessel-bundle

structure of plant vascular networks in which area-preserving arises automatically

because each branch is assumed to be a bundle of nN−k elementary vessels of the

same radius. They recognized that this is not the situation with vascular blood

flow where the beating of the heart produces a pulsating flow that generates a very

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On Allometry Relations

different kind of scaling. Area-preserving is also not true in the mammalian lung

where there is a distribution of radii at each level of branching.

A physical property that the area preserving condition violates is that blood

slows down in going from the aorta to the capillary bed. Here WBE return to the

principle of energy minimization and as stated by West248 assert that to sustain a

given metabolic rate in an organism of fixed mass, with a given volume of blood,

the cardiac output is minimized subject to a space-filling geometry. This variation

is essentially equivalent to minimizing the total impedance since the flow rate is

constant and again yields the Hess–Murray law β = n−1/3 corresponding to areaincreasing branching.6,190 This change in scaling from the area-preserving n−1/2

to the area-increasing n−1/3 solves the problem of slowing down blood flow to

accommodate diffusion at the capillary level. Moreover, the variation also leads to

an allometry exponent b = 1. Such an isometric scaling (b = 1) suggests that plants

and animals follow different allometry scaling relations as was found.174,201

A detailed treatment of pulsate flow is not straightforward and will not be

presented here, but see Savage et al.190; Silva et al.201 and Apol et al.6 for details

and commentary in the context of the WBE model. We do note that for blood flow

the walls of the tubes are elastic and consequently the impedance is complex, as is

the dispersion relation that determines the velocity of the wave and its frequency.

Consequently pulsate flow is attenuated32,60 and WBE argue that the impedance

changes its r-dependence from r−4 for large tubes to r−2 for small tubes. The

variation therefore changes from area-preserving flow β = n−1/2 for large vessels to

dissipative flow β = n−1/3 for small vessels where blood flow is forced to slow. Thus

βk is k-dependent in the WBE model for pulsate flow and at an intermediate value of

k the scaling changes and this changeover value is species dependent. These results

are contradicted in the more extensive analysis of pulsate flow by Apol et al.,

6 who

conclude that Kleiber’s law remains theoretically unexplained.

Kozlowski and Konarzewski113 critique the apparent limitations of the WBE

model assumptions. The size-invariance assumption has been interpreted by them

to mean that NT ∝ M, that is, the terminal number of vessels scales isometrically

with size and consequently causes the number of levels to be a function of body

size since more levels are required to fill a larger volume with the same density of

final vessels. However Brown et al.28 assert that NT VT ∝ M so that

NT ∝ M3/4 (36)

to which Kozlowski and Konarzewski114 respond that such mass dependence is

an arbitrary assumption and is not proven, see also Dawson.43 Etienne et al.54

reconstruct the WBE model without making the self-similarity assumption and in

so doing satisfy the concerns of Kozlowski and Konarzewske.113,114 The arguments

remain unresolved and Cyr and Walker38 refer to the assumption embodied in

Eq. (36) as the illusion of mechanistic understanding and maintain that after a

century of work the jury is still out on the magnitude of the allometry exponents.

Riisg ̊ard180 argues that respiration and growth are integrated through the energetic

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costs of growth and that this explains why the b value is not a “natural constant”

and that a “3/4 power scaling law cannot be deduced from the interplay between

pure physical and geometric constraints of the transport of oxygen.

A quite different critique comes from Savage et al.190 who emphasize that the

WBE model is only valid in the limit N → ∞, that is, for infinite network size (body

mass) and that the actual allometry exponent predicted depends on the sizes of the

organisms considered. They calculate that the AR between BMR and TBM have

corrections for finite N given by:

M = a1B + a2B4/3 (37)

from which it is clear that b = 3/4 only occurs when a2B1/3 ≫ a1. This inequality is not satisfied for bodies of finite size. In their original publication WBE acknowledged the potential importance of such finite-size effects, especially for small

animals, but the magnitude of the effect remained unspecified. Using explicit expressions for the coefficients in the WBE model Savage et al.

190 show that when

accounting for these corrections over a size range spanning the eight orders of magnitude observed in mammals a scaling exponent of b = 0.81 is obtained. Moreover in

addition to this strong deviation from the desired value of 3/4 there is a curvilinear

relation between the TBM and the BMR in the WBE model given by:

ln M = ln a2 +

4

3

ln B + ln 1 + a1

a2

B−1/3



,

whose curvature is opposite to that observed in the data. Consequently they conclude that the WBE model needs to be amended and/or the data analysis needs

reassessment to resolve this discrepancy. A start in this direction has been made

by Kolokotrones et al.110 Agutter and Tuszynski5 also review the evidence that the

fractal network theory for the two-variable AR is invalid.

Another variation on this theme was made by Price et al.168 who relax the

fractal scaling assumptions of WBE and show that allometry exponents are highly

constrained and covary according to specific quantitative functions. Their results

emphasize the importance of network geometry in determining the allometry exponents and supports the hypothesis that natural selection minimizes hydrodynamic

resistance. Moreover they extended McMahon’s elastic similarity model and apply it to plant scaling exponents showing its consistency with their modification of

WBE.

Prior to WBE there was no unified theoretical explanation of quarter-power

scaling. Banavar et al.12 show that the 3/4 exponent emerges naturally as an upper

bound for the scaling of metabolic rate in the radial explosion network and in

the hierarchical branching networks models and they point out that quarter-power

scaling can arise even when the underlying network is not fractal.

Finally, Weibel224 presents a simple and compelling argument on the limitations

of the WBE model in terms of transitioning from BMR to the maximal metabolic

rate (MMR) induced by exercise. The AR for MMR has an exponent b = 0.86

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rather than 3/4, so that a different approach to determining the exponent is needed.

Painter158 demonstrates that the empirical allometry exponent for MMR can be

obtained in the manner pioneered by WBE by using the Hess–Murray law for the

scaling of branch sizes between levels.

Weibel224 argues that a single cause for the power function arising from a fractal

network is not as reasonable as a model involving multiple causes, see also Agutter and Tuszynski.5 Darveau et al.41 propose such a model recognizing that the

metabolic rate is a complex property resulting from a combination of functions.

West et al.251 and Banavar et al.11 demonstrate that the mathematics in the distributed control model of Darveau et al.41 is fundamentally flawed. In their reply

Darveau et al.42 do not contest the mathematical criticism and instead point out

consistency of the multiple-cause model of metabolic scaling with what is known

from biochemical205 and physiological101 analysis of metabolic control. The notion

of distributed control remains an attractive alternative to the single cause models of metabolic AR. A mathematically rigorous development of AR with fractal

responses from multiple causes was recently given by Vlad et al.218 in a general

context. This latter approach may answer the formal questions posed by many of

these critics.

4.3. Why fractal transport?

Why are fractals important in the design of allometry networks? Barenblatt and

Monin14 suggested that metabolic scaling might be a consequence of the fractal nature of biology and WBE determined that fractal geometry maximizes the efficiency

of nutrient transport in biological networks. Weibel223 maintains that the fractal

design principle can be observed in all manner of physiologic networks quantifying

the observations and speculations of Mandelbrot,131,132 as does West.238

West230 conjectured that fractals are more adaptive to internal changes and to

changes in the environment than are classical processes and structures. Consider a

network property characterized by classical scaling at the level k such as the length

or diameter of a branch Fk ∝ e−λk compared with a fractal scaling characterization

of the same property Fk ∝ k−λ. What is significant in these two functional forms for

the present argument is the dependence on the parameter λ. The exponential has

emerged from a large number of optimization arguments and the inverse power-law

results from RG scaling arguments.

Assume the parameter λ is the sum of a constant part λ0 and a random part ξ.

The random part can arise from unpredictable changes in the environment during

morphogenesis, non-systematic errors in the code generating the physiologic structure or any of a number of other causes of irregularity. Thus, regardless of whether

the errors are induced internally or externally, the average is taken over an ensemble

of zero-centered Gaussian fluctuations ξ with variance σ2/2. Note that the choice

of Gauss statistics has no special significance here except to provide closed form

expressions for the averages to facilitate discussion. The relative error generated by

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Fig. 8. The error between the model prediction and the prediction averaged over a noisy parameter is shown for the classical model (upper curve) and the fractal model (lower curve).

the fluctuations is given by the ratio of the average value to the function in the absence of fluctuations, yielding the relative error for classical scaling εk = exp[σ2k2]

and for fractal scaling εk = exp[σ2(ln k)2].

The two error functions are graphed in Fig. 8 for fluctuations with a variance

σ2 = 0.01. At k = 15 the error in classical scaling is 9.5. This enormous relative error

means that the perturbed average property at generation 15 differs by nearly an

order of magnitude from what it would be in an unperturbed network. A biological

network with this sensitivity to error would not survive for very long in the wild. For

example, the diameter of a bronchial airway in the human lung could not survive

this level of sensitivity. However, the property of the fractal network only changes

by 10% at the distal point k = 20. The implication is that the fractal network is

relatively unresponsive to fluctuations.

A fractal network is consequently very tolerant of variability. This error tolerance

can be traced back to the broadband nature of the distribution in scale sizes of a

fractal object. This distribution ascribes many scales to each generation in the

network. The scales introduced by the errors are therefore already present in a

fractal object. Thus, the fractal network is preadapted to variation and is therefore

insensitive to change.230,231 These conclusions do not vary with modification in the

assumed statistics of the errors. Therefore let us review how fractal statistics have

been employed in the understanding of some nonmetabolic physiologic ARs.

4.3.1. WBG bronchial tree

The energy minimization argument applied to a branching network resulted in

Horton’s law. However, the theoretical arguments justifying this empirical law and

others like it assume that the network is geometrically self-similar. Real networks,

such as the bronchial tree in the mammalian lung, do not have such deterministic regularity. At any generation of the bronchial airway there is a distribution of

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lengths and diameters, with a reduction in the average length and average diameter between successive generations. Consequently, the simple characteristic scale

governing the decrease in bronchial dimensions across generations given by the

Hess–Murray law must be reexamined. West, Barghava and Goldberger228 (WBG)

assumed the mammalian lung to be a fractal structure having a distribution of

scales contributing to the variability in tube size at each generation. This model

yields an average bronchial diameter that decreases with generation number, not

as an exponential as suggested by Weibel and Gomez,222 but as an inverse power

law.229,232,233

WBG assume that the variability in the diameters of the bronchial tubes at

generation k is given by d(γk), where γ is a random variable that determines the

size of the diameter. In the energy minimization argument we determined that

γ = ln 2/3 and the classical reduction in diameter with generation number was

exponential d(γk) = d0e−γk as shown in Fig. 7. However with real data this is the

reduction in the average diameter just as it was for other allometry networks so that

the random variation in diameter at generation k is smoothed over to determine the

average. Formally the behavior of the average diameter as a function of generation

number is given by:

d(k) = d(γk)p(γ)dγ , (38)

with p(γ) the pdf in scale size. Rather than prescribe a particular functional form to

this pdf in the formal definition WBG use a RG argument originally developed in an

economic context by Montroll and Shlesinger.148 Assuming the original distribution

of scales has an average value  ̄γ the RG argument constructs a new distribution with

an infinite number of new scales each a factor of β larger than the preceding scale

and each occurring with a relative frequency ζ less often. The distribution resulting

from the RG relation gives rise to an empirical average denoted by brackets in terms

of the formal average Eq. (38) as follows234,235:

�d(k)� = ζ�d(βk)� + (1 − ζ)d(k). (39)

The dominant behavior for the solution to the RG relation Eq. (39) is determined

by the singular part of the solution to the equation, which for simplicity WBG write

with a subscript �d(k)�s = ζ�d(βk)�s. Shlesinger and West200 note that the solution

to this scaling equation that provides the best fit to the data can be written as the

real part of:

�d(k)�s = ∞

n=−∞

An

kμn ; μn = b + i2πn/ ln β ; (40)

μn is the complex fractal dimension of order n for the branching process, with the

exponent b = − ln ζ/ ln β > 0 and the An are fit by experimental data. Of course

there is in addition to Eq. (40) an analytic part to the solution that WBG calculated

but which becomes negligible with increasing k.

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Thus, the average diameter is an inverse power law in the generation number

modulated by a slowly oscillating function just as observed in the data depicted in

Fig. 9. This fractal model of the bronchial airway provides an excellent fit to the

average bronchial tube diameter data in four distinct species: dogs, rats, hamsters

Fig. 9. Top: Reploted data from Fig. 7 on log–log scales for humans. Center: The harmonic

variation in measurements for dogs, rats and hamsters taken by Raabe et al. (1976). Bottom: The

average diameters for the bronchial tubes of a hamster and RG (fractal) model prediction using

Eq. (40). [Adapted from West and Goldberger228].

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and humans.152 The quality of the fit is shown for hamsters in Fig. 9, which restricts

the summation index in Eq. (40) to n = 0, 1, −1 and A1 = A−1 yields a single

oscillation of the modulation function fit to the data.

The form of Eq. (40) strongly suggests that the RG relation for the average diameter captures a fundamental property of the structure of the lung that is distinct

from classical scaling. There is an AR at each level of the bronchial tree with the

average diameter decreasing as k−b with the level being a measure of size. Moreover

the data show the same type of scaling for bronchial tube lengths and consequently

volume.

4.3.2. Distribution of bronchial diameters

Kitaoka and Suki107 use a related but distinct analysis of the lung data depicted

in Fig. 9 in which they investigate the statistical variability of the bronchial airway

diameters. Their analysis was based on the assumed relation between the airflow

rate and the airway diameter d:

Q = Cdα ,

a relation discussed in terms of energy optimization by Murry150 yielding α = 3.

The relation was also considered in a bronchial airway context by a number of other

investigators.82,105,163 The data of Raabe et al.171 reveals that the flow rate is a

random variable and that the cumulative probability for the flow rate to exceed Q

is given by:

Pr(≥ Q) ∝ 1/Q

with an inverse power-law index of −1 to three significant figures. Note that from

the additive form of the flow rate between successive generations and the da Vinci

relation between parent and daughter branches that the pdf for the diameter of the

bronchial airway is determined to be:

p(d) = Cd−(α+1) , C = α(d0dT )α

dα

0 − dα

T

(41)

with d0 the diameter of the thorax and dT the diameter of the smallest bronchial

airway. Here again the empirical distribution for the observable, the bronchial tube

diameter rather than the allometry coefficient, is an inverse power-law (Pareto) pdf.

Note that in the previous subsection the average diameter was expressed as an

inverse power law in the generation number. The distribution in the size of the

bronchial airway Eq. (41) does not directly depend on generation number and is an

inverse power law in diameter size, that is, in millimeters as depicted in Fig. 10. The

data used to construct Fig. 9 was also used to determine the cumulative distribution

of airway diameters and yields the inverse power-law distribution with α = 3.1 with

correlation coefficient of r2 = 0.988. Note that this value of the exponent is slightly

different from the optimum as we discuss in the next section. It is also noteworthy

that the oscillations around the inverse power law given by the straight line are the

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Fig. 10. Cumulative distribution of diameters on log–log plot. The regression model over the full

range of diameters yields α = 3.1 with r2 = 0.988. [Reproduced from Kitaoka and Suki107 with

permission].

same periodic variation explained using WBG theory except that the oscillations

are now a function of distance rather than generation number.

4.3.3. The optimum can be dangerous

A question of interest is whether optimal design criteria are ever realized in nature

and whether they are even desirable. The WBE model suggests that the allometry

exponent value of 3/4 is proof of the optimally of fractal design of networks for nutrient transport. However the controversy over the empirical value of the allometry

exponent calls this implication into question. The present discussion regarding the

mammalian lung and whether the bronchial tree is optimal has been investigated

by Mauroy et al.138 They maintain that the bronchial tree of most mammalian

lungs is a good example of an efficient distribution network with an approximate

fractal structure.152,228 They state that physical optimization is critical in that

small variations in the geometry can induce large variations in the net air flux and

consequently optimality cannot be a sufficient criterion for physiologic design of the

bronchial tree. The slight deviations observed in the parameters presumed to be

optimized are a manifestation of a safety factor being encorporated into the design

and into the capacity for regulating airway caliber.

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In the present context the size ratio h of successive airway segments are homothetic with h = 2−1/3 ≈ 0.79 as discussed earlier. Homothetic scaling means that the

lengths and diameters have the same ratios between successive generations. Using

the resistance minimization argument for the bronchial network Mauroy et al.138

show that the “best” bronchial tree is fractal with constant reduction factor given

by the Hess–Murray law. Do the data support this optimal value and if not what

does that imply about the efficiency of bronchial airways?

The fractal dimension for a bronchial airway is D = − ln 2/ ln h so that the

Hess–Murray law implies D = 3 whereas h > 0.79 implies D > 3. In the human

lung it is found that the homothety ratio is h ≈ 0.85224 and the bronchial network

is therefore not optimized: its volume is too large and its overall resistance is too

small. Mauroy et al.138 emphasize that this deviation from optimal is, in fact, a

safety margin for breathing with respect to possible bronchial constrictions.

Sapoval187 has argued that without regulation of the airway caliber170 there

would be a multifractal spatial distribution of air within the lungs, resulting in

strongly nonuniform ventilation with some regions of the lung being poorly fed

with fresh air. Expanding on this theme using inhomogeneity of the homothety

ratio Mauroy et al.138 show how the optimal network is dangerously sensitive to

physiological variability and consequently design of the bronchial tree must incorporate more than just physical optimality. This argument has clear implications for

other allometric networks.

5. Probability Calculus and ARs

A fractal processes is one that is rich in scales with no one scale being dominant.

Thus, information in fractal phenomena is coupled across multiple scales manifesting long-time memory, as for example, observed in the architecture of the mammalian lung,152,223,228 manifest in the long-range correlations in human gait84,235

and measured in the human cardiovascular network,161 all of which are discussed in

West.238 These phenomenological characterizations of fractal time series relate back

to the observation made earlier that if X(t) and Y (t) are stochastic time series then

Y (X) must be a random function of its random argument as well. Consequently

the general approach to determining the relation between such variables is through

their pdf ’s.

In this section we examine pdf ’s that can produce the interspecies metabolic

ARs as a relation between moments. To do this requires a brief review of some

methods from nonequilibrium statistical physics118,175 and their extension to fractional equations of motions.108,236

5.1. Stochastic differential and Fokker–Planck equations

There are two major techniques available in statistical physics for the modeling of

stochastic phenomena. The first technique uses the dynamic equations constructed

by Langevin117 who introduced uncertainty through a random force in the equations

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of motion. The second technique is based on the phase space evolution of the pdf

using the Fokker–Planck equation (FPE). The conditions under which these two

methods are equivalent have been shown in a number of places, see, for example,

Lindenberg and West.118

5.1.1. Additive fluctuations

Consider the dynamics of a simple exponential relaxation process Z(t) that is disrupted by a random force ξ(t):

dZ(t)

dt = −λZ(t) + ξ(t). (42)

We assume the statistics of the random force are normal with a delta correlated

correlation of strength D

�ξ(t)ξ(t

′

)� = 2Dδ(t − t

′

). (43)

An alternative description of the dynamics is given by the evolution of the probability density in phase space where P(z, t)dz is the probability that the dynamic

variable Z(t) has a value in the interval (z,z + dz) at time t given a value q(t) at

the origin z = 0. The phase space dynamics are formally expressed by the equation:

G

 ∂

∂t, ∂∂z 

P(z, t) = q(t)δ(z). (44)

The analytic function G(·, ·) of the indicated operators along with the inhomogeneous term q(t) determine the dynamics of the process modeled by the pdf.

A phase space equation for a classical diffusion process with linear dissipation

equivalent to Eq. (42) is determined by:

G

 ∂

∂t, ∂∂z 

= ∂

∂t − ∂∂z 

λz + D ∂

∂z 

and q(t)=1 . (45)

The corresponding FPE is given by217:

∂P(z, t|z0, t0)

∂t = ∂∂z 

λz + D ∂

∂z 

P(z, t|z0, t0), (46)

where P(z, t|z0, t0)dz is the probability that the dynamic variable lies in the phase

space interval (z + dz, z) at time t conditional on Z(0) = z0 at time t = t0. The

solution to Eq. (46) for t0 = 0 is given by:

P(z, t|z0) = 1

4πσ2(t)

exp −(z − �z;t�)2

2σ2(t)



(47)

with average value �z;t� = z0e−λt and variance σ2(t) = D/λ(1 − e−λt). In the

absence of dissipation the average value is constant and the variance grows linearly

in time. However for finite dissipation the average decays exponentially in time

and the variance approaches a constant value given by the ratio of the diffusion

coefficient and the dissipation rate. If Z is the velocity of a Brownian particle the

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On Allometry Relations

variance would be proportional to the temperature of the surrounding fluid resulting

in the Einstein relation D/λ = kBT , see, for example, F ̈urth.61

Multiplicative fluctuations can be addressed by examining the FPE associated

with a linear stochastic dissipation parameter.118,193 Consider the nonlinear rate

equation with multiplicative fluctuations

dX

dt = −λX ln X + ξ(t)X (48)

for which the logarithmic transformation Z = ln X yields the rate equation with

additive fluctuations ξ(t) given by Eq. (42). The FPE for the transformed equation

is Eq. (46) with the solution Eq. (47). In terms of the original variable the pdf is a

log-normal distribution, for more details see Goel et al.76

5.1.2. Stochastic ontogenetic growth model

A more interesting rate equation was first considered by von Bertalanffy219 who

postulated a simple nonlinear rate equation to describe the growth of TBM m of

the form

dm

dt = Aηmη − Cβmβ (49)

at time t where η and β are unspecified exponents, and Aη and Cβ are positives

constants. The solution to this equation was studied by von Bertalanffy219 with

η = 2/3 and β = 1. More recently West et al.250 constructed Eq. (49), called an

ontogenetic growth model (OGM), from a conservation of energy argument and

obtained a universal curve from the solution and fit it to data by using η = 3/4

and β = 1. In the OGM the parameters are given by Cβ = Bm/Em with Bm the

metabolic rate required to maintain an existing unit of biomass; Em the metabolic

energy required to create a unit biomass and Aη = a/Em. Banavar et al.10 obtained

an equivalent universal curve and fit the same data for both η = 2/3, 3/4 and β = 1.

The OGM equation has the scaling form:

dm

dt = mηf(m/M0), (50)

where M0 is chosen to be the maximum TBM that solves the stationary equation

dm/dt = 0. The data from 13 organisms graphed as r = (m/M0)1−η versus τ =

− ln(1 − r) collapse onto a single universal curve250 that is fit equally well with

η = 2/3 or η = 3/4.10

West and West246 generalized the OGM to incorporate the disordering influence of entropy by including the statistical fluctuations of the measured mass. The

dynamics of such a stochastic process eventually erases the influence of the initial

state and this gradual loss of information is a manifestation of increasing entropy.

They refer to this as the stochastic ontogenetic growth model (SOGM). Consider a

phenomenological Langevin equation with multiplicative fluctuations for the TBM

where to avoid confusion we use the dummy variable Z = m, which is actually the

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TBM for species i and individual j and individual j given by Mij . Thus, we replace

Eq. (49) for η = b and β = 1 for the SOGM stochastic differential equation:

dZ

dt = AbZb − C0Z + Zξ(t), (51)

with ξ(t) given by a delta correlated random process of strength D with normal

statistics and correlations given by Eq. (43). Rather than assuming the rate of

creating a unit biomass is a single number C1 they assume it has a stochastic

component: C1 → C0 + ξ(t), with average value C0.

An equivalent description of the SOGM is given by the corresponding dynamics

of the pdf in phase space. The FPE corresponding to Eq. (51) is118,193:

∂P(z, t)

∂t = ∂∂z 

−(Abzb − C0z) + Dz ∂

∂z z



P(z, t). (52)

The general solution to Eq. (52) remains a mystery, but the steady-state solution

(t → ∞) denoted by the ss subscript obtained by setting the probability flux to

zero is given by:

Pss(z) = βγ μ−1

β

Γ

μ − 1

β



exp[−γz−β]

zμ ; (53)

normalized on the interval (0, ∞) with the parameter values γ = Ab/βD, μ =

1+(C0/D) and β = 1 − b. Note that the steady-state solution to the SOGM FPE

replaces the deterministic steady-state solution obtained from the OGM.10,250

West and West246 fit the parameters in Eq. (53) to a data set of mammalian

species tabulated by Heusner91 and this fit is depicted in Fig. 11. They constructed

a histogram by partitioning the mass axis into intervals of 20 g and counting the

number of species within each interval. The vertical axis is the relative number of

species as a function of TBM. The allometry exponet is fixed at b = 3/4 so that

β = 1/4 and the remaining parameters are determined to have the best values of

γ = 13.4 and μ = 2.04 with a quality of fit measured by the correlation coefficient

r2 = 0.96.

Figure 11 shows the fit to the low-mass species out to 500 g. The remaining fit

out to three million grams is not shown but is in fact the more ineresting part of the

distribution. To capture this information on a single graph we246 construct a second

histogram, this one for the asymptotic region from approximately one thousand to

three million grams. Figure 12 depicts the fit to the logarithmic histogram data

points indicated by the dots starting at a TBM of 1.1 kg. An inverse power-law

distribution would be a straight line with a negative slope on this log–log graph.

Inserting the parameter values μ = 1.67 and γ − 8.96 into the steady-state TBM

pdf given by Eq. (53) yields the solid cuve in Fig. 12 which fits the data extremely

well. The curve is quite clearly an inverse power law in the interspecies TBM.

This coarse-grained description of the interspecies TBM statistics indicates great

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Fig. 11. The histogram constructed from the average TBM data for 391 mammalian species91

is given by the dots. The mass data has been divided into intervals of 20 g each and the number

of species within each such interval counted. The horizontal axis is the relative frequency in each

interval and the solid line segment is the least squares fit of Eq. (53) to the data points. The

quality of fit is r2 = 0.96.246

Fig. 12. The average TBM data for 391 mammalian species91 are used to construct a histogram.

The mass interval is divided into 20 equally spaced intervlas on a logarithm scale and the number

of species within each interval counted. The least-square fit to the nine data points is then made

using logrimically transformed distribution, see Ref. 246 for details. The quality of the fit is

r2 = 0.998.

variability particularly since μ < 2 indicating that the variance of the interspecies

TBM diverges.

5.1.3. Interspecies AR derivation

We241,243 explicitly constructed a statistical theory of the fluctuations in the measured species metabolic rate Bi and TBM Mi to calculate the interspecies metabolic

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AR. The strategy was to related the average BMR to the average TBM through the

AR. This approach can be followed here from the long-time or steady-state average

SOGM TBM using the steady-state pdf:

�z� =

 ∞

0

zPss(z)dz = βγ μ−1

β

Γ

μ − 1

β



 ∞

0

z1−μ exp[−γz−β]dz

and replacing z with the TBM for species i reduces the average to:

�Mi� = γ1/β

Γ

μ − 1

β − 1β



Γ

μ − 1

β

 ≈



γ

β

μ − 1

1/β

. (54)

Here we interpret the ensemble average in terms of an average over an ensemble of

individual members of species i. In this equation we have approximated the ratio

of gamma functions by an asymptotic value and substituting the parameter values

into the right-hand side of Eq. (54) yields:

�Mi� ≈ Ab

C0

1/β

=

 a

Bm

1/β

. (55)

The equality in Eq. (55) is obtained by substituting the values of the parameters

from OGM and using b = 3/4 and β = 1/4 to yield

Bm ≈ a�Mi�

−1/4 . (56)

Thus, the average metabolic rate necessary to maintain a unit biomass is dependent

on the inverse quarter power of the average TBM of the species. This is the same

expression obtained by Moses et al.149 if we identify their adult TBM with the

average TBM obtained from the steady-state pdf of the SOGM.

Note that the average TBM for species i replaces M0 obtained from the

dm/dt = 0 solution to Eq. (50). On the other hand the average BMR can be determined from the bth moment of the TBM

�Bi� = a�Mb

i � = a

 ∞

0

zbPss(z)dz

= aγ−1+1/β

Γ

μ − 1

β + 1 − 1β



Γ

μ − 1

β

 ≈ a



γ

β

μ − 1

1/β−1

. (57)

Comparing this last equation with Eq. (54) allows us to write:

�Bi� ≈ a�Mi�

1−β = a�Mi�b , (58)

which is the interspecies metabolic AR. We emphasize246 that the interspecies AR

is a pheomenological equation that without the SOGM does not have a formal

theoretical basis.

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On Allometry Relations

The SOGM constitutes the first theoretical construction of an interspecies

metabolic AR starting from a fundamental dynamic equation and relating the

proper averages. It is true that we used a nonlinear multiplicative Langevin equation, but the original dynamics stem from the conservation of energy argument250

and the fluctuations are a consequence of the dynamics being degraded by entropy.243 This suggests a possible untraveled path for future research in this area.

5.2. Scaling solution for AR

A phase space equation for anomalous diffusion containing the influence of longterm memory on the dynamics of the pdf was first discussed by West and Seshadri227

in term of the fractional calulus. The resulting fractional diffusion equation108,236

(FDE) has subsequently found a variety of uses in the physical sciences. The pdf

that solves a class of FDEs satisfies the scaling equation

P(z, t) = 1

tμz Fz

 z

tμz



. (59)

Note that this scaling of the pdf is also a realization of the RG relation for the

random variable Z(bt) = bμZ(t).

The function Fz(·) in Eq. (59) is left unspecified but it is analytic in the similarity

variable z/tμ. As mentioned in the introduction a standard diffusion process Z(t)

is the displacement of the diffusing particle from its initial position at time t, and

for vanishing small dissipation the scaling parameter is μz = 1/2 and the functional

form of Fz(·) is a normal distribution. However, for general complex phenomenon

there is a broad class of distributions for which the functional form of Fz(·) is

not Gaussian and the scaling index μz �= 1/2. For example, the α-stable L ́evy

process147,184,198,264 scales in this way and the L ́evy index is in the range 0 < α ≤

2, with the equality holding for the Gauss distribution and the scaling index is

related to the L ́evy index by μz = 1/α, see West, Geneston and Grigolini239 and

Uchaikin216 for very different discussions of this scaling.

The Gibbs entropy (Shannon information),23,197 can be defined using the pdf

for the variable Z(t) Eq. (59) to obtain:

Sz(t) = −



P(z, t) lnP(z, t)dz = S0

z + μz ln t , (60)

so the entropy deviation from the reference state S0

z ≡ − 

Fz(q) ln Fz(q)dq integrated with respect to the scaled variable q = z/tμx , increases logarithmically in

the independent variable t.

Huxley98 assumed the differential growth equations in an organism to have

the same form with proportional growth rates. West and West241,243 adapted this

assumption and presumed the two parts of an organism share the same class of

probability densities that describe their interacting observables. Here t is clock time,

the independent variable, and the network measures are the dependent variables,

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D. West & B. J. West

the average of a generic scaling observable using the pdf given by Eq. (59) yields:

Z(t) =



zP(z, t)dz = t

μz



qFz(q)dq = t

μz Z .  ̄ (61)

Note that Z ̄ = qFz(q)dq is a finite, time-independent constant. Comparing

Eqs. (61) and (60) enables us to write for the change in entropy relative to a

reference state:

∆Sz = ln[Z(t)/Z ̄] (62)

so that the change in the average value of an observable is determined by the change

in entropy

Z(t) = Ze ̄ ∆Sz(t) . (63)

Consequently Z may be identified with either of the network variables to obtain the

relation between entropies from Eq. (60)

∆Sx/μx = ∆Sy/μy

and using Eq. (63) we can write:

Yi = Y e  ̄ ∆Sy = Y e  ̄ μy

μx ∆Sx = aXi

b (64)

with the allometry coefficient given by a = Y /  ̄ X ̄ b and the allometry exponent by

b = μy/μx. We have again introduced the index for the single species i to emphsize

that this is the interspecies AR. This derivation of the empirical interspecies AR is

solely a consequence of the scaling properties of the pdf ’s.

The rate of entropy generation is determined by the time derivative of Eq. (63)

for X and Y and substituting from the entropy balance equation yields:

1

μxXi

dXi

dt = 1μyYi

dYi

dt . (65)

Equation (65) shows the correspondence of the probabilistic approach to AR with

that of Huxley, with the dynamic variables of individual species members replaced

by averages over an ensemble of individuals. In the present case the allometry exponent is now the ratio of the power-law indices in the pdf ’s, which from Eq. (65)

can also be interpreted as the ratio of growth rates. Moreover, the allometry coefficient is determined by the scaling functions in the pdf ’s. It is the average response

or adaptation of the subnetwork to the average behavior of the host network that

is captured by the interspecies AR through the balance of the entropy generated.

Moreover it is the scaling in the pdf and not necessarily geometric scaling of fractal

networks that is the origin of the physiologic AR. Underlying this derivation is the

assumption made earlier that this is an information-dominated network and is consequently driven by information (entropy) gradients and not the energy gradient

necessary in the fractal geometry derivation of nutrient transport.

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On Allometry Relations

6. Conclusion

The statistical analysis of the metabolic data presented herein show that empirical ARs exist across multiple species and consequently the form of ARs are not

solely dependent on species specific mechanisms. Moreover we have shown there is

no universal value for the allometry exponent for three reasons. First there is the

variability in the values of the allometry exponent obtained from data analysis in

Sec. 3. We could rehash the arguments here, but an objective observer would have

to conclude that although there are indications that a particular value might be

strongly indicated in a specific context, the evidence for universality is not compelling. Even in the case of metabolic ARs one could reasonably make a case of

b = 2/3 for small animals, b = 3/4 for large animal and b = 1 for plants, but no

one value of b that spans the total range of animal and plant sizes. Second, the

only theories that predict a universal value of the allometry exponent do so to explain the intraspecies AR and not the interspecies AR. There is no first principles

theory that derives the empirical AR between the averaged observables. The phenomenological theory presented in Section 4 successfully derives the interspecies

AR and treats the allometry coefficient and allometry exponent as empirical parameters determined by the parameters in the pdf ’s. Third and last the allometry

exponent and coefficient are determined to co-vary using theory243 and statistical

data analysis.72

However let us not be too critical of phenomenological theories. Recall that

thermodynamics is a phenomenological theory that has enjoyed remarkable success in explaining a plethora of complex physical phenomena. Even so the physics

community has not yet been able to derive thermodynamics from the more reductionist (fundamental) statistical mechanics, so that a mechanistic understanding

of thermodynamics remains controversial. By the same token the phenomenological theory of AR presented herein takes the discussion of AR out of the domain

of the reductionist approaches previously used to derive the intraspecies ARs and

refocuses it on the statistical properties of the empirical interspecies ARs. These

analyses indicate a new avenue for the study of physiologic ARs, one that systematically includes both deterministic and statistical aspects using the probability

calculus. This new perspective indicates that the origins of physiologic AR reside

in the scaling properties of the pdf ’s for nested complex networks.

In simple physical phenomena the dynamics of a network coupled to the environment is described by a Langevin equation, which is a stochastic differential

equation. This mechanical description of the dynamics is equivalent to the phase

space dynamics of the pdf such as given by the FPE. As the network becomes more

complex its past history becomes more and more influential until finally the FPE

must be replaced with a FDE and the dynamics are the domain of the fractional

calculus. As we saw in Sec. 5 the general solution to a class of FDE’s is a scaling pdf

implying that ARs can result from scaling in the statistical dynamics of the phenomenon. The influence of the history of the growth of an organism on the growth

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D. West & B. J. West

of an organ leads to intraspecies AR; the history of the size of an organism on the

metabolic rate of the organism across species leads to interspecies AR.

The arguments in Sec. 5 imply that empirical ARs are a consequence of information transfer between complex information-dominated networks. Of course we

have not rigorously proven that the entropy decrease (information increase) in the

host network and the increase in entropy (information decrease) in the subnetwork

have the relations assumed, this remains to be done. However the empirical relations are consistent with the Principle of Complexity Management in which the

transfer of information between two complex information-dominated networks has

been shown7,239 to proceed from the network with the greater information content

to the network with the lesser.

The phenomenological theory of empirical AR presented herein incorporates

both stochastic and reductionistic mechanisms. From this approach we conclude

that the empirical AR for the BMR, the life time of an organism and the myriad

of other complex phenomena are not completely explained by reductionistic mechanisms. In particular the allometry parameters are determined not to be universal.

This was supported by the demonstration that the allometry exponent and coefficient co-vary. On the other hand, the laws from the theory of probability can have

universal forms without having universal parameters, for example the Law of Frequency of Errors and the Law of Large Numbers. The empirical AR is in part a

consequence of the generic statistical behavior of complex networks that depend on

the infinitely divisible nature of the pdf.

74,75 The phenomenological theory of AR

presented herein is the application of the probability calculus to the understanding

of the origins of the empirical relations between averages of different parts of a complex network. Consequently empirical AR is a manifestation of a law whose origin

can be traced back to the probability calculus and the balancing of deterministic

and stochastic mechanisms.

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Research@

173

Distributed Algorithms for Learning 

and Cognitive Medium Access 

with Logarithmic Regret

Animashree Anandkumar, Nithin Michael, Ao Kevin Tang, and Ananthram Swami

IEEE Journal on Selected Areas in Communications, 29(4), 731-745 (2011)
174 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 29, NO. 4, APRIL 2011 731

Distributed Algorithms for Learning and

Cognitive Medium Access with Logarithmic Regret

Animashree Anandkumar, Member, IEEE, Nithin Michael, Student Member, IEEE,

Ao Kevin Tang, Member, IEEE, and Ananthram Swami, Fellow, IEEE

Abstract—The problem of distributed learning and channel access is considered in a cognitive network with multiple secondary

users. The availability statistics of the channels are initially

unknown to the secondary users and are estimated using sensing

decisions. There is no explicit information exchange or prior

agreement among the secondary users and sensing and access

decisions are undertaken by them in a completely distributed

manner. We propose policies for distributed learning and access which achieve order-optimal cognitive system throughput

(number of successful secondary transmissions) under self play,

i.e., when implemented at all the secondary users. Equivalently,

our policies minimize the sum regret in distributed learning and

access, which is the loss in secondary throughput due to learning

and distributed access. For the scenario when the number of

secondary users is known to the policy, we prove that the total

regret is logarithmic in the number of transmission slots. This

policy achieves order-optimal regret based on a logarithmic lower

bound for regret under any uniformly-good learning and access

policy. We then consider the case when the number of secondary

users is xed but unknown, and is estimated at each user through

feedback. We propose a policy whose sum regret grows only

slightly faster than logarithmic in the number of transmission

slots.

Index Terms—Cognitive medium access control, multi-armed

bandits, distributed algorithms, logarithmic regret.

I. INTRODUCTION

T HERE has been extensive research on cognitive radio

network in the past decade to resolve many challenges not

encountered previously in traditional communication networks

(see e.g., [2]). One of the main challenges is to achieve

coexistence of heterogeneous users accessing the same part of

the spectrum. In a hierarchical cognitive network, there are two

classes of transmitting users, viz., the primary users who have

priority in accessing the spectrum and the secondary users who

opportunistically transmit when the primary user is idle. The

secondary users are cognitive and can sense the spectrum to

detect the presence of a primary transmission. However, due

Manuscript received 1 December 2009; revised 4 June 2010. During the

stint of this work, the rst author was supported by MURI through AFOSR

Grant FA9550-06-1-0324. The second and the third authors are supported in

part through NSF grant CCF-0835706. Parts of this paper were presented at

[1].

A. Anandkumar is with the Center for Pervasive Communications

and Computing at the Department of Electrical Engineering and Computer Science, University of California, Irvine, CA, 92697, USA (e-mail:

a.anandkumar@uci.edu).

N. Michael and A. K. Tang are with the School of Electrical and

Computer Engineering, Cornell University, Ithaca, NY 14853, USA (e-mail:

nm373@,atang@ece.cornell.edu).

A. Swami is with the Army Research Laboratory, Adelphi, MD 20783,

USA (e-mail: a.swami@ieee.org).

Digital Object Identier 10.1109/JSAC.2011.110406.

to resource and hardware constraints, they can sense only a

part of the spectrum at any given time.

We consider a slotted cognitive system where each secondary user can sense and access only one orthogonal channel

in each transmission slot (see Fig. 1). Under sensing constraints, it is thus benecial for the secondary users to select

channels with higher mean availability, i.e., channels which

are less likely to be occupied by the primary users. However,

in practice, the channel availability statistics are unknown to

the secondary users at the start of the transmissions.

Since the secondary users are required to sense the medium

before transmission, can these sensing decisions be used to

learn the channel availability statistics? If so, using these

estimated channel availabilities, can we design channel access

rules which maximize the transmission throughput? Designing

provably efcient algorithms to accomplish the above goals

forms the focus of our paper. Such algorithms need to be

efcient, both in terms of learning and channel access.

For any learning algorithm, there are two important performance criteria: consistency and regret bounds [3]. A learning

algorithm is said to be consistent if the learnt estimates

converge to the true values as the number of samples goes

to innity. The regret1 of a learning algorithm is a measure

of the speed of convergence, and is thus meaningful only for

consistent algorithms. In our context, the users need to learn

the channel availability statistics consistently in a distributed

manner. The regret in this case is dened as the loss in

secondary throughput due to learning when compared to the

ideal scenario where the channel statistics are known perfectly

(see (2)). It is thus desirable to design distributed learning

algorithms with small regret.

Additionally, we consider a distributed framework where

there is no information exchange or prior agreement among

the secondary users. This introduces additional challenges:

it results in loss of throughput due to collisions among the

secondary users, and there is now competition among the

secondary users since they all tend to access channels with

higher availabilities. It is imperative that the channel access

policies overcome the above challenges. Hence, a distributed

learning and access policy experiences regret both due to

learning the unknown channel availabilities as well as due

to collisions under distributed access.

1Note that the sum regret is a ner measure of performance of a

distributed algorithm than the time-averaged total throughput of the users

since any sub-linear regret (with respect to time) implies optimal average

throughput. On the other hand, regret is equivalent to total throughput and

hence, optimal regret is equivalent to achieving optimal total throughput.

0733-8716/11/$25.00 c 2011 IEEE
175

732 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 29, NO. 4, APRIL 2011

Primary User

Secondary User

Fig. 1. Cognitive radio network with U = 4 secondary users and C = 5

channels. A secondary user is not allowed to transmit if the accessed channel

is occupied by a primary user. If more than one secondary user transmits in

the same free channel, then all the transmissions are unsuccessful.

A. Our Contributions

The main contributions of this paper are two fold. First,

we propose two distributed learning and access policies under

multiple secondary users in a cognitive network. Second, we

provide performance guarantees for these policies in terms of

regret. Overall, we prove that one of our proposed algorithms

achieves order-optimal regret and the other achieves nearly

order-optimal regret, where the order is with respect to the

number of transmission slots.

The rst policy we propose assumes that the total number

of secondary users in the system is known while our second

policy removes this requirement. We provide bounds on the

total regret experienced by the secondary users under self

play, i.e., when implemented at all the secondary users. For

the rst policy, we prove that the regret is logarithmic, i.e.,

O(log n) where n in the number of transmission slots. For the

second policy, the regret grows slightly faster than logarithmic,

i.e., O(f(n) log n), where we can choose any function f(n)

satisfying f(n) → ∞, as n → ∞. Hence, we provide

performance guarantees for the proposed distributed learning

and access policies.

A lower bound on regret under any uniformly-good distributed learning policy has been derived in [4], which is also

logarithmic in the number of transmission slots (See (6) for

denition of uniformly-good policies). Thus, our rst policy

(which requires knowledge of the number of secondary users)

achieves order-optimal regret. The effects of the number of

secondary users and the number of channels on regret are

also explicitly characterized and veried via simulations.

The exploration-exploitation tradeoff for learning, combined with the cooperation-competition tradeoffs among multiple users for distributed medium access have not been

sufciently examined in the literature (see Section I-B for a

discussion). Our analysis in this paper provides important engineering insights towards dealing with learning, competition,

and cooperation in practical cognitive systems.

Remark on Model Assumptions: We note some of the

limitations of our approach. We prove order-optimal regret for

our proposed algorithms under an i.i.d. primary transmission

model2, where the probability of a channel being occupied

by any primary user in any slot is assumed to be i.i.d.

This is indeed is idealistic and in practice, a Markovian

model may be more appropriate [5], [6]. The i.i.d. model

leads to simple learning schemes for channel availabilities

but is not crucial towards deriving regret bounds for our

proposed schemes. Extensions of the classical multi-armed

bandit problem to a Markovian model are considered in [7].

In principle, our results on achieving order-optimal regret for

distributed learning and access can be similarly extended to

a Markovian channel model but this entails more complex

estimators and rules for evaluating the exploration-exploitation

tradeoffs of different channels and is a topic of interest for

future investigation.

B. Related Work

Several results on the multi-armed bandit problem will be

used and generalized to study our problem. Detailed discussion

on multi-armed bandits can be found in [8]–[11]. Cognitive

medium access is a topic of extensive research; see [12]

for an overview. The connection between cognitive medium

access and the multi-armed bandit problem is explored in

[13], where a restless bandit formulation is employed. Under

this formulation, indexability is established, Whittle’s index

for channel selection is obtained in closed-form, and the

equivalence between the myopic policy and the Whittle’s

index is established. However, this work assumes known

channel availability statistics and does not consider competing

secondary users. The work in [14] considers allocation of two

users to two channels under a Markovian channel model using

a partially observable Markov decision process (POMDP)

framework. The use of collision feedback information for

learning, and spatial heterogeneity in spectrum opportunities

were investigated. However, the difference from our work

is that [14] assumes that the availability statistics (transition

probabilities) of the channels are known to the secondary users

while we consider learning of unknown channel statistics. The

works in [15], [16] consider centralized access schemes in

contrast to distributed access here, the work in [17] considers

access through information exchange and studies the optimal

choice of the amount of information to be exchanged given

the cost of negotiation. The work in [18] considers access

under Q-learning for two users and two channels where users

can sense both the channels simultaneously. The work in

[19] discusses a game-theoretic approach to cognitive medium

access. In [20], learning in congestion games through multiplicative updates is considered and convergence to weaklystable equilibria (which reduces to the pure Nash equilibrium

for almost all games) is proven. However, the work assumes

xed costs (or equivalently rewards) in contrast to random

rewards here, and that the players can fully observe the actions

of other players.

2By i.i.d. primary transmission model, we do not mean the presence of

a single primary user, but rather, this model is used to capture the overall

statistical behavior of all the primary users in the system.
176

ANANDKUMAR et al.: DISTRIBUTED ALGORITHMS FOR LEARNING AND COGNITIVE MEDIUM ACCESS WITH LOGARITHMIC REGRET 733

Recently, the work in [21] considers combinatorial bandits,

where a more general model of different (unknown) channel

availabilities is assumed for different secondary users, and a

matching algorithm is proposed for jointly allocating users

to channels. The algorithm is guaranteed to have logarithmic

regret with respect to the number of transmission slots and

polynomial storage requirements. A decentralized implementation of the proposed algorithm is proposed but it still

requires information exchange and coordination among the

users. In contrast, we propose algorithms which removes this

requirement albeit in a more restrictive setting of identical

channel availabilities for all users.

Recently, Liu and Zhao [4] proposed a family of distributed

learning and access policies known as time-division fair share

(TDFS), and proved logarithmic regret for these policies. They

established a lower bound on the growth rate of system regret

for a general class of uniformly-good decentralized polices.

The TDFS policies in [4] can incorporate any order-optimal

single-player policy while our work here is based on the

single-user policy proposed in [11]. Another difference is

that in [4], the users orthogonalize via settling at different

offsets in their time-sharing schedule, while in our work here,

users orthogonalize into different channels. Moreover, the

TDFS policies ensure that each player achieves the same timeaverage reward while our policies here achieve probabilistic

fairness, in the sense that the policies do not discriminate between different users. In [22], the TDFS policies are extended

to incorporate imperfect sensing.

In our prior work [1], we rst formulated the problem

of decentralized learning and access for multiple secondary

users. We considered two scenarios: one where there is initial

common information among the secondary users in the form of

pre-allocated ranks, and the other where no such information

is available. In this paper, we analyze the distributed policy in

detail and prove that it has logarithmic regret. In addition, we

also consider the case when the number of secondary users is

unknown, and provide bounds on regret in this scenario.

Organization: Section II deals with the system model,

Section III deals with the special case of single secondary

user and of multiple users with centralized access which

can be directly solved using the classical results on multiarmed bandits. In Section IV, we propose distributed learning

and access policy with provably logarithmic regret when the

number of secondary users is known. Section V considers the

scenario when the number of secondary users is unknown.

Section VI provides a lower bound for distributed learning.

Section VII includes some simulation results for the proposed

schemes and Section VIII concludes the paper. Most of the

proofs are found in the Appendix.

Since Section III mostly deals with a recap of the classical results on multi-armed bandits, those familiar with the

literature (e.g., [8]–[11]) may skip this section without loss of

continuity.

II. SYSTEM MODEL & FORMULATION

Notation: For any two functions f(n), g(n), f(n) =

O(g(n)) if there exists a constant c such that f(n) ≤ cg(n)

for all n ≥ n0 for a xed n0 ∈ N. Similarly, f(n) = Ω(g(n))

if there exists a constant c� such that f(n) ≥ c�

g(n) for all

n ≥ n0 for a xed n0 ∈ N, and f(n) = Θ(g(n)) if f(n) =

Ω(g(n)) and f(n) = O(g(n)). Also, f(n) = o(g(n)) when

f(n)/g(n) → 0 and f(n) = ω(g(n)) when f(n)/g(n) → ∞

as n → ∞.

For a vector μ, let |μ| denote its cardinality, and let

μ = [μ1, μ2,...,μ|μ|]

T . We refer to the U highest entries

in a vector μ as the U-best channels and the rest as the Uworst channels, where 1 ≤ U ≤ |μ|. Let σ(a; μ) denote

the index of the ath highest entry in μ. Alternatively, we

abbreviate a∗:=σ(a; μ) for ease of notation. Let B(μ) denote

the Bernoulli distribution with mean μ. With abuse of notation,

let D(μ1, μ2):=D(B(μ1); B(μ2)) be the Kullback-Leibler

distance between the Bernoulli distributions B(μ1) and B(μ2)

[23] and let Δ(1, 2):=μ1 − μ2.

A. Sensing & Channel Models

Let U ≥ 1 be the number of secondary users3 and C ≥ U

be the number4 of orthogonal channels available for slotted

transmissions with a xed slot width. In each channel i and

slot k, the primary user transmits i.i.d. with probability 1 −

μi > 0. In other words, let Wi(k) denote the indicator variable

if the channel is free

Wi(k) =  0, channel i occupied in slot k

1, o.w,

and we assume that Wi(k) i.i.d.

∼ B(μi).

The mean availability vector μ consists of mean availabilities μi of all channels, i.e., is μ:=[μ1,...,μC ], where all

μi ∈ (0, 1) and are distinct. μ is initially unknown to all the

secondary users and is learnt independently over time using

the past sensing decisions without any information exchange

among the users. We assume that channel sensing is perfect

at all the users.

Let Ti,j (k) denote the number of slots where channel i is

sensed in k slots by user j (not necessarily being the sole

user to sense that channel). Note that C

i=1 Ti,j(k) = k for

all users j, since each user senses exactly one channel in every

time slot. The sensing variables are obtained as follows: at the

beginning of the kth slot, each secondary user j ∈ U selects

exactly one channel i ∈ C for sensing, and hence, obtains the

value of Wi(k), indicating if the channel is free. User j then

records all the sensing decisions of each channel i in a vector

Xk

i,j :=[Xi,j (1),...,Xi,j (Ti,j (k))]T . Hence, ∪Ci=1Xki,j is the

collection of sensed decisions for user j in k slots for all the

C channels.

We assume the collision model under which if two or

more users transmit in the same channel then none of the

transmissions go through. At the end of each slot k, each

user j receives acknowledgement Zj (k) on whether its transmission in the kth slot was received. Hence, in general, any

policy employed by user j in the (k + 1)th slot, denoted by

ρ(∪C

i=1Xki,j , Zkj ), is based on all the previous sensing and

feedback results.

3A user refers to a secondary user unless otherwise mentioned.

4When U ≥ C, learning availability statistics is less crucial, since all

channels need to be accessed to avoid collisions. In this case, design of

medium access is more crucial.
177 734 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 29, NO. 4, APRIL 2011

Algorithm 1 Single User Policy ρ1(g(n)) in [10].

Input: {X ̄i(n)}i=1,...,C : Sample-mean availabilities after n

rounds, g(i; n): statistic based on X ̄i,j (n),

σ(a; g(n)): index of ath highest entry in g(n).

Init: Sense in each channel once, n ← C, Curr_Sel ← C.

Loop: n ← n + 1

Curr_Sel ← channel corresponding to highest entry in g(n)

for sensing. If free, transmit.

B. Regret of a Policy

Under the above model, we are interested in designing

policies ρ which maximize the expected number of successful

transmissions of the secondary users subject to the noninterference constraint for the primary users. Let S(n; μ, U, ρ)

be the expected total number of successful transmissions after

n slots under U number of secondary users and policy ρ.

In the ideal scenario where the availability statistics μ are

known a priori and a central agent orthogonally allocates the

secondary users to the U-best channels, the expected number

of successful transmissions after n slots is given by

S∗(n; μ, U):=n



U

j=1

μ(j∗), (1)

where j∗ is the jth-highest entry in μ.

It is clear that S∗(n; μ, U) > S(n; μ, U, ρ) for any policy

ρ and any nite n ∈ N. We are interested in minimizing the

regret in learning and access, given by

R(n; μ, U, ρ):=S∗(n; μ, U) − S(n; μ, U, ρ) > 0. (2)

We are interested in minimizing regret under any given μ ∈

(0, 1)C with distinct elements.

By incorporating the collision channel model assumption

with no avoidance mechanisms5, the expected throughput

under the policy ρ is given by

S(n; μ, U, ρ) = 

C

i=1



U

j=1

μ(i)E[Vi,j (n)],

where Vi,j (n) is the number of times in n slots where user

j is the sole user to sense channel i. Hence, the regret in (2)

simplies as

R(n; ρ) = 

U

k=1

nμ(k∗) −

C

i=1



U

j=1

μ(i)E[Vi,j (n)]. (3)

III. SPECIAL CASES FROM KNOWN RESULTS

We recap the bounds for the regret under the special cases

of a single secondary user (U = 1) and multiple users with

centralized learning and access by appealing to the classical

results on the multi-armed bandit process [8]–[10].

A. Single Secondary User (U = 1)

When there is only one secondary user, the problem of

nding policies with minimum regret reduces to that of a

5If the users employ CSMA-CA to avoid collisions then the regret is

reduced. The bounds derived in this paper are applicable for this case as

well.

multi-armed bandit process. Lai and Robbins [8] rst analyzed

schemes for multi-armed bandits with asymptotic logarithmic

regret based on the upper condence bounds on the unknown

channel availabilities. Since then, simpler schemes have been

proposed in [10], [11] which compute a statistic or an index for

each arm (channel), henceforth referred to as the g-statistic,

based only on its sample mean and the number of slots where

the particular arm is sensed. The arm with the highest index is

selected in each slot in these works. We summarize the policy

in Algorithm 1 and denote it ρ1(g(n)), where g(n) is the

vector of scores assigned to the channels after n transmission

slots.

The sample-mean based policy in [11, Thm. 1] proposes an

index for each channel i and user j at time n is given by

gMEAN

j (i; n):=X ̄i,j (Ti,j (n)) + 

2 log n

Ti,j (n)

, (4)

where Ti,j (n) is the number of slots where user j selects

channel i for sensing,

X ̄i,j (Ti,j (n)):=

Ti,j



(n)

k=1

Xi,j (k)

Ti,j(n)

is the sample-mean availability of channel i, as sensed by user

j.

The statistic in (4) captures the exploration-exploitation

tradeoff between sensing the channel with the best predicted

availability to maximize immediate throughput and sensing

different channels to obtain improved estimates of their availabilities. The sample-mean term in (4) corresponds to exploitation while the other term involving Ti,j(n) corresponds to

exploration since it penalizes channels which are not sensed

often. The normalization of the exploration term with log n

in (4) implies that the term is signicant when Ti,j (n) is

much smaller than log n. On the other hand, if all the channels

are sensed Θ(log n) number of times, the exploration terms

become unimportant in the g-statistics of the channels and the

exploitation term dominates, thereby, favoring sensing of the

channel with the highest sample mean.

The regret based on the above statistic in (4) is logarithmic

for any nite number of slots n but does not have the optimal

scaling constant. The sample-mean based statistic in [10,

Example 5.7] leads to the optimal scaling constant for regret

and is given by

gOPT

j (i; n):=X ̄i,j (Ti,j (n)) + min 

log n

2Ti,j(n)

, 1



. (5)

In this paper, we design policies based on the gMEAN statistic

since it is simpler to analyze than the gOPT statistic. This is

because gMEAN is a continuous function in Ti,j (n) while gOPT

has a threshold on Ti,j (n).

We now recap the results which show logarithmic regret in

learning the best channel. In this context, we dene uniformly

good policies ρ [8] as those with regret

R(n; μ, U, ρ) = o(nα), ∀α > 0, μ ∈ (0, 1)C. (6)
178 ANANDKUMAR et al.: DISTRIBUTED ALGORITHMS FOR LEARNING AND COGNITIVE MEDIUM ACCESS WITH LOGARITHMIC REGRET 735

Theorem 1 (Logarithmic Regret for U = 1 [10], [11]):

For any uniformly good policy ρ satisfying (6), the expected

time spent in any suboptimal channel i �= 1∗ satises

limn→∞ P

�

Ti,1(n) ≥ (1 − �) log n

D(μi, μ1∗ ) ; μ

�

= 1, (7)

where 1∗ is the channel with the best availability. Hence, the

regret satises

lim inf

n→∞

R(n; μ, 1, ρ)

log n

≥ �

i∈1-worst

Δ(1∗, i)

D(μi, μ1∗ )

. (8)

The regret under the gOPT statistic in (5) achieves the above

bound.

limn→∞

R(n; μ, 1, ρ1(gOPT

j ))

log n = �

i∈1-worst

Δ(1∗, i)

D(μi, μ1∗ )

. (9)

The regret under gMEAN statistic in (34) satises

R(n; μ, 1, ρ1(gMEAN

j )) ≤ �

i�=1∗

Δ(1∗, i)

� 8 log n

Δ(j∗, i)2 +1+ π23

�

.

B. Centralized Learning & Access for Multiple Users

We now consider multiple secondary users under centralized

access policies where there is joint learning and access by a

central agent on behalf of all the U users. Here, to minimize

the sum regret, the centralized policy allocates the U users to

orthogonal channels to avoid collisions. Let ρCENT(X k), with

X k := ∪U

j=1 ∪Ci=1 Xki,j , denote a centralized policy based

on the sensing variables of all the users. The policy under

centralized learning is a simple generalization of the singleuser policy and is given in Algorithm 2. We now recap the

results of [9].

Theorem 2 (Regret Under Centralized Policy ρCENT [9]):

For any uniformly good centralized policy ρCENT satisfying

(6), the expected times spent in a U-worst channel i satises

limn→∞ P

⎡

⎣

�

U

j=1

Ti,j (n) ≥ (1 − �) log n

D(μi, μU∗ ) ; μ

⎤

⎦ = 1, (10)

where U∗ is the channel with the Uth best availability. Hence,

the regret satises

lim inf

n→∞

R(n; μ, 1, ρCENT)

log n

≥ �

i∈U-worst

Δ(U∗, i)

D(μi, μU∗ )

. (11)

The scheme in Algorithm 2 based on gOPT achieves the above

bound.

limn→∞

R(n; μ, 1, ρCENT(gOPT)

log n = �

i∈U-worst

Δ(U∗, i)

D(μi, μU∗ )

. (12)

The scheme in Algorithm 2 based on the gMEAN satises for

any n > 0,

R(n; μ, U, ρCENT(gMEAN))

≤ �

U

m=1

�

i∈U-worst

�

U

k=1

Δ(m∗, i)

U

� 8 log n

Δ(m∗, i)2 +1+ π23

�

.

(13)

Proof: See Appendix A. ✷

Algorithm 2 Centralized Learning Policy ρCENT in [9].

Input: Xn := ∪U

j=1 ∪Ci=1 Xni,j : Channel availability after n

slots, g(n): statistic based on X n,

σ(a; g(n)): index of ath highest entry in g(n).

Init: Sense in each channel once, n ← C

Loop: n ← n + 1

Curr_Sel ← channels with U-best entries in g(n). If free,

transmit.

Notice that for both the single user case U = 1 (Theorem 1)

and the centralized multi-user case (Theorem 2), the number

of time slots spent in the U-worst channels is Θ(log n) and

hence, the regret is also Θ(log n).

IV. MAIN RESULTS

Armed with the classical results on multi-armed bandits, we

now design distributed learning and allocation policies.

A. Preliminaries: Bounds on Regret

We rst provide simple bounds on the regret in (3) for any

distributed learning and access policy ρ.

Proposition 1 (Lower and Upper Bounds on Regret): The

regret under any distributed policy ρ satises

R(n; ρ) ≥�

U

j=1

�

i∈U-worst

Δ(U∗, i)E[Ti,j (n)], (14)

R(n; ρ) ≤μ(1∗)

⎡

⎣

�

U

j=1

�

i∈U-worst

E[Ti,j (n)] + E[M(n)]

⎤

⎦, (15)

where Ti,j (n) is the number of slots where user j selects

channel i for sensing, M(n) is the number of collisions faced

by the users in the U-best channels in n slots, Δ(i, j) =

μ(i) − μ(j) and μ(1∗) is the highest mean availability.

Proof: See Appendix B. ✷

In the subsequent sections, we propose distributed learning

and access policies and provide regret guarantees for the

policies using the upper bound in (15). The lower bound in

(14) can be used to derive lower bound on regret for any

uniformly-good policy.

The rst term in (15) represents the lost transmission

opportunities due to selection of U-worst channels (with lower

mean availabilities), while the second term represents the

performance loss due to collisions among the users in the

U-best channels. The rst term in (15) decouples among the

different users and can be analyzed solely through the marginal

distributions of the g-statistics at the users. This in turn, can

be analyzed by manipulating the classical results on multiarmed bandits [10], [11]. On the other hand, the second term

in (15), involving collisions in the U-best channels, requires

the joint distribution of the g-statistics at different users which

are correlated variables. This is intractable to analyze directly

and we develop techniques to bound this term.
179

736 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 29, NO. 4, APRIL 2011

Algorithm 3 Policy ρRAND(U, C, gj (n)) for each user j under

U users, C channels and statistic gj (n).

Input: {X ̄i,j(n)}i=1,...,C : Sample-mean availabilities at

user j after n rounds, gj (i; n): statistic based on X ̄i,j (n),

σ(a; gj (n)): index of ath highest entry in gj (n).

ζj (i; n): indicator of collision at nth slot at channel i

Init: Sense in each channel once, n ← C, Curr_Rank ← 1,

ζj (i; m)←0

Loop: n ← n + 1

if ζj (Curr_Sel; n − 1) = 1 then

Draw a new Curr_Rank ∼ Unif(U)

end if

Select channel for sensing. If free, transmit.

Curr_Sel ← σ(Curr_Rank; gj(n)).

If collision ζj (Curr_Sel; m) ← 1, Else 0.

B. ρRAND : Distributed Learning and Access

We present the ρRAND policy in Algorithm 3. Before describing this policy, we make some simple observations. If

each user implemented the single-user policy in Algorithm 1,

then it would result in collisions, since all the users target

the best channel. When there are multiple users and there

is no direct communication among them, the users need to

randomize channel access in order to avoid collisions. At

the same time, accessing the U-worst channels needs to be

avoided since they contribute to regret. Hence, users can avoid

collisions by randomizing access over the U-best channels,

based on their estimates of the channel ranks. However, if the

users randomize in every slot, there is a nite probability of

collisions in every slot and this results in a linear growth of

regret with the number of time slots. Hence, the users need

to converge to a collision-free conguration to ensure that the

regret is logarithmic.

In Algorithm 3, there is adaptive randomization based

on feedback regarding the previous transmission. Each user

randomizes only if there is a collision in the previous slot;

otherwise, the previously generated random rank for the user

is retained. The estimation for the channel ranks is through

the g-statistic, along the lines of the single-user case.

C. Regret Bounds under ρRAND

It is easy to see that the ρRAND policy ensures that the

users are allocated orthogonally to the U-best channels as

the number of transmission slots goes to innity. The regret

bounds on ρRAND are however not immediately clear and we

provide guarantees below.

We rst provide a logarithmic upper bound6 on the number

of slots spent by each user in any U-worst channel. Hence, the

rst term in the bound on regret in (15) is also logarithmic.

Lemma 1 (Time Spent in U-worst Channels): Under the

ρRAND scheme in Algorithm 3, the total time spent by any user

6Note that the bound on E[Ti,j (n)] in (16) holds for user j even if the

other users are using a policy other than ρRAND. But on the other hand, to

analyze the number of collisions E[M(n)] in (19), we need every user to

implement ρRAND.

j = 1,...,U, in any i ∈ U-worst channel is given by

E[Ti,j (n)] ≤ 

U

k=1

 8 log n

Δ(i, k∗)2 +1+ π23



. (16)

Proof: The proof is similar along the lines of Theorem 2,

given in Appendix A. The only difference is that here, we

need to consider the probability of transmission of each user

in each U-worst channel while in Appendix A, the probability

of transmission by all users is considered. ✷

We now focus on analyzing the number of collisions M(n)

in the U-best channels. We rst give a result on the expected

number of collisions in the ideal scenario where each user has

perfect knowledge of the channel availability statistics μ. In

this case, the users attempt to reach an orthogonal (collisionfree) conguration by uniformly randomizing over the U-best

channels.

The stochastic process in this case is a nite-state Markov

chain. A state in this Markov chain corresponds to a con-

guration of U number of (identical) users in U number of

channels. The number of states in the Markov chain is the

number of compositions of U, given by 2U−1

U



[24, Thm. 5.1].

The orthogonal conguration corresponds to the absorbing

state. There are two other classes of states in the Markov

chain. One class consists of states in which each channel either

has more than one user or no user at all. The second class

consists of the remaining states where some channels may

have just one user. For the rst class, the transition probability

to any state of the Markov chain (including self transition and

absorption probabilities) is uniform. For a state in the second

class, where certain channels have exactly one user, there are

only transitions to states which have identical conguration of

the single users and the transition probabilities are uniform.

Let Υ(U, U) denote the maximum time to absorption in the

above Markov chain starting from any initial distribution. We

have the following result

Lemma 2 (No. of Collisions Under Perfect Knowledge):

The expected number of collisions under ρRAND scheme in

Algorithm 3, assuming that each user has perfect knowledge

of the mean channel availabilities μ, is given by

E[M(n); ρRAND(U, C, μ)] ≤ UE[Υ(U, U)]

≤ U

2U − 1

U



−1



. (17)

Proof: See Appendix C. ✷

The above result states that there is at most a nite number

of expected collisions, bounded by UE[Υ(U, U)] under perfect

knowledge of μ. In contrast, recall from the previous section,

that there are no collisions under perfect knowledge of μ

in the presence of pre-allocated ranks. Hence, UE[Υ(U, U)]

represents a bound on the additional regret due to the lack

of direct communication among the users to negotiate their

ranks.

We use the result of Lemma 2 for analyzing the number of collisions under distributed learning of the unknown

availabilities μ as follows: if we show that the users are

able to learn the correct order of the different channels with

only logarithmic regret then only an additional nite expected
180

ANANDKUMAR et al.: DISTRIBUTED ALGORITHMS FOR LEARNING AND COGNITIVE MEDIUM ACCESS WITH LOGARITHMIC REGRET 737

number of collisions occur before reaching an orthogonal

conguration.

Dene T �

(n; ρRAND) as the number of slots where any one

of the top U-estimated ranks of the channels at some user is

wrong under the ρRAND policy. Below we prove that its expected

value is logarithmic in the number of transmissions.

Lemma 3 (Wrong Order of g-statistics): Under the ρRAND

scheme in Algorithm 3,

E[T �

(n; ρRAND)] ≤ U 

U

a=1



C

b=a+1

 8 log n

Δ(a∗, b∗)2 +1+ π23



.(18)

Proof: See Appendix D. ✷

We now provide an upper bound on the number of collisions

M(n) in the U-best channels by incorporating the above result

on E[T �

(n)], the result on the average number of slots E[Ti,j ]

spent in the i ∈ U-worst channels in Lemma 1 and the average

number of collisions UE[Υ(U, U)] under perfect knowledge

of μ in Lemma 2.

Theorem 3 (Logarithmic Number of Collisions Under ρRAND):

The expected number of collisions in the U-best channels

under ρRAND(U, C, gMEAN) scheme satises

E[M(n)] ≤ U(E[Υ(U, U)] + 1)E[T �

j(n)]. (19)

Hence, from (16), (18) and (17), M(n) = O(log n).

Proof: See Appendix E. ✷

Hence, there are only a logarithmic number of expected

collisions before the users settle in the orthogonal channels.

Combining this result with Lemma 1 that the number of

slots spent in the U-worst channels is also logarithmic, we

immediately have one of the main results of this paper

that the sum regret under distributed learning and access is

logarithmic.

Theorem 4 (Logarithmic Regret Under ρRAND): The policy

ρRAND(U, C, gMEAN) in Algorithm 3 has Θ(log n) regret.

Proof: Substituting (19) and (16) in (15). ✷

Hence, we prove that distributed learning and channel

access among multiple secondary users is possible with logarithmic regret without any explicit communication among

the users. This implies that the number of lost opportunities

for successful transmissions at all secondary users is only

logarithmic in the number of transmissions, which is negligible

when there are a large number of transmissions.

We have so far focused on designing schemes that maximize

system or social throughput. We now briey discuss the

fairness for an individual user under ρRAND. Since ρRAND does

not distinguish any of the users, in the sense that each user

has equal probability of “settling" down in one of the U-best

channels while experiencing only logarithmic regret in doing

so. Simulations in Section VII (in Fig. 4) demonstrate this

phenomenon.

We also note that the bound on regret under ρRAND grows

rapidly with the number of users U, due to the bound in

(17). This is due to uniform channel access by the users

without any coordination or information exchange. It is of

interest to explore better distributed access schemes, which

when combined with learning algorithms yield low regret in

the number of users.

Algorithm 4 Policy ρEST(n, C, gj (m), ξ) for each user j under

n transmission slots (horizon length), C channels, statistic

gj (m) and threshold function ξ.

1) Input: {X ̄i,j (n)}i=1,...,C : Sample-mean availabilities

at user j, gj (i; n): statistic based on X ̄i,j (n),

σ(a; gj (n)): index of ath highest entry in gj (n).

ζj (i; n): indicator of collision at nth slot at channel i

U: current estimate of the number of users.

n: horizon (total number of slots for transmission)

ξ(n; k): threshold functions for k = 1,...,C.

2) Init: Sense each channel once, m ← C, Curr_Sel ←

C, Curr_Rank ← 1, U ← 1, ζj (i; m) ← 0 for all

i = 1,...,C

3) Loop 4 to 6: m ← m+ 1, stop when m = n or U = C.

4) If ζj (Curr_Sel; m − 1) = 1 then

Draw a new Curr_Rank ∼ Unif(U). end if

Select channel for sensing. If free, transmit.

Curr_Sel ← σ(Curr_Rank; gj(m))

5) ζj (Curr_Sel; m) ← 1 if collision, 0 o.w.

6) If m

a=1

U

k=1 ζj (σ(k; gj (m)); a) > ξ(n;U)) then

U ← U + 1, ζj (i; a) ← 0, i = 1,...C, a = 1,...,m.

end if

V. DISTRIBUTED LEARNING AND ACCESS UNDER

UNKNOWN NUMBER OF USERS

We have so far assumed that the number of secondary

users is known, and is required for the implementation of

the ρRAND policy. In practice, this entails initial announcement

from each of the secondary users to indicate their presence in

the cognitive network. However, in a truly distributed setting

without any information exchange among the users, such an

announcement may not be possible.

In this section, we consider the scenario, where the number

of users U is unknown (but xed throughout the duration of

transmissions and U ≤ C, the number of channels). In this

case, the policy needs to estimate the number of secondary

users in the system, in addition to learning the channel

availability statistics and designing channel access rules based

on collision feedback. Note that if the policy assumed the

worst-case scenario that U = C, then the regret grows linearly

since there is a positive probability that the U-worst channels

are selected for sensing in any time slot.

A. Description of ρEST Policy

We now propose a policy ρEST in Algorithm 4. This policy

incorporates two broad aspects in each transmission slot, viz.,

execution of the ρRAND policy in Algorithm 3, based on the

current estimate of the number of users U, and updating of

the estimate U based on the number of collisions experienced

by the user.

The updating is based on the idea that if there is underestimation of U at all the users (Uj < U at all the users j),

collisions necessarily build up and the collision count serves

as a criterion for incrementing U. This is because after a long

learning period, the users learn the true ranks of the channels,

and target the same set of channels. However, when there is
181

738 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 29, NO. 4, APRIL 2011

under-estimation, the number of users exceeds the number of

channels targeted by the users. Hence, collisions among the

users accumulate, and can be used as a test for incrementing

U.

Denote the collision count used by the ρEST policy as

Φk,j (m) := m

a=1



k

b=1

ζj (σ(b; gj (m)); a). (20)

which is the total number of collisions experienced by user j

so far (till the mth transmission slot) in the top Uj -channels,

where the ranks of the channels are estimated using the gstatistics. The collision count is tested against a threshold

ξ(n;Uj ), which is a function of the horizon length7 and

current estimate Uj. When the threshold is exceeded, Uj is

incremented, and the collision samples collected so far are

discarded (by setting them to zero) (line 6 in Algorithm 4).

The choice of the threshold for incrementing Uj is critical; if

it is too small it can result in over-estimation of the number

of users. On the other hand, if it is too large, it can result in

slow learning and large regret. Threshold selection is studied

in detail in the subsequent section.

B. Regret Bounds under ρEST

We analyze regret bounds under the ρEST policy, where the

regret is dened in (3). Let the maximum threshold function

for the number of consecutive collisions under the ρEST policy

be denoted by

ξ∗(n;U):= max k=1,...,U ξ(n; k). (21)

We prove that the ρEST policy has O(ξ∗(n;U)) regret when

ξ∗(n;U) = ω(log n), and where n is the number of transmission slots.

The proof for the regret bound under the ρEST policy consists

of two main parts: we prove bounds on regret conditioned

on the event that none of the users over-estimate U. Second,

we show that the probability of over-estimation at any of the

users goes to zero asymptotically in the horizon length. Taken

together, we obtain the regret bound for the ρEST policy.

Note that in order to have small regret, it is crucial that none

of the users over-estimate U. This is because when there is

over-estimation, there is a nite probability of selecting the

U-worst channels even upon learning the true ranks of the

channels. Note that regret is incurred whenever a U-worst

channel is selected since under perfect knowledge this channel

would not be selected. Hence, under over-estimation, the regret

grows linearly in the number of transmissions.

In a nutshell, under the ρEST policy, the decision to increment

the estimate U reduces to a hypothesis-testing problem with

hypotheses H0: number of users is less than or equal to the

current estimate and H1: number of users is greater than

the current estimate. In order to have a sub-linear regret,

the false-alarm probability (deciding H1 under H0) needs to

decay asymptotically. This is ensured by selecting appropriate

7In this section, we assume that the users are aware of the horizon length

n for transmission. Note that this is not a limitation and can be extended

to case of unknown horizon length as follows: implement the algorithm by

xing horizon lengths to n0, 2n0, 4n0 ... for a xed n0 ∈ N and discarding

estimates from previous stages.

thresholds ξ(n) to test against the collision counts obtained

through feedback.

Conditional Regret: We now give the result for the rst

part. Dene the “good event” C(n;U) that none of the users

over-estimates U under ρEST as

C(n;U):={



U

j=1

U EST

j (n) ≤ U}. (22)

The regret conditioned on C(n;U), denoted by

R(n; μ, U, ρEST)|C(n;U), is given by

n



U

k=1

μ(k∗) −

C

i=1



U

j=1

μ(i)E[Vi,j (n)|C(n;U)],

where Vi,j (n) is the number of times that user j is the sole

user of channel i. Similarly, we have conditional expectations

of E[Ti,j (n)|C(n;U)] and of the number of collisions in Ubest channels, given by E[M(n)|C(n;U)]. We now show that

the regret conditioned on C(n;U) is O(max(ξ∗(n;U), log n)).

Lemma 4: (Conditional Regret): When all the U secondary

users implement the ρEST policy, we have for all i ∈ U-worst

channel and each user j = 1,...,U,

E[Ti,j (n)|C(n)] ≤ 

U

k=1

 8 log n

Δ(i, k∗)2 +1+ π23



. (23)

The conditional expectation on number of collisions M(n) in

the U-best channel satises

E[M(n)|C(n;U)] ≤ U 

U

k=1

ξ(n; k) ≤ U2ξ∗(n;U). (24)

From (15), we have R(n)|C(n;U) is O(max(ξ∗(n;U), log n))

for any n ∈ N.

Proof: See Appendix F. ✷

Probability of Over-estimation: We now prove that none

of the users over-estimates8 U under the ρEST policy, i.e., the

probability of the event C(n;U) in (22) approaches one as

n → ∞, when the thresholds ξ(n;U) for testing against

the collision count are chosen appropriately (see line 6 in

Algorithm 4). Trivially, we can set ξ(n; 1) = 1 since a single

collision is enough to indicate that there is more than one user.

For any other k > 1, we choose function ξ satisfying

ξ(n; k) = ω(log n), ∀k > 1. (25)

We prove that the above condition ensures that over-estimation

does not occur.

Recall that T �

(n; ρEST) is the number of slots where any

one of the top U-estimated ranks of the channels at some

user is wrong under the ρEST policy. We show that E[T �

(n)] is

O(log n).

Lemma 5 (Time spent with wrong estimates): The

expected number of slots where any of the top U-estimated

8Note that the ρEST policy automatically ensures that all the users do not

under-estimate U, since it increments U based on collision estimate. This

implies that the probability of the event that all the users under-estimate U

goes to zero asymptotically.
182

ANANDKUMAR et al.: DISTRIBUTED ALGORITHMS FOR LEARNING AND COGNITIVE MEDIUM ACCESS WITH LOGARITHMIC REGRET 739

ranks of the channels at any user is wrong under the ρEST

policy satises

E[T �

(n)] ≤ U 

U

a=1



C

b=a+1

 8 log n

Δ(a∗, b∗)2 +1+ π23



. (26)

Proof: The proof is on the lines of Lemma 3 ✷

Recall the denition of Υ(U, U) in the previous section,

as the maximum time to absorption starting from any initial

distribution of the nite-state Markov chain, where the states

correspond to different user congurations and the absorbing

state corresponds to the collision-free conguration. We now

generalize the denition to Υ(U, k), as the time to absorption

in a new Markov chain, where the state space is the set of

congurations of U users in k channels, and the transition

probabilities are dened on similar lines. Note that Υ(U, k)

is almost-surely nite when k ≥ U and ∞ otherwise (since

there is no absorbing state in the latter case).

We now bound the maximum value of the collision count

Φk,j (m) under the ρEST policy in (20) using T �

(m), the total

time spent with wrong channel estimates, and Υ(U, k), the

time to absorption in the Markov chain. Let st

≤ denote the

stochastic order for two random variables [25].

Proposition 2: The maximum collision count in (20) over

all users under the ρEST policy satises

max

j=1,...,U

Φk,j (m)

st

≤ (T �(m) + 1)Υ(U, k), ∀m ∈ N. (27)

Proof: The proof is on the lines of Theorem 3. See Appendix G. ✷

We now prove that the probability of over-estimation goes

to zero asymptotically.

Lemma 6 (No Over-estimation Under ρEST): For threshold

function ξ satisfying (25), the event C(n;U) in (22) satises

limn→∞ P[C(n;U)] = 1, (28)

and hence, none of the users over-estimates U under the ρEST

policy.

Proof: See Appendix H. ✷

We now give the main result of this section that ρEST has

slightly more than logarithmic regret asymptotically and this

depends on the threshold function ξ∗(n;U) in (21).

Theorem 5 (Asymptotic Regret Under ρEST): With

threshold function ξ satisfying conditions in (25), the

policy ρEST(n, C, gj (m), ξ) in Algorithm 4 satises

lim sup

n→∞

R(n; μ, U, ρEST)

ξ∗(n;U) < ∞. (29)

Proof: From Lemma 4 and Lemma 6. ✷

Hence, the regret under the proposed ρEST policy is

O(ξ∗(n;U)) under fully decentralized setting without the

knowledge of number of users when ξ∗(n;U) = ω(log n).

Hence, O(f(n) log n) regret is achievable for all functions

f(n) → ∞ as n → ∞. The question of whether logarithmic

regret is possible under unknown number of users is of

interest.

Note the difference between the ρEST policy in Algorithm 4

under unknown number of users with the ρRAND policy with

known number of users in Algorithm 3. The regret under ρEST

is O(f(n) log n) for any function f(n) = ω(1), while it is

O(log n) under the ρRAND policy. Hence, we are able to quantify

the degradation of performance when the number of users is

unknown.

VI. LOWER BOUND & EFFECT OF NUMBER OF USERS

A. Lower Bound For Distributed Learning & access

We have so far designed distributed learning and access

policies with provable bounds on regret. We now discuss the

relative performance of these policies, compared to the optimal

learning and access policies. This is accomplished by noting

a lower bound on regret for any uniformly-good policy, rst

derived in [4] for a general class of uniformly-good timedivision policies. We restate the result below.

Theorem 6 (Lower Bound [4]): For any uniformly good

distributed learning and access policy ρ, the sum regret in

(2) satises

lim inf

n→∞

R(n; μ, U, ρ)

log n

≥ 

i∈U-worst



U

j=1

Δ(U∗, i)

D(μi, μj∗ )

. (30)

The lower bound derived in [9] for centralized learning and

access holds for distributed learning and access considered

here. But a better lower bound is obtained above by considering the distributed nature of learning. The lower bound for

distributed policies is worse than the bound for the centralized

policies in (11). This is because each user independently learns

the channel availabilities μ in a distributed policy, whereas

sensing decisions from all the users are used for learning in a

centralized policy.

Our distributed learning and access policy ρRAND matches

the lower bound on regret in (15) in the order (log n) but the

scaling factors are different. It is not clear if the regret lower

bound in (30) can be achieved by any policy under no explicit

information exchange and is a topic for future investigation.

B. Behavior with Number of Users

We have so far analyzed the sum regret under our policies

under a xed number of users U. We now analyze the behavior

of regret growth as U increases while keeping the number of

channels C>U xed.

Theorem 7 (Varying Number of Users): When the number

of channels C is xed and the number of users U<C is

varied, the sum regret under centralized learning and access

ρCENT in (12) decreases as U increases while the upper bounds

on the sum regret under ρRAND in (15) monotonically increases

with U.

Proof: The proof involves analysis of (12) and (15). To

prove that the sum regret under centralized learning and access

in (12) decreases with the number of users U, it sufces to

show that for i ∈ U-worst channel,

Δ(U∗, i)

D(μi, μU∗ )

decreases as U increases. Note that μ(U∗) and D(μi, μU∗ )

decrease as U increases. Hence, it sufces to show that

μ(U∗)

D(μi, μU∗ )
183

740 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 29, NO. 4, APRIL 2011

decreases with U. This is true since its derivative with respect

to U is negative.

For the upper bound on regret under ρRAND in (15), when U

is increased, the number of U-worst channels decreases and

hence, the rst term in (15) decreases. However, the second

term consisting of collisions M(n) increases to a far greater

extent. ✷

Note that the above results are for the upper bound on regret

under the ρRAND policy and not the regret itself. Simulations in

Section VII reveal that the actual regret also increases with U.

Under the centralized scheme ρCENT, as U increases, the number

of U-worst channels decreases. Hence, the regret decreases,

since there are less number of possibilities of making bad

decisions. However, for distributed schemes although this

effect exists, it is far outweighed by the increase in regret

due to the increase in collisions among the U users.

In contrast, the distributed lower bound in (30) displays

anomalous behavior with U since it fails to account for

collisions among the users. Here, as U increases there are

two competing effects: a decrease in regret due to decrease

in the number of U-worst channels and an increase in regret

due to increase in the number of users visiting these U-worst

channels.

VII. NUMERICAL RESULTS

We present simulations for the algorithms developed in

the paper, varying the number of users and channels to

verify the performance of the algorithms detailed earlier. We

consider C= 9 channels (or a subset of them when the number of channels is varying) with probabilities of availability

characterized by Bernoulli distributions with evenly spaced

parameters ranging from 0.1 to 0.9.

Comparison of Different Schemes: Fig. 2a compares the

regret under the centralized and random allocation schemes in

a scenario with U = 4 cognitive users vying for access to the

C = 9 channels. The theoretical lower bound for the regret

in the centralized case from Theorem 2 and the distributed

case from Theorem 6 are also plotted. The upper bounds on

the random allocation scheme from Theorem 4 is not plotted

here, since the bounds are loose especially as the number of

users U increases. Finding tight upper bounds is a subject of

future study.

As expected, centralized allocation has the least regret.

Another important observation is the gap between the lower

bounds on the regret and the actual regret in both the distributed and the centralized cases. In the centralized scenario,

this is simply due to using the gMEAN statistic in (34) instead

of the optimal gOPT statistic in (5). However, in the distributed

case, there is an additional gap since we do not account for

collisions among the users. Hence, the schemes under consideration are O(log n) and achieve order optimality although

they are not optimal in the scaling constant.

Performance with Varying U and C: Fig. 3a explores the

impact of increasing the number of secondary users U on the

regret experienced by the different policies with the number

of channels C xed. With increasing U, the regret decreases

for the centralized schemes and increases for the distributed

schemes, as predicted in Theorem 7. The monotonic increase

0 1 2 3 4 5

x 104

0

50

100

150

200

250

300

350

400

450

500

Random Allocation Scheme

Central Allocation Scheme

Distributed Lower Bound

Centralized Lower Bound

(a) Normalized regret R(n)

log n vs. n slots.

U = 4 users, C = 9 channels.

0 2000 4000 6000 8000 10000 0

50

100

150

200

250

300

350

400

450

500

ρRAND with gMEAN

ρRAND with gOPT

Distributed Lower Bound

(b) Normalized regret R(n)

log n vs. n slots.

U = 4 users, C = 9 channels.

20 40 60 80 100 120 140 0

20

40

60

80

100

120

140

160

180

200

μ unknown

μ known: UE[Υ(U, U)]

Upper Bound on UE[Υ(U, U)]

(c) No. of collisions M(n) vs. n slots.

U = 4 users, C = 9 channels, ρRAND

policy.

Fig. 2. Simulation Results. Probability of Availability μ =

[0.1, 0.2,..., 0.9].

of regret under random allocation ρRAND is a result of the

increase in the collisions as U increases. On the other hand,

the monotonic decreasing behavior in the centralized case is

because as the number of users increases, the number of U-
184

ANANDKUMAR et al.: DISTRIBUTED ALGORITHMS FOR LEARNING AND COGNITIVE MEDIUM ACCESS WITH LOGARITHMIC REGRET 741

1 2 3 4 5 6 0

50

100

150

200

250

300

350

400

Random Allocation Scheme

Central Allocation Scheme

Distributed Lower Bound

Centralized Lower Bound

(a) Normalized regret R(n)

log n vs. U users.

C = 9 channels, n = 2500 slots.

2 3 4 5 6 7 8 9 0

10

20

30

40

50

60

70

80

90

100

Random Allocation Scheme

Central Allocation Scheme

Distributed Lower Bound

Centralized Lower Bound

(b) Normalized regret R(n)

log n vs. C channels.

U = 2 users, n = 2500 slots.

1 2 3 4 5 6 0

50

100

150

200

250

300

350

400

Random Allocation Scheme

Central Allocation Scheme

Distributed Lower Bound

Centralized Lower Bound

(c) Normalized regret R(n)

log n vs. U users.

User-channel ratio U/C = 0.5, n = 2500 slots.

Fig. 3. Simulation Results. Probability of Availability μ =

[0.1, 0.2,..., 0.9].

worst channels decreases resulting in lower regret. Also, the

lower bound for the distributed case in (30) initially increases

and then decreases with U. This is because as U increases

there are two competing effects: decrease in regret due to

decrease in number of U-worst channels and increase in regret

due to increase in number of users visiting these U-worst

channels.

Fig. 3b evaluates the performance of the different algorithms

as the number of channels C is varied while xing the number

of users U. The probability of availability of each additional

channel is set higher than those already present. Here, the

regret monotonically increases with C in all cases. When the

number of channels increases along with the quality of the

channels, the regret increases as a result of an increase in the

number of U-worst channels as well as the increasing gap in

quality between the U-best and U-worst channels.

Also, the situation where the ratio U/C is xed to be 0.5

and both the number of users and channels along with their

quality increase is considered in Fig. 3c and we nd that

the regret grows in this case as well. Once again, this is in

agreement with theory since the number of U-worst channels

increases as U and C increase while keeping U/C xed.

Collisions and Learning: Fig. 2c veries the logarithmic

nature of the number of collisions under the random allocation scheme ρRAND. Additionally, we also plot the number of

collisions under ρRAND in the ideal scenario when the channel

availability statistics μ are known to see the effect of learning

on the number of collisions. The low value of the number

of collisions obtained under known channel parameters in

the simulations is in agreement with theoretical predictions,

analyzed as UE[Υ(U, U)] in Lemma 2. As the number of

slots n increases, the gap between the number of collisions

under the known and unknown parameters increases since the

former converges to a nite constant while the latter grows as

O(log n). The logarithmic behavior of the cumulative number

of collisions can be inferred from Fig. 2a. However, the curve

in Fig. 2c for the unknown parameter case appears linear in

n due to the small value of n.

Difference between gOPT and gMEAN: Since the statistic gMEAN

used in the schemes in this paper differs from the optimal

statistic gOPT in (5), a simulation is done to compare the performance of the schemes under both the statistics. As expected, in

Fig. 2b, the optimal scheme has better performance. However,

the use of gMEAN enables us to provide nite-time bounds, as

described earlier.

Fairness: One of the important features of ρRAND is that

it does not favor any one user over another. Each user has

an equal chance of settling down in any one of the U-best

channels. Fig. 4 evaluates the fairness characteristics of ρRAND.

The simulation assumes U = 4 cognitive users vying for

access to C = 9 channels. The graph depicts which user

asymptotically gets the best channel over 1000 runs of the

random allocation scheme. As can be seen, each user has

approximately the same frequency of being allotted the best

channel indicating that the random allocation scheme is indeed

fair.

VIII. CONCLUSION

In this paper, we proposed novel policies for distributed

learning of channel availability statistics and channel access

of multiple secondary users in a cognitive network. The rst
185

742 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 29, NO. 4, APRIL 2011

1 2 3 4 0

50

100

150

200

250

300

Fig. 4. Simulation Results. Probability of Availability μ =

[0.1, 0.2,..., 0.9]. No. of slots where user has best channel vs. user. U = 4,

C = 9, n = 2500 slots, 1000 runs, ρRAND.

policy assumed that the number of secondary users in the

network is known, while the second policy removed this

requirement. We provide provable guarantees for our policies

in terms of sum regret. By noting the lower bound on regret for

any uniformly-good learning and access policy, we nd that

our rst policy achieves order-optimal regret while our second

policy is also nearly order optimal. Our analysis in this paper

provides insights on incorporating learning and distributed

medium access control in a practical cognitive network.

The results of this paper open up an interesting array of

problems for future investigation. Simulations suggest that our

lower and upper bounds are not tight in terms of the scaling

constant and that better bounds are needed. Our assumptions

of an i.i.d. model for primary user transmissions and perfect

sensing at the secondary users need to be relaxed. Our policy

allows for an unknown but xed number of secondary users,

and it is of interest to incorporate users dynamically entering

and leaving the system. Moreover, our model ignores dynamic

trafc at the secondary nodes and extension to a queueingtheoretic formulation is desirable. We consider the worst-case

scenario that there is no information exchange among the secondary users. Extension to the case with limited information

exchange is of interest. The proposed schemes are designed

to be simple and tractable for mathematical analysis. It is

desirable to obtain more robust schemes with additional design

considerations for real-world deployment.

Acknowledgement

The authors thank the guest editors and anonymous reviewers for valuable comments that vastly improved this paper, and

for pointing out an error in Proposition 1. The authors thank

Keqin Liu and Prof. Qing Zhao for extensive discussions,

feedback on the proofs in an earlier version of the manuscript,

and for sharing their simulation code. The authors also thank

Prof. Lang Tong and Prof. Robert Kleinberg at Cornell, Prof.

Bhaskar Krishnamachari at USC and Prof. John Tsitsiklis and

Dr. Ishai Menache at MIT for helpful comments.

APPENDIX

A. Proof of Theorem 2

The result in (13) involves extending the results of [11,

Thm. 1]. Dene Ti(n):=U

j=1 Ti,j (n) as the number of times

a channel i is sensed in n rounds for all users. We will show

that

E[Ti(n)] ≤ 

k∈U-best

 8 log n

Δ(k∗, i)2 +1+ π23



, ∀i ∈ U-worst.

(31)

We have

P[Tx. in i in nth slot] = P[g(U∗; n) ≤ g(i; n)],

=P[A(i; n) ∩ (g(U∗; n) ≤ g(i; n))]

+ P[Ac

(i; n) ∩ (g(U∗; n) ≤ g(i; n))],

where

A(i; n):= 

k∈U-best

(g(k; n) ≤ g(i; n))

is the event that at least one of the U-best channels has gstatistic less than i. Hence, from union bound we have

P[A(i; n)] ≤ 

k∈U-best

P[g(k; n) ≤ g(i; n)].

We have for C>U,

P[Ac

(i; n) ∩ (g(U∗; n) ≤ g(i; n))] = 0,

Hence,

P[Tx. in i in nth round] ≤ 

k∈-best

P[g(k; n) ≤ g(i; n)].

On the lines of [11, Thm. 1], we have ∀k, i :

k is U-best, i is U-worst

n

l=1

I[g(k; l) ≤ g(i; l)] ≤ 8 log n

Δ(k∗, i)2 +1+ π23 .

Hence, we have (31). For the bound on regret, we can break

R in (2) into two terms

R(n; μ, U, ρCENT) = 

i∈U-worst

 1

U



U

l=1

Δ(l

∗, i)



E[Ti(n)]

+ 

i∈U-best

 1

U



U

l=1

Δ(l

∗, i)



E[Ti(n)].

For the second term, we have



i∈U-best

 1

U



U

l=1

Δ(l

∗, i)



E[Ti(n)]

≤ E[T ∗(n)] 

i∈U-best

 1

U



U

l=1

Δ(l

∗, i)



= 0,

where T ∗(n):= max i∈U-best

Ti(n). Hence, we have the bound.
186 ANANDKUMAR et al.: DISTRIBUTED ALGORITHMS FOR LEARNING AND COGNITIVE MEDIUM ACCESS WITH LOGARITHMIC REGRET 743

B. Proof of Proposition 1

For convenience, let Ti(n) := �U

j=1 Ti,j (n), Vi(n) :=

�U

j=1 Vi,j (n). Note that �Ci=1 Ti(n) = nU, since each user

selects one channel for sensing in each slot and there are U

users. From (3),

R(n) =n

�

U

i=1

μ(i

∗) −�

C

i=1

μ(i)E[Vi(n)],

≤ �

i∈U-best

μ(i)(n − E[Vi(n)])

≤μ(1∗)(nU − �

i∈U-best

E[Vi(n)]) (32)

=μ(1∗)(E[M(n)] + �

i∈U-worst

E[Ti(n)]), (33)

where Eqn.(32) uses the fact that Vi(n) ≤ n since total number

of sole occupancies in n slots of channel i is at most n, and

Eqn.(33) uses the fact that M(n) = �

i∈U-best(Ti(n)−Vi(n)).

For the lower bound, since each user selects one channel

for sensing in each slot, �C

i=1

�U

j=1 Ti,j(n) = nU. Now

Ti,j (n) ≥ Vi,j (n).

R(n; μ, U, ρ) ≥

1

U

⎡

⎣

�

U

k=1

�

U

j=1

�

C

i=1

Δ(U∗, i)E[Ti,j (n)]

⎤

⎦ ,

≥�

U

j=1

�

i∈U-worst

Δ(U∗, i)E[Ti,j (n)].

C. Proof of Lemma 2

Although, we could directly compute the time to absorption

of the Markov chain, we give a simple bound E[Υ(U, U)] by

considering an i.i.d process over the same state space. We term

this process as a genie-aided modication of random allocation

scheme, since this can be realized as follows: in each slot, a

genie checks if any collision occurred, in which case, a new

random variable is drawn from Unif(U) by all users. This is

in contrast to the original random allocation scheme where a

new random variable is drawn only when the particular user

experiences a collision. Note that for U = 2 users, the two

scenarios coincide.

For the genie-aided scheme, the expected number of slots to

hit orthogonality is just the mean of the geometric distribution

�∞

k=1

k(1 − p)

kp = 1 − p

p

< ∞, (34)

where p is the probability of having an orthogonal conguration in a slot. This is in fact the reciprocal of the number of

compositions of U [24, Thm. 5.1], given by

p =

�2U − 1

U

�−1

. (35)

The above expression is nothing but the reciprocal of number

of ways U identical balls (users) can be placed in U different

bins (channels): there are 2U − 1 possible positions to form

U partitions of the balls.

Now for the random allocation scheme without the genie,

any user not experiencing collision does not draw a new

variable from Unif(U). Hence, the number of possible congurations in any slot is lower than under genie-aided scheme.

Since there is only one conguration satisfying orthogonality9

,

the probability of orthogonality increases in the absence of the

genie and is at least (35). Hence, the number of slots to reach

orthogonality without the genie is at most (34). Since in any

slot, at most U collisions occur, (17) holds.

D. Proof of Lemma 3

Let cn,m:=�2 log n

m .

Case 1: Consider U = C = 2 rst. Let

A(t, l):={gMEAN

j (1∗;t − 1) ≤ gMEANj (2∗;t − 1), T �

j(t − 1) ≥ l}.

On lines of [11, Thm. 1],

T �

(n) ≤ l +�n

t=2

I[A(t, l)],

≤ l +�∞

t=1

�t

m+h=l

I

�

X ̄1∗,j (h) + ct,h ≤ X ̄2∗,j (m) + ct,m�

.

The above event is implied by

X ̄1∗,j(h) + ct,h ≤ X ̄2∗,j (h) + ct,h+m

since ct,m > ct,h+m.

The above event implies at least one of the following events

and hence, we can use the union bound.

X ̄1∗,j (h) ≤ μ1∗ − ct,h,

X ̄2∗,j (m) ≥ μ2∗ + ct,h+m,

μ1∗ < μ2∗ + 2ct,h+m.

From the Chernoff-Hoeffding bound,

P[X ̄1∗,j (t) ≤ μ1∗ − ct,h] ≤ t

−4,

P[X ̄2∗,j ≥ μ2∗ + ct,h+m] ≤ t

−4,

and the event that μ1∗ < μ2∗ + 2ct,h+m implies that

h + m < �

8 log t

Δ2

1∗,2∗

�

.

Since

�∞

t=1

�t

m=1

�t

h=1

2t

−4 = π2

3 ,

,

E[T �

(n;U = C = 2)] ≤ 8 log n

Δ2

1∗,2∗

+1+ π2

3 .

Case 2: For min(U, C) > 2, we have

T �

(n) ≤ U �

U

a=1

�

C

b=a+1

�n

m=1

I(gMEAN

j (a∗; m) < gMEANj (b∗; m)),

where a∗ and b∗ represent channels with ath and bth highest

availabilities. On lines of the result for U = C = 2, we can

show that

�n

m=1

EI[gMEAN

j (a∗; m) < gMEANj (b∗; m)] ≤ 8 log n

Δ2

a∗,b∗

+1+ π2

3 .

Hence, (18) holds.

9since all users are identical for this analysis.
187

744 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 29, NO. 4, APRIL 2011

E. Proof of Theorem 3

Dene the good event as all users having correct top Uorder of the g-statistics, given by

G(n):= 

U

j=1

{Top-U entries of gj (n) are same as in μ}.

The number of slots under the bad event is

n

m=1

I[Gc(m)] = T �(n),

by denition of T �

(n). In each slot, either a good or a bad

event occurs. Let γ be the total number of collisions in Ubest channels between two bad events, i.e., under a run of

good events. In this case, all the users have the correct top

U-ranks of channels and hence,

E[γ|G(n)] ≤ UE[Υ(U, U)] < ∞,

where E[Υ(U, U)] is given by (17). Hence, each transition

from the bad to the good state results in at most UE[Υ(U, U)]

expected number of collisions in the U-best channels. The

expected number of collisions under the bad event is at most

UE[T �

(n)]. Hence, (19) holds.

F. Proof of Lemma 4

Under C(n;U), a U-worst channel is sensed only if it is

mistaken to be a U-best channel. Hence, on lines of Lemma 1,

E[Ti,j (n)|C(n;U)] = O(log n), ∀i ∈ U-worst, j = 1, . . . , U.

For the number of collisions M(n) in the U-best channels,

there can be at most U a

k=1 ξ(n; k) collisions in the U-best

channels where a := maxj=1,...,U Uj is the maximum estimate

of number of users. Conditioned on C(n;U,), a ≤ U, and

hence, we have (24).

G. Proof of Proposition 2

Dene the good event as all users having correct top Uorder, given by

G(n):= 

U

j=1

{Top-U entries of gj (n) are same as in μ}.

The number of slots under the bad event is

n

m=1

I[Gc(m)] = T �(n),

by denition of T �

(n). In each slot, either a good or a bad

event occurs. Let γ be the total number of collisions in k-best

channels between two bad events, i.e., under a run of good

events. In this case, all the users have the correct top U-ranks

of channels and hence,

γ|G(n)

st

≤ UΥ(U, k),

The number of collisions under the bad event is at most T �

(n).

Hence, (27) holds.

H. Proof of Lemma 6

We are interested in

P[Cc

(n);U] = P[∪U

j=1U EST

j (n) > U],

= P[

n

m=1



U

j=1

{ΦU,j(m) > ξ(n;U)}],

= P[ max

j=1,...,U

ΦU,j (n) > ξ(n;U)],

where Φ is given by (20). For U = 1, we have P[Cc(n);U]=0

since no collisions occur.

Using (27) in Proposition 2,

P[ k

max

j=1

Φk,j (n) > ξ(n; k)]

≤ P[kΥ(U, k)(T �

(n) + 1) > ξ(n; k)]

≤ P[k(T �

(n) + 1) > ξ(n; k)

αn

] + P[Υ(U, k) > αn]

≤

kαn(E[T �

(n)] + 1)

ξ(n; k) + P[Υ(U, k) > αn], (36)

using Markov inequality. By choosing αn = ω(1), the second

term in (36), viz., P[Υ(U, k) > αn] → 0 as n → ∞, for

k ≥ U. For the rst term, from (26) in Lemma 5, E[T �

(n)] =

O(log n). Hence, by choosing αn = o(ξ∗(n; k)/ log n), the

rst term decays to zero. Since ξ∗(n;U) = ω(log n), we can

choose αn satisfying both the conditions. By letting k = U in

(36), we have P[Cc(n);U] → 0 as n → ∞, and (28) holds.

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Animashree Anandkumar (S’ 02, M’ 09) received

her B.Tech in Electrical Engineering from the Indian

Institute of Technology Madras in 2004 and her

Ph.D. in Electrical and Computer Engineering with

a minor in Applied Math from Cornell University

in 2009. She was a post-doctoral researcher at the

Stochastic Systems Group at Massachusetts Institute of Technology between 2009 and 2010. She

is currently an assistant professor at the Electrical

Engineering and Computer Science Department at

University of California, Irvine. She is also a member of the center for pervasive communications and computing at UCI.

She is the recipient of the 2008 IEEE Signal Processing Society (SPS)

Young Author award for her paper co-authored with Lang Tong which

appeared in the IEEE Transactions on Signal Processing. She is the recipient

of the Fran Allen IBM Ph.D fellowship 2008-09, presented in conjunction

with the IBM Ph.D. Fellowship Award. She received the Best Thesis Award

2009 by the ACM Sigmetrics society. Her research interests are in the area

of statistical-signal processing, information theory and networking with a

focus on distributed inference and learning of graphical models. She serves

on the Technical Program Committee of IEEE INFOCOM 2011, ACM

MOBIHOC 2011, IEEE ICC 2010, MILCOM 2010 and IEEE PIMRC 2010.

She has served as a reviewer for IEEE Transactions on Signal Processing,

IEEE Journal on Selected Areas in Communications, IEEE Transactions on

Information Theory, IEEE Transactions on Wireless Communications and

IEEE Signal Processing Letters.

Nithin Michael (S ‘06) received the B.S. degree

summa cum laude from Drexel University, Philadelphia, PA, in 2008 in electrical engineering. He is

currently pursuing the M.S./Ph.D. joint degree program in electrical engineering at Cornell University.

Mr. Michael was the recipient of rst Honors in ECE

and the Highest Academic Achievement Award at

Drexel University in 2008. At Cornell University,

he has been awarded the Jacobs Fellowship in 2008

and 2010.

Ao Tang (S’ 01 M’ 07) received the B.E. in

Electronics Engineering from Tsinghua University,

Beijing, China, and the Ph.D. in Electrical Engineering with a minor in applied and computational mathematics from the California Institute of Technology,

Pasadena, CA, in 1999 and 2006, respectively.

He is an Assistant Professor in the School of

Electrical and Computer Engineering at Cornell University, where his current research is on control and

optimization of networks including communication

networks, power networks and on-chip networks. Dr.

Tang was the recipient of the 2006 George B. Dantzig Best Dissertation

Award, the 2007 Charles Wilts best dissertation Prize, and a 2009 IBM Faculty

Award.

Ananthram Swami (F’08) received the B.Tech.

degree from the Indian Institute of Technology (IIT),

Bombay; the M.S. degree from Rice University,

Houston, TX, and the Ph.D. degree from the University of Southern California (USC), Los Angeles,

all in electrical engineering.

He has held positions with Unocal Corporation,

USC, CS-3 and Malgudi Systems. He was a Statistical Consultant to the California Lottery, developed HOSAT, a MATLAB-based toolbox for nonGaussian signal processing, and has held visiting

faculty positions at INP, Toulouse, France. He is the ST for Network Science

at the U.S. Army Research Laboratory. His work is in the broad areas of signal

processing, wireless communications, sensor and mobile ad hoc networks. He

is the co-editor of the book Wireless Sensor Networks: Signal Processing &

Communications Perspectives (New York: Wiley, 2007).

Dr. Swami is a member of the IEEE Signal Processing Society’s (SPS)

Technical Committee on Sensor Array and Multichannel systems and serves

on the Senior Editorial Board of the IEEE Journal on Selected Topics in Signal

Processing. He was a tutorial speaker on Networking Cognitive Radios for

Dynamic Spectrum Access at IEEE ICC 2010 and co-chair of IEEE SPAWC

2010.
Research@

189

A Survey on Trust Management for 

Mobile Ad Hoc Networks

Jin-Hee Cho, Ananthram Swami, and Ing-Ray Chen

IEEE Communications Surveys & Tutorials, 13(4), 562-583 (2011)
190 562 IEEE COMMUNICATIONS SURVEYS & TUTORIALS, VOL. 13, NO. 4, FOURTH QUARTER 2011

A Survey on Trust Management for Mobile

Ad Hoc Networks

Jin-Hee Cho, Member, IEEE, Ananthram Swami, Fellow, IEEE, and Ing-Ray Chen, Member, IEEE

Abstract—Managing trust in a distributed Mobile Ad Hoc Network (MANET) is challenging when collaboration or cooperation

is critical to achieving mission and system goals such as reliability,

availability, scalability, and recongurability. In dening and

managing trust in a military MANET, we must consider the

interactions between the composite cognitive, social, information

and communication networks, and take into account the severe

resource constraints (e.g., computing power, energy, bandwidth,

time), and dynamics (e.g., topology changes, node mobility, node

failure, propagation channel conditions). We seek to combine

the notions of “social trust” derived from social networks with

“quality-of-service (QoS) trust” derived from information and

communication networks to obtain a composite trust metric. We

discuss the concepts and properties of trust and derive some

unique characteristics of trust in MANETs, drawing upon social

notions of trust. We provide a survey of trust management

schemes developed for MANETs and discuss generally accepted

classications, potential attacks, performance metrics, and trust

metrics in MANETs. Finally, we discuss future research areas

on trust management in MANETs based on the concept of social

and cognitive networks.

Index Terms—Trust management, mobile ad hoc networks,

social networks, cognitive networks, trust, trust metrics.

I. INTRODUCTION

I

N AN INCREASINGLY networked world, increased connectivity could lead to improved information sharing, facilitate collaboration, and enable distributed decision making,

which is the underlying concept in Network Centric Operations. In mobile ad hoc networks (MANETs), the distributed

decision making should take into account trust in the elements:

the sources of information, the processors of information, the

elements of the communications network across which the

information is transmitted, etc. This trust must often be derived

under time-critical conditions, and in a distributed way.

A. Design Challenges in MANET Protocols

A mobile ad hoc network [1] consists of wireless mobile

nodes forming a temporary network without the help of centralized infrastructure, and where nodes communicate through

multi-hops.

Security protocol designers for MANETs face technical

challenges due to severe resource constraints in bandwidth,

Manuscript received 29 September 2009; revised 1 March 2010, 28 June

2010, and 7 July 2010.

Jin-Hee Cho and Ananthram Swami are with the Computational and

Information Sciences Directorate, U.S. Army Research Laboratory, 2800

Powder Mill Rd., Adelphi, Maryland 20783, USA (e-mail: {jinhee.cho,

ananthram.swami}@us.army.mil).

Ing-Ray Chen is with the Department of Computer Science, Virginia

Polytechnic Institute and State University, 7054 Haycock Road, Falls Church,

VA 22043, USA (e-mail: irchen@vt.edu).

Digital Object Identier 10.1109/SURV.2011.092110.00088

memory size, battery life, computational power, and unique

wireless characteristics such as openness to eavesdropping,

lack of specic ingress and exit points, high security threats,

vulnerability, unreliable communication, and rapid changes

in topologies or memberships because of user mobility or

node failure [1][2][3]. In addition, compared with designing

security protocols for civilian MANETs, designing security

protocols for military MANETs requires additional caution,

since battleeld communication networks must cope with

hostile environments, node heterogeneity, often stringent performance constraints, node subversion, high tempo operations

leading to rapid changes in network topology and service

requirements, and dynamically formed communities of interest wherein participants may not have predened trust

relationships [4]. To cope with these dynamics, networks

must be able to recongure seamlessly, via low-complexity

distributed network management schemes [3]. Security in a

tactical network includes notions of communication security

which can be easily quantied as opposed to the perception

of security which is hard to quantify.

B. Motivation for Trust Management in MANETs

The concept of ”Trust” originally derives from social sciences and is dened as the degree of subjective belief about

the behaviors of a particular entity [5]. Blaze et al. [6] rst

introduced the term ”Trust Management” and identied it as

a separate component of security services in networks and

claried that ”Trust management provides a unied approach

for specifying and interpreting security policies, credentials,

and relationships.”

Trust management in MANETs is needed when participating nodes, without any previous interactions, desire to

establish a network with an acceptable level of trust relationships among themselves. Examples would be in building

initial trust bootstrapping [7], coalition operations without

predened trust, and authentication of certicates generated

by another party when links are down or ensuring safety

before entering a new zone [8]. In addition, trust management

has diverse applicability in many decision making situations

including intrusion detection, authentication, access control,

key management, isolating misbehaving nodes for effective

routing, and other purposes.

Trust management, including trust establishment, trust update, and trust revocation, in MANETs is also much more

challenging than in traditional centralized environments. For

example, collecting trust information or evidence to evaluate trustworthiness is difcult due to changes in topology

induced by node mobility or node failure. Further, resource

constraints often conne the trust evaluation process only

U.S. Government Work Not Protected by U.S. Copyright.
191

CHO et al.: A SURVEY ON TRUST MANAGEMENT FOR MOBILE AD HOC NETWORKS 563

Fig. 1. The multidisciplinary concept of trust and its application in communications and networking.

to local information. The dynamic nature and characteristics

of MANETs result in uncertainty and incompleteness of the

trust evidence, which is continuously changing over time [8]

[9]. Despite a couple of surveys of trust management [10]

[11] [12], a comprehensive survey of trust management in

MANETs does not exist and is the main aim of this paper.

A short version of this paper was presented at ICCRTS 2009

[13]. The contributions of this paper are: (1) to give a clear

denition of trust in the communication and networking eld,

drawing upon denitions from different disciplines; (2) to

extensively survey the existing trust management schemes

developed for MANETs and investigate their general trends;

and (3) to discuss future research areas based on the concept

of social and cognitive networks.

The rest of this paper is organized as follows. In Section 2,

we discuss the concept of trust in diverse disciplines, give a

clear distinction between trust and trustworthiness, and discuss

the relationship between trust and risk. We also introduce

the main properties of trust in MANETs. Section 3 surveys

generally accepted classications of trust management, attacks considered in existing trust management schemes for

MANETs, and metrics used to measure the performance

of existing MANET trust management schemes. Section 4

surveys trust management schemes that have been developed

for specic purposes, including secure routing, authentication,

intrusion detection, access control, key management, and trust

evidence distribution and evaluation. In Section 5, we discuss

design concepts that designers of MANET trust management

systems should keep in mind and suggest trust metrics based

on the concepts of social trust and quality-of-service (QoS)

trust. Section 6 concludes this paper.

II. CONCEPTS AND PROPERTIES OF TRUST

In this section, we review how trust is dened in different

disciplines and how these trust concepts can be applied in

modeling trust in MANETs. Further, we examine the relationship between trust and risk, and how trust should be dened

in order to realistically reect the unique characteristics of

MANETs.

A. Multidisciplinary Concept of Trust

According to Merriam Webster’s Dictionary [14], trust is

dened as ”assured reliance on the character, ability, strength,

or truth of someone or something.” Despite the subjective

nature of trust, the concept of trust has been very attractive

to network security protocol designers because of its diverse

applicability as a decision making mechanism. We examine

the literature to study how trust is dened in various disciplines including sociology, economics, philosophy, psychology, organizational management, and autonomic computing

in industrial and system engineering. Finally, we also examine

how trust can be dened in communications and networking

with the help of denitions in other elds.

Trust in sociology: Gambetta’s notion of trust [15] is popularly called sociological trust and is dened as an assessor’s a

priori subjective probability that a person (or agent, or group)

will perform specic actions that affect the assessor. That is,

Gambetta [15] describes the nature of trust as subjectivity,

an indicator for future actions, and dynamicity based on

continuous interactions between two entities. Luhmann [16]

also emphasized the importance of trust in society as a

mechanism for building cooperation among people to extend

human interactions for future collaboration. Adams et al. [17]

rephrased Gambetta’s trust concept in applying the sociological concept of trust in computer science; they represented

trust as a continuous variable, quantifying trust in the light

of context or acceptance of risk. They further stressed that

risking betrayal is an important aspect in building trust. To

be useful, network trust models must capture this subjective

aspect of social trust.

Trust in economics: Economists distinguish between the

personal, informal trust that comes from being friendly with

your neighbors and the impersonal, institutionalized trust that

lets you give your credit card number out over the Internet

[18]. Both notions of trust are important in military MANETs.

In economics, trust is represented as an expectation that

applies to situations in which trustors take risky actions under

uncertainty or information incompleteness [19]. However, as

illustrated in the Prisoner’s Dilemma (PD) game [20], trust in

economics is based on the assumption that humans are rational

and strict utility maximizers of their own interest or incentives.

In this sense, when we apply a human trust model to a network

trust model, the assumption of selsh nodes seems reasonable.

But altruistic behaviors can emerge from mechanisms that may

be initially purely selsh [21], and thus making an argument

for redemption mechanisms. Economic models are used in
192

564 IEEE COMMUNICATIONS SURVEYS & TUTORIALS, VOL. 13, NO. 4, FOURTH QUARTER 2011

conjunction with trust-based encryption primitives in [22] to

develop a trust management paradigm for securing information

ows across organizations.

Trust in philosophy: According to the Stanford Encyclopedia of Philosophy [23], trust is important but dangerous. Since

trust allows us to form relationships with others and to rely on

others for love, advice, help, etc., trust is regarded as a very

important factor in our life that compels others to give us such

things with no outside force such as the law. On the other

hand, since trust requires taking a risk that the trustee may

not behave as the trustor expects, trust is dangerous implying

the possible betrayal of trust. In his comments on Lagerspetz’s

book titled Trust: The Tacit Demand, Lahno [24] describes the

author’s view on trust as a moral relationship in human society.

Langerspetz believes that investigations of trust reveal that

”human individuals, their beliefs, desires and actions are only

intelligible against the background of existing social practices

and social ties” [24]. This implies that depending on the nature

of personal relationships between a trustor and a trustee (i.e.,

moral relationship between them), trustful actions or betrayal

can occur.

Trust in psychology: According to the Wikipedia denition

of trust in psychology [25], trust starts from the birth of the

child. As the child grows older, trust also grows stronger.

However, the root of trust derives from the relationship between mother (or caregiver) of the child since the strength of

the family relies on trust, if the child is raised in a family

which is very accepting and loving, the child also returns

those feelings to others by trusting them. But if trust is lost,

it is hard to regain it. In this sense, trust in psychology

emphasizes the cognitive process that human beings learn

trust from their experiences. Deutsch [26] denes trust as the

condence that one will nd what is desired from another

rather than what is feared. An individual may be said to

have trust in the occurrence of an event if he expects its

occurrence and his expectation leads to the behavior which

he perceives to have greater negative consequences if the

expectation is not conrmed than positive consequences if it is

conrmed. In addition, Hardin [27] and Rotter [28] observed

in their experiments that past experience may strikingly affect

later capacity for trust. For example, bad experience with

people will lower the trust level, leading to fewer trusted

relationships with people, and thus fewer opportunities for

mutual gain. Further, they recognized that the gains obtained

by having high trust relationships exceed the loss by having

low trust relationships. For instance, high trustors are less

likely to lie or cheat or steal. Also they are less likely to be

unhappy, conicted, or unstable, and sought by more friends.

Even though high trustors are deceived more often in novel

situations, low trustors are also fooled equally by distrusting

trustworthy people, thereby losing the advantages that high

trustors may have [28].

Trust in organizational management: In this eld, the

concept of trust is also dened as the extent to which one party

is willing to count on someone or something with a feeling

of relative security in spite of possible negative consequences,

emphasizing the possibility of facing risk [29]. Schoorman

et al. [30] dened trust as the willingness to take a risk or

willingness to be vulnerable in the relationship in terms of

ability, integrity, and benevolence. They also explained that

trust is not necessarily mutual and is not reciprocal. Trust

concepts in organizational management can give us insights

on how to measure trust by investigating methods to measure

ability, integrity, and benevolence of each networked node,

as well as on assessing risk. They can also give us insights

on dening group trust (i.e., between a person and a group or

between groups) which is important for dynamic communities

of interest.

Trust in autonomic computing: As technology becomes

more complex, fully understanding automation becomes infeasible, if not impossible, and trust in automation becomes

critical, particularly when unexpected situations arise and

system responses cannot be predicted. Researchers studying

autonomic computing in industrial systems engineering have

sought to develop models of trust to understand how trust in

automation develops and how it may be misplaced. Lee and

See [31] dene trust as the attitude that an agent will help

accomplish an individual’s goals in a situation with uncertainty

and vulnerability. In this sense, an agent can be automation or

another person that actively interacts with the environment on

behalf of the person. Parasuraman [32] links the level of trust

with automation reliability stating that ”Trust often determines

automation usage. Operators may not use a reliable automated

system if they believe it to be untrustworthy.” The notion of

automation reliability as a trust metric is one that is applicable

in MANETs, where the user’s trust in reliability on technology

is an important aspect.

Trust in communications and networking: The concept of

trust also has been attractive to communication and network

protocol designers where trust relationships among participating nodes are critical in building cooperative and collaborative environments to optimize system objectives in terms of

scalability, recongurability, and reliability (i.e., survivability),

dependability, or security. According to Eschenauer et al. [9],

trust is dened as ”a set of relations among entities that

participate in a protocol. These relations are based on the

evidence generated by the previous interactions of entities

within a protocol. In general, if the interactions have been

faithful to the protocol, then trust will accumulate between

these entities.” Capra [34] proposes to use a human trust

model based on human interactions in a trust model for fully

distributed network environments such as MANETs. Capra

denes trust as the degree of a belief about the behavior of

other entities (or agents). Li and Singhal [35] dene trust

as the belief that an entity is capable of performing reliably,

dependably, and securely in a particular case; hence, different

levels of trust exist in different contexts. For example, Alice

may trust her physician to give her advice on her health

concerns but may not trust her physician’s advice on xing

her car. Aivaloglou et al. [36] describe trust as the quantied

belief of a trustor regarding competence, honesty, security, and

dependability of a trustee in a specic context.

Recently, researchers have recognized the importance of

social networks in building trust relationships among entities.

Golbeck [37][38][39] introduces the concept of social trust

by suggesting the use of social networks as a bridge to

build trust relationships among entities. Golbeck proposes the

application of a trust concept derived from a sociological
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Fig. 2. Trust level [42].

viewpoint to computer science, and describes trust as a welldened descriptor of security and encryption as a metric to

reect security goals. Wong and Sycara [40] introduce security

mechanisms to establish trust in multi-agent systems. They are

concerned with both authenticating agents as well as ensuring

that agents do not misbehave. Trustworthiness emerges from

the security features in their system.

From the denitions of trust derived from various elds

as reviewed above, we can construct a trust metric having

the following characteristics: (1) trust should be established

based on potential risks; (2) trust should be context-dependent;

(3) trust should be based on each party’s own interest (e.g.,

selshness); (4) trust is learned (i.e., a cognitive process); and

(5) trust may represent system reliability.

B. Trust, Trustworthiness, and Risk

In the literature, the terms trust and trustworthiness seem to

be used interchangeably without clear distinction. Josang et al.

[41] claried the difference between trust and trustworthiness

based on denitions provided by Gambetta [15]. Level of

trust is dened as the belief probability varying from 0

(complete distrust) to 1 (complete trust) [41]. In this sense,

trustworthiness is a measure of the actual probability that the

trustees will behave as expected. Solhaug et al. [42] dene

trustworthiness as the objective probability that the trustee

performs a particular action on which the interests of the

trustor depend.

Figure 2 [42] explains how trust (i.e., subjective probability

of trust level) and trustworthiness (i.e., objective probability of

trust level) can differ and how the difference affects the level

of risk the trustor needs to take. The diagonal dashed line is

assumed to be marks of well-founded trust in which trust is

equivalent to trustworthiness.

Depending on the extent to which the trustor is ignorant

about the difference between the believed (i.e., trust) and the

actual (i.e., trustworthiness) probability, there is a miscalculation of the involved risk. That is, the subjective aspect of

trust results in incorrect risk estimation and improper risk

management accordingly. Figure 2 shows the cases in which

the probability is miscalculated. In the area below the diagonal

line, there is misplaced trust to various degrees that the

perceived trust is higher than the actual trustworthiness. Even

though risk is an intrinsic characteristic of trust even in wellfounded trust, misplaced trust increases risk and thus enhances

the chance of deceit as well, as shown in the example marked

with b in Figure 2. On the other hand, when the perceived

trust is lower than the actual trustworthiness as shown in the

example marked with a, the trustee is distrusted more than

warranted. In this case, the trustor may lose potentially good

opportunities to cooperate with partners with high trustworthiness.

From the above discussions, we can conclude that careful

risk estimation is closely linked with building accurate trust

relations among participating entities in networks. However,

Josang et al. [41] argue that objective trust may not be applicable to decision making in real situations. They dene two

interesting types of trust: 1) a context independent reliability

trust which measures the perceived reliability by another party

regardless of the situations which the trustor might face by

recognizing possible risk; 2) decision trust as ”the extent

to which a given party is willing to depend on something

or somebody in a given situation with a feeling of relative

security even though negative consequences are possible.”

Decision trust deals with components such as utility and risk

attitude. As an example, one may not trust an old rope for

climbing down from the 3rd oor of a building during a re

exercise (i.e., reliability trust) while trusting the rope in a real

re (i.e., decision trust).

The relationship between trust and risk has been investigated in [41][42]. Figure 3 shows an example of three different

risk values: low, medium, and high. The value of risk is low

for all trust values when the stake is close to zero. Similarly,

if the stake is too high, risk is regarded as high regardless of

the estimated trust value. Risk is generally low when the trust

value is high. However, the risk value should be determined

based on the value at stake (e.g., risk probability) since as

shown in Figure 3, high risk exists even for the case of trust

value = 1. Also important are the aspects (or probability) of

opportunity and prospect (or the positive consequence of an

opportunity) [41][42]. To buy rubber is to do risky business,

but it also gives the opportunity of selling rened products

with net prot. The purchaser of rubber should estimate her/his

acceptable risk level in terms of the calculated prospects.

Josang et al. [41] and Solhaug et al. [42] conclude that trust

is generally neither proportional nor inversely proportional to

risk.

Some researchers have commented that trust and uncertainty

are intimately linked - trust is a mechanism to cope with

uncertainty. The level of uncertainty in the information used

as trust evidence will also considerably inuence the accuracy

of trust evaluation [43].

C. Trust Properties in MANETs

Due to the unique characteristics of MANET environments

and the inherent unreliability of the wireless channel, the

concept of trust in MANETs should be carefully dened.

The main properties of trust in MANET environments can

be summarized as follows (see Figure 4):

First, trust is dynamic, not static. Trust establishment in

MANETs should be based on temporally and spatially local
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566 IEEE COMMUNICATIONS SURVEYS & TUTORIALS, VOL. 13, NO. 4, FOURTH QUARTER 2011

Fig. 3. Risk and trust [41].

Fig. 4. Trust properties in MANETs.

information: due to node mobility or failure, information is

typically incomplete and can change rapidly [8][32]. Adams

et al. [44] point out that in order to capture the dynamicity of

trust, trust should be expressed as a continuous variable, rather

than as a binary or even discrete-valued entity. A continuous

valued variable can represent uncertainty better than a binary

variable.

Second, trust is subjective [45]. In MANET environments,

a trustor node may determine a different level of trust against

the same trustee node due to different experiences with the

node derived from a dynamically changing network topology.

Third, trust is not necessarily transitive [46]. For example,

if A trusts B, and B trusts C, it does not guarantee A trusts C.

In order to use the transitivity of trust between two entities to

a third party, a trustor should maintain two types of trust: trust

in a trustee and trust in the trustee’s recommendation of the

third party. For example, Alice may trust Bob about movies,

but not trust him at all to recommend other people whose

opinion about movies is worth considering or not trust other

people that Bob recommended as much as she trusts Bob.

Fourth, trust is asymmetric, not necessarily reciprocal [44].

In heterogeneous MANETs, nodes with higher capability (e.g.,

more energy or computational power) may not trust nodes

Fig. 5. Trust properties in trust management schemes in MANETs.

with lower capability at the same level that nodes with lower

capability trust nodes with higher capability. As a typical

example in organizational management, a supervisor tends to

trust an employee less than the employee trusts the supervisor.

Fifth, trust is context-dependent [33]. For example, A may

trust B as a wine expert but not as a car xer. Similarly in

MANETs depending on the given task, different types of trust

(e.g., trust in computational power or trust in unselshness,

trust in forwarding versus trust in reporting) are required.

Figure 5 shows how several trust properties are considered

in the literature. Dynamicity and weighted transitivity are

most often considered. However, we notice that some existing

work does not even consider trust properly; some represent

trust as a discrete variable, while others assume that trust

is symmetric or completely transitive. As such they do not

capture characteristics of trust in a MANET. Further, we could

not nd any prior work that comprehensively considers all ve

properties of trust shown in Figure 5. Note that Figure 5 is

based on 36 papers and each work may consider multiple trust

properties.

In order to properly take into account these unique characteristics of trust in MANETs as described above, any trustbased framework for MANETs should consider the following

as well:

First, a decision procedure to determine the trust of an entity

should be fully distributed based on cooperative evaluation

with uncertain and incomplete evidence, since one cannot

rely on a trusted third party such as a trusted centralized

certicate authority to take care of trust management as in

wired networks [8][9][34].

Second, trust should be determined in a highly customizable

way (e.g., exible to membership changes and to deployment

scenarios) without causing disruption to the device computation and communication resources while capturing the various

and complicated natural components of an individual’s trust

into a network model [34][47].

Third, a trust decision framework should not assume that

all nodes are cooperative [34]. In resource-restricted environments, selshness is likely to be prevalent over cooperation, for

example, in order to save battery life or computational power.

Thomas et al. [48] discuss the tradeoff between selshness

and altruism of participating nodes in MANETs in terms of

prolonging system lifetime (e.g., with system lifetime dened
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CHO et al.: A SURVEY ON TRUST MANAGEMENT FOR MOBILE AD HOC NETWORKS 567

as the time to a node’s death due to energy exhaustion) versus

reducing selsh behaviors to enhance system throughput.

Finally, trust should be established in a self-organized

recongurable way in order not to be disrupted by the dynamics of MANET environments [8][48]. In addition to the

characteristics mentioned above, trust-based frameworks for

MANETs should consider the tradeoff issues between security

and performance including reliability, fault tolerance, scalability, and energy consumption where resources are restricted but

security vulnerability is relatively high.

III. CLASSIFICATIONS, POTENTIAL ATTACKS, AND

METRICS FOR MANET TRUST MANAGEMENT

This section discusses classications, attacks and performance metrics for MANET trust management. Before reviewing the literature, we would like to clarify some terminologies

that have been used interchangeably but sometimes confusingly in the context of trust management.

In general, the term trust management is interchangeably

used with the term reputation management [35]. However,

there is a slight difference between trust and reputation.

According to Liu et al. [49], trust is active while reputation is

passive. That is, trust is a node’s belief in the trust qualities

of a peer, thus being extended from a node to its peer.

Reputation is the perception that peers form about a node.

Further, Ruhomaa et al. [10] distinguish trust from reputation,

noting that trust puts an emphasis on risk and incentives

while reputation focuses on a perception that a party creates

through past actions about its intentions in the context of the

norms effective within a community. Also, recommendation

is frequently used as a way to measure trust or reputation.

Recommendation is simply an attempt at communicating a

party’s reputation from one community context to another

[45][10].

A working denition of trust for Internet applications, and

a survey of trust management schemes for such applications

may be found in [12].

In most of the literature, reputation management is regarded as part of trust management. Further, the terms trust

management and trust establishment are also interchangeably

used. To clarify these two terms, according to Aivaloglou et

al. [36], trust establishment is a process to deal with the

representation, evaluation, maintenance, and distribution of

trust among nodes.

Trust management deals with problems such as the formulation of evaluation rules and policies, representation of trust

evidence, and evaluation and management of trust relationships among nodes. As Figure 6 explains, trust establishment

is one of several trust management tasks.

A. Classications

According to Solhaug et al. [42], trust management is a

special case of risk management with a particular emphasis

on authentication of entities under uncertainty and decision

making on cooperation with unknown entities. However, the

application of trust management has been extended from

authentication to various aspects of communications and

networking, including secure routing for isolating malicious

or selsh nodes, intrusion detection, key management, access control, and other decision making mechanisms. Trust

management includes trust establishment (i.e., collection of

appropriate trust evidence, trust generation, trust distribution,

trust discovery, and evaluation of trust evidence), trust update,

and trust revocation [50] [42]. This section surveys popularly

used classications of trust management (or establishment).

Li et al. [51] and Li et al. [52] classify trust management

as reputation-based framework and trust establishment framework. A reputation-based framework uses direct observations

and second-hand information distributed among nodes in a

network to evaluate a node. A trust establishment framework

evaluates neighboring nodes based on direct observations

while trust relations between two nodes without prior direct

interactions are built through a combination of opinions from

intermediate nodes.

Yonfang [53] suggests two different approaches to evaluate

trust: policy-based trust management and reputation-based

trust management. Policy-based trust management is based

on strong and objective security schemes such as logical rules

and veriable properties encoded in signed credentials for

access control of users to resources. In addition, the access

decision is usually on the basis of mechanisms having a welldened trust management language that has strong verication

and proof support. Such a policy-based trust management

approach usually makes a binary decision according to which

the requester is trusted or not, and accordingly the access

request is allowed or not. Due to the binary nature of trust

evaluation, policy-based trust management has less exibility.

Furthermore, the availability of (or access to) trusted certicate

authorities (CA) cannot always be guaranteed, particularly

for distributed systems such as MANETs. On the other

hand, reputation-based trust management utilizes numerical

and computational mechanisms to evaluate trust. Typically, in

such a system, trust is calculated by collecting, aggregating,

and disseminating reputation among the entities.

According to Li and Singhal [35], trust management

can be classied as evidence-based trust management and

monitoring-based trust management. Evidence-based trust

management considers anything that proves trust relationships

among nodes: these could include public key, address, identity,

or any evidence that any node can generate for itself or other

nodes through a challenge and response process. Monitoringbased trust management rates the trust level of each participating node based on direct information (e.g., observing the

benign or malicious behaviors of neighboring nodes, such

as packet dropping, and packet ooding leading to excessive

resource consumption in the network, or denial of service attacks) as well as indirect information (e.g., reputation ratings,

such as recommendations forwarded from other nodes).

Aivaloglou et al. [36] classify two types of trust establishment frameworks for MANETs: certicate-based framework

versus behavior-based framework. In the former, mechanisms

are dened for pre-deployment knowledge of trust relationships within the network, using certicates which are distributed, maintained and managed, either independently or

cooperatively by the nodes. Trust decisions can be made based

on a valid certicate that proves trustworthiness of the target

node by a certicate authority or by other nodes that the issuer
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568 IEEE COMMUNICATIONS SURVEYS & TUTORIALS, VOL. 13, NO. 4, FOURTH QUARTER 2011

Fig. 6. Denition of trust management.

trusts. In behavior-based framework, each node continuously

monitors behaviors of its neighboring nodes in order to

evaluate trust. The behavior-based framework is a reactive

approach, operating under the assumption that the identities of

nodes in the network are ensured by preloaded authentication

mechanisms. For example, if a node uses network resources

in an unauthorized way, it will be regarded as a selsh or

malicious node, and will nally be isolated from other nodes.

Aivaloglou et al. [36] also classify trust establishment

schemes in terms of the type of architectures used: hierarchical framework versus distributed framework. In the former, a

hierarchy exists among the nodes based on their capabilities

or levels of trust. In this framework, centralized certicate authorities or trusted third parties are usually provided for on-line

or off-line evidence. Such a centralized infrastructure does not

exist in a distributed framework; hence, each node has some,

possibly equal, responsibility for acquiring, maintaining, and

distributing trust evidence.

Even though reputation management is part of trust management, many researchers further classify reputation management schemes. Adams et al. [44] propose three types of

reputation systems: positive reputation, negative reputation,

and a combination of the two. Positive reputation systems only

consider observations or feedback of the positive behaviors of

a node. Negative reputation systems only record complaints

or observations of the negative behaviors of a node. Peers are

assumed to be trusted and so feedback on behaviors is used

to negatively reect a node’s reputation. To complement the

drawbacks of these mechanisms, hybrid reputation systems

have been proposed [53]. For more information on reputation

management, the readers may refer to [11].

B. Potential Attacks

It is important to ensure that a trust management system

itself should not be easily subverted, attacked or compromised. In this section, we discuss various common attacks

and describe features important from the viewpoint of trust

management. A survey of threat models and specic attacks

on ad hoc routing protocols are described by Argyroudis et

al. [54] and Djenouri et al. [55].

Liu et al. [49] describe the characteristics of attacks in

MANETs by both the nature of attacks and the type of

attackers. One classication of attacks is passive attack versus

active attack. A passive attack occurs when an unauthorized

party gains access to an asset but does not modify its content.

Passive attacks include eavesdropping and trafc analysis (e.g.,

trafc ow analysis). Eavesdropping indicates that the attacker

monitors transmissions of message content. Trafc analysis

refers to analyzing patterns of data transmission. An active

attack occurs when an unauthorized party modies a message,

data stream, or le. Active attacks usually take the form of

one of the following four types or combinations: masquerading (i.e., impersonation attack), replay (i.e., retransmitting

messages), message modication, and denial-of-service (DoS)

(leading to excessive resource consumption in the network).

Yet another way to characterize attacks is based on the

legitimacy of an entity in a network: insider attack versus

outsider attack [56]. If an entity is authorized to access system

resources but employs them in a malicious way (e.g., in a

way not approved by the authorizer), it is classied as an

insider attack. More specically, inside attackers exploit bugs

in privileged system programs or poorly congured privileges,

and then they may install backdoors or Trojan horses or

other such mechanisms to facilitate subsequent acquisition of

privileged access. On the other hand, an outsider attack is

initiated by an unauthorized or illegitimate user. They usually

acquire access to an authorized account and try to perpetrate

insider attacks. Both attackers may spoof network protocols

to effectively acquire access to an authorized account.

Many trust management schemes are devised to detect

misbehaving nodes, both selsh nodes as well as malicious

nodes. Specic attack examples are described as follows (the

list is representative, not exhaustive):

• Routing loop attacks: A malicious node may modify

routing packets in such a way that packets traverse a

cycle and so do not reach the intended destination [56].

• Wormhole attacks: A group of cooperating malicious

nodes can pretend to connect two distant points in the

network with a low-latency communication link called a

wormhole link, causing disruptions in normal trafc load

and ow [57][58][59].

• Blackhole attacks: A malicious node, the so called

black hole node, may always respond positively to route

requests even when it does not have proper routing information. The black hole can drop all packets forwarded

to it [60].

• Grayhole attacks: A malicious node may selectively drop

packets [61], as a special case of a black hole attack. For
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CHO et al.: A SURVEY ON TRUST MANAGEMENT FOR MOBILE AD HOC NETWORKS 569

example, the malicious node may forward routing packets

but not data packets. Similarly, a sinkhole attacker attracts

nodes to route through it and then selectively routes

packets [49].

• DoS attacks: A malicious node may block the normal

use or management of communications facilities, for example, by causing excessive resource consumption [62].

• False information or false recommendation: A malicious node may collude and provide false recommendations/information to isolate good nodes while keeping

malicious nodes connected. In the stacking attack, a

malicious node keeps complaining about a peer node and

creates the peer’s negative reputation [44][63].

• Incomplete information: A malicious node may not

cooperate in providing proper or complete information.

Usually compromised nodes collude to perform this attack. However, node mobility or link failure, prevalent

in MANETs, may also result in the same phenomenon

[8][34].

• Packet modication/insertion: A malicious node may

modify packets or insert malicious packets such as packets with incorrect routing information [64].

• Newcomer attacks: A malicious node may discard its bad

reputation or distrust by registering as a new user. The

malicious node simply leaves the system and joins again

for trust revocation, ushing out its previous bad history

and starting to accumulate new trust [65].

• Sybil attacks: A malicious node can use multiple network identities which can affect topology maintenance

and fault tolerant schemes such as multi-path routing

[61][49][46].

• Blackmailing: A malicious node can blackmail another

node by disseminating false information that another

node is malicious or misbehaving. This can generate

signicant amount of trafc and ultimately disrupt the

functionality of the entire network [49]. This attack can

be seen as false accusation plus DoS attacks in the

sense that false information is disseminated leading to

a signicant amount of resource consumption.

• Replay attacks: A malicious node may replay earlier

transmitted packets. If the packets include data, this

should not cause trouble, and the receiving node just

discards erroneous packets. However, if the adversary

replays route requests, routing table information would

become erroneous, and old locations and routing information might make nodes unreachable [56].

• Selective misbehaving attacks: A malicious node behaves

badly but selectively to other nodes [66].

• On-off attacks: A malicious node may alternatively behave well and badly to stay undetected while disrupting

services [66].

• Conicting behavior attacks: A malicious node may

behave differently to nodes in different groups to make

the opinions from the different good groups conicting,

and ultimately lead to non-trusted relationships [52].

Figure 7 shows various attacks considered in a survey of

43 papers. Note that the ”general selsh” category means

no specic information is given in the work except that it

Fig. 7. Attacks considered in existing trust management systems in

MANETs.

deals with selsh nodes. Also papers in the ”general misbehaving” category deal with a broad range of misbehaving

nodes, including malicious and selsh nodes, but do not

provide detailed information. ”Packet related” attacks include

packet dropping, packet modication, packet insertion, and

selective packet forwarding. ”Identity related” attacks include

impersonation, masquerading, and Sybil attacks. Except for

the ”general selsh” and ”general misbehaving” categories,

we notice that ”false information” (e.g., including false recommendation or reputation) and ”packet related” attacks are

dominantly considered in the literature on trust management

schemes for MANETs. Figure 7 illustrates that most of the

attacks considered in the literature on trust management are

general attacks often targeted at other aspects of MANETs.

Hence, the trust evaluation engine should be robust and

degrade gracefully if some information or evidence does not

provide a certain level of trust based on partial or potentially

corrupted information.

C. Metrics for MANET Trust Management

Although many trust management schemes have been proposed to evaluate trust values, no work clearly addresses what

should be measured to evaluate network trust. Liu et al.

[49] dened trust in their model as reliability, timeliness, and

integrity of message delivery to the intended next-hop. Also

most trust-based protocols for secure routing calculated trust

values based on the characteristics of nodes behaving properly

at the network layer. Trust measurement can be applicationdependent and will be different based on the design goals of

proposed schemes.

Based on 31 papers, Figure 8 shows various performance

metrics that have been used to evaluate trust management

schemes for MANETs. Note that a single work may use

multiple performance metrics. Figure 8 shows standard system

performance metrics typically used to evaluate trust management systems; these metrics include overhead (e.g., control
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570 IEEE COMMUNICATIONS SURVEYS & TUTORIALS, VOL. 13, NO. 4, FOURTH QUARTER 2011

Fig. 8. Metrics considered by MANET trust management systems.

packet overheads), throughput, goodput, packet dropping rate,

and delay. ”Route usage” refers to the number of routes

selected particularly when the purpose is for secure routing.

”Trust level” is a recently used system metric. Example

metrics using the trust level include condence level of

the trust value, trustworthiness, opinion values about other

nodes, and trust level per session. ”Others” indicates metrics

that consider system tolerance based on incorrect reputation

threshold, availability, convergence time to reach steady state

in trustworthiness of all participating nodes, and percentage

of malicious nodes.

IV. MANET TRUST MANAGEMENT SCHEMES

This section summarizes trust management schemes that

have been developed for MANETs.

We describe trust management schemes based on specic

design purposes such as secure routing, authentication, intrusion detection, access control (authorization), and key management. Further, we also describe existing general frameworks

for trust (or reputation) evidence distribution and evaluation.

Figure 9 summarizes 45 trust management schemes proposed

for MANETs during 2000-2009 based on their design purposes. Note that under each research category, we will survey

existing works in chronological order.

A. Secure Routing

Most reputation-based trust management schemes are devised for collaborative secure routing by detecting misbehaving nodes, both selsh and malicious ones. Marti et al. [67]

proposed a reputation-based trust management scheme that

consists of a watchdog that monitors node behaviors and a

pathrater that collects reputation and takes response actions

(e.g., isolating misbehaving nodes as a result of misbehavior

detection). This work is an initiative to dynamically incorporate direct observations into trust values for secure routing. It

extends DSR (Dynamic Source Routing) but trust evaluation

is based only on direct observations.

Buchegger et al. [68] initiated a new design to develop

a routing protocol by introducing a ”trust manager” in their

scheme. They determined trust levels based on self-monitored

information while employing reputation collected from both

direct and indirect observations and experiences. They did not

show any experimental results, but pose several interesting

questions such as what is a sustainable relationship between

the total number of nodes in the network, the maximum

number of malicious nodes the system can tolerate, and the

minimum number of friends per node needed to achieve

high tolerance, and a prescribed level of trust. Buchegger et

al. [69] also developed a reputation-based trust management

scheme called CONFIDANT (Cooperation Of Nodes-Fairness

In Dynamic Ad-hoc NeTworks) based on both direct and

indirect observations to detect misbehaving nodes. The unique

feature in this work is an incentive mechanism for altruistic

nodes to be paid as a result of cooperation.

Paul and Westhoff [70] proposed a context-aware mechanism for detecting selsh nodes by extending DSR with a

context-aware inference scheme to punish the accused and the

malicious accuser. However, the use of digital signatures to

disseminate information about the accused and the malicious

accuser may not be viable in a resource-constrained MANET

environment.

Michiardi et al. [71] proposed CORE (COllaborative REputation) that has a monitoring mechanism complemented by

a reputation functionality that differentiates between direct

reputation, indirect reputation, and functional reputation (taskspecic behavior). The proposed protocol is developed to

make decisions about cooperation or gradual isolation of a

node. A unique characteristic of this mechanism is that it

exchanges only positive reputation information. However, this

may limit its reliance on positive reports without the facility

to submit negative feedback.

He et al. [72] proposed a reputation-based trust management

scheme using an incentive mechanism, called SORI (Secure

and Objective Reputation-based Incentive). This scheme encourages packet forwarding and discourages selsh behaviors based on quantied objective measures and reputation

propagation by a one-way hash chain based authentication.

The performance of this scheme in the presence of malicious

nodes, as may be expected in a hostile environment, has not

been investigated.

Nekkanti and Lee [73] extended AODV (Ad hoc Ondemand Distance Vector) using trust factor and security level

at each node. Their approach deals differently with each route

request based on the node’s trust factor and security level.

In a typical scheme, routing information for every request

would be encrypted leading to large overheads; they propose

to use different levels of encryption based on the trust factor

of a node, thus reducing overhead. This approach adjusts

the security level based on the recognized hostility level and

hence can conserve resources; however, the approach does not

treat evaluation of trust itself. Li et al. [74] also extended

AODV and adopted a trust model to guard against malicious

behaviors of nodes at the network layer. They represented

trust as opinion stemming from subjective logic. The opinion

reects the characteristics of trust in MANETs, particularly

dynamicity. The key feature is to consider system performance

aspects by dealing with each query based on its level of

trust. Depending on the level of trust of nodes involved

in the query, there is no need for a node to request and
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Fig. 9. Metrics considered by MANET trust management systems.

verify certicates all the time, thereby leading to signicant

reduction of computation and communication overhead. This

work advances trust management by considering a generic

trust management framework for MANETs.

Pisinou et al. [75] devised a secure AODV-based routing

protocol for multi-hop ad hoc networks for discovering a

secure end-to-end route free of any compromised nodes. Their

trust-based routing protocol calculates trust values based only

on direct observations, assuming that trust is transitive. As

a continuation of [68], Buchegger et al. [76] also proposed

a fully distributed reputation system in order to cope with

false information propagation. The proposed design maintains

a reputation and trust rating system about individual nodes

by designing a modied Bayesian approach. Recognizing the

dynamic nature of trust and reputation, the authors introduced

reevaluation and reputation fading as well as redemption

mechanisms. Nevertheless, no other characteristics of trust are

addressed except for dynamicity.

Ghosh et al. [77] enhanced trust management by considering the condence level of trust. Their use of the condence

level as a weight on the computed trust value and the method

for calculating trust in a fully distributed way provide a

general framework that can be applied to non-trust-aware

routing protocols. In [77], SORI [72] is extended to alleviate

the problem of selsh nodes, by considering the number of

forwarding packets to evaluate the condence level.

Wang et al. [78] proposed a mechanism to distinguish

selsh peers from cooperative ones based solely on local

observations of AODV routing protocol behaviors. They use a

nite state machine model of locally observed AODV actions

to construct a statistical description of each peer’s behavior. In

order to distinguish between selsh and cooperative peers, a

series of well-known statistical tests are applied to features

obtained from the observed AODV actions. An interesting

extension of this work would be to consider various patterns

of node mobility which can give additional insights.

Zouridaki et al. [79] proposed a trust establishment mechanism for MANETs called Herms to improve the reliability

of packet forwarding over multi-hop routes in the presence

of potentially malicious nodes. Essentially, direct observations

are used to evaluate opinions about others. Also, condence

level is used as a weight to evaluate trust of other nodes based

on a Bayesian approach. They also introduced a windowing

scheme to systematically expire old data to maintain accuracy

of the opinion metric in the face of dynamics. However, this

scheme is vulnerable to attacks that can exploit the windowing

scheme to disseminate false information to accuse good nodes

and to keep bad nodes in the system (such as badmouthing

attacks).

As an extension, Zouridaki et al. [80] employed both

rst-hand trust information based on direct observations and

second-hand trust information forwarded from neighboring

nodes about non-neighboring nodes. This trust establishment

scheme can cope with more attacks, including propagation of

false recommendations or information, identifying bad nodes

among neighboring nodes, colluding attacks, replay attacks,

and duplicate attacks. It is noteworthy that they used only

security related metrics to evaluate their scheme, such as

trustworthiness and the percentage of nodes recognized as bad.

Pirzada et al. [81] proposed and examined the efcacy

of trust-based reactive routing protocols in the presence of

attacks. This work only considers rst hand information to

evaluate other nodes’ trust values. Thus, trust evaluation is

restricted to direct neighboring nodes.

Sun et al. [46] proposed trust modeling and evaluation

methods for secure ad hoc routing and malicious node detection. The unique part of their design is to consider trust as

a measure of uncertainty that can be calculated using entropy.

In their denition, trust is a continuous variable, and does not

need to be transitive, thus capturing some of the characteristics

of trust in MANETs. However, this work considers packet

dropping as the only component of direct observations to

evaluate trust.

Abusalah et al. [82] proposed a trust-aware routing protocol

(TARP) and developed a trust metric based on six trust components including software conguration, hardware conguration, battery power, credit history, exposure and organizational

hierarchy. However, no consideration was given to trust decay

over time and space to reect uncertainty due to dynamics and

incomplete information in MANET environments.

Sen et al. [83] proposed a trust-based mechanism to detect malicious packet dropping nodes based on reputation of

neighboring nodes, and take into account the decay of trust

over time. This work assumes that a pair of public/private keys

can be preloaded to prevent identity-related attacks. However,

this may not be scalable for a large network.

Soltanali et al. [84] proposed a distributed mechanism to

deal with selsh nodes as well as to encourage cooperation in
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MANETs based on the combination of reputation-based and

currency-based incentive mechanism mitigating their defects

and improving their advantages. Compared to existing works,

this work considers more aspects of trust such as dynamicity,

weighted transitivity, and subjectivity. However, it used only

packet forwarding behaviors to evaluate a node’s trust and

standard performance metrics to evaluate the proposed trust

scheme.

Balakrishnnan et al. [85] developed a trust model to

strengthen the security of MANETs and to deal with the issues

associated with recommendations. Their model utilizes only

trusted routes for communication, and isolates malicious nodes

based on the evidence obtained from direct interactions and

recommendations. Their protocol is described as robust to the

recommender’s bias, honest-elicitation, and free-riding. This

work uniquely considered a context-dependency characteristic

of trust in extending DSR.

Li et al. [52] stated that using only a reputation-based trust

framework gives only an incomplete partial solution for trust

management. They proposed an objective trust management

framework (OTMF) for MANETs based on both direct and

indirect information for reputation management and showed

the effectiveness of OTMF. This work used the term ”objective” trust to refer to trust evaluated based on secondhand information. However, this work did not consider node

collusion in obtaining second-hand information, which may

lead to incorrect recommendations.

Mundinger and Boudec [86] were the rst to analyze the

robustness of a reputation system based on a deviation test.

Using a mean-eld approach in their stochastic model, they

showed that liars have no impact unless the number of liars

exceeds a certain threshold (a phase transition). They provided

precise formulas for the critical values and guidelines for an

optimal choice of parameters. This work is unique in that it

evaluates a system’s tolerance to untrusted nodes; however, the

reputation evaluation is based only on the ”fake” information.

Moe et al. [87] proposed a trust-based routing protocol as

an extension of DSR based on an incentive mechanism that

enforces cooperation among nodes and reduces the benets

that selsh nodes can enjoy (e.g., saving resources by selectively dropping packets). This work is unique in that they used

a hidden Markov model (HMM) to quantitatively measure the

trustworthiness of nodes. In this work, selsh nodes are benign

and selectively drop packets. Performance characteristics of

the protocol when malicious nodes perform active attacks

such as packet modications, identity attacks, etc., need to

be investigated further.

In quorum or threshold schemes, a node must successfully

interact with at least k of n distributed trusted authority (TA)

nodes. Finding k such nodes can be resource intensive. Reidt

et al. [88] prioritize the TA nodes and nd a route to connect

to k desirable TA nodes so as to minimize a performance

metric such as overhead, taking into account reliability and

energy consumption of individual nodes. Signicant savings

over a standard system were shown. An interesting aspect, not

considered yet, would be to incorporate trustworthiness into

the TA selection and routing scheme.

Ayachi et al. [89] formalized implicit trust relations in

AODV and demonstrated that a node can utilize these trust

relations to isolate malicious nodes for secure routing. Nodes

overhear neighbors’ transmissions from which they can build

a neighbor routing table and check for deviation from normal behaviors for AODV. This scheme can detect malicious

behaviors such as message replication, message forgery and

some instances of message modication. However, it is not

amenable to incorporation of other trust metric components,

such as intimacy and competence but monitored behaviors

could feed into a trust evaluation scheme.

Adnane et al. [90] proposed trust-based countermeasures

to isolate malicious nodes extending OLSR (Optimized Link

State Routing). Their protocol provides secure routing paths

by identifying malicious nodes. The focus of the protocol is

to prevent usurpation of node identities. Performance analysis

under other types of attacks remains to be investigated.

Although many researchers have developed secure routing

protocols using trust, most of the approaches have focused on

monitoring routing behaviors and the evaluation of trust has

been in the context of communication networks. Further steps

should be taken to rene issues such as (1) how to quantify

trust in a MANET node; (2) how to employ (a continuousvalued) trust in a routing decision; and (3) how to develop a

composite trust metric incorporating task performance goals,

taking into account the social aspects of a MANET node.

B. Authentication

There have been efforts to establish trust relationships to

ensure authentication in MANETs. Weimerskirch et al. [91]

developed a trust model based on human behavior, noting that

society can be properly considered as an ad hoc network.

They used recommendations from a distributed trust model to

construct trust relationships and extended it by a request for

recommendations. Based on models derived from observations

of human society, recommendations are used to calculate trust,

with weights based on the distance of relationships. Their

denition does not assume symmetry or complete transitivity,

thus capturing essential features of trust in MANETs. The

assumption of low-value transactions does not require any

evidence-based mechanism to ensure trust such as authentications using public/private keys. Consequently, it is not

applicable to systems where hostility may be high, or where

consequences of misplaced trust can be severe.

Verma et al. [92] presented an overview of a trust negotiation scheme using DSR and ZRP (Zone Routing Protocol).

Their scheme consists of two components. The peer-to-peer

component deals with secure communications with neighbors

in a lightweight manner. The heavyweight remote component

performs trust negotiation and establishes secure end-to-end

communication. The main goal of this work is to add robustness in the process of trust negotiation, rather than trust

evaluation.

Pirzada and McDonald [93] proposed a trust-based communication model that, based on a notion of a belief, provides a dynamic measure of reliability and trustworthiness

in MANETs. The merit of this work is to incorporate utility

as general trust and time as situational trust into the overall

trust metric to evaluate an agent in the network. However, the

situational trust considered is limited to monitoring dynamics

of packet forwarding behaviors.
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Davis [47] proposed a reliable and structured hierarchical

model for trust management in MANETs that is robust to

malicious accusation exploits. The scheme deals with explicit

revocation of certicates in a distributed way, eliminating the

case in which revoked certicates can be accepted as valid.,

This work assumes that the initial certicates and public keys

of all nodes are distributed by a centralized trust authority to

each node before the network is deployed which may not be

scalable in a large scale MANET. The paper does not discuss

the issue of false positives which can lead to continual eviction

of nodes, and eventually loss of network connectivity. To

counteract this, dynamic reissue of certicates may be needed

which may incur extra overhead.

Ngai and Lyu [94] proposed a secure public key authentication service based on their trust model to prevent propagation

of false public keys in the presence of malicious nodes.

Trust is evaluated based on direct monitoring as well as

recommendation. However, this work does not consider group

membership changes, the distance from the evaluator, and their

effect on the performance of their trust management scheme.

In summary, there has been quite a bit of work on using

trust for authentication. However, as in the case of trust-based

secure routing, the models and protocols used are based solely

on monitoring packet forwarding behaviors.

C. Intrusion Detection

Trust can be used as a basis for developing an intrusion

detection system (IDS). Also, IDS itself can help nodes

measure trust of other nodes when they cooperate with each

other to detect malicious nodes. Albers et al. [95] proposed

a general architecture for an intrusion detection system (IDS)

called a Local IDS (LIDS) such that intrusion detection can

be performed locally among trustworthy participating nodes.

Here, trust is used to detect intrusions in the system. In Ahmed

et al. [96], IDS provides audit and monitoring capabilities

that offer local security to a node and helps perceive the

specic trust levels of other nodes. Hence, evaluating trust and

identifying intrusions may not be totally separated processes.

D. Access Control

Trust also can be applied in determining whether or not to

grant access to certain resources or rights. Gray et al. [97]

integrated trust-based admission control with standard rolebased access control. By doing this, an access control decision

is effectively made without being affected by incomplete information collected in MANETs. A simple distributed blackjack

card game application is described, in which the trust-based

admission control system is used to assign roles to users based

on their trust-based admission rights. It is not clear how the

approach can be extended to a general framework applicable

to MANETs.

Luo et al. [98] presented a ubiquitous and robust access

control solution (URSA) for MANETs based on a localized

group trust model so that only well behaving nodes will have

access rights to network resources. Their localized group trust

model for MANETs is based on threshold cryptography: a

node is globally trusted only if it is individually trusted by

any k trusted nodes where k is a system-wide trust threshold.

This work assumes that the node density is large enough so

that any node can nd k trusted nodes, perhaps by moving

to another location. Interesting extensions of the work include

consideration of mobility models other than random waypoint,

and trust evaluation under high node mobility situations.

Adams and Davis [17] presented a decentralized access

control system implementing sociological trust constructs in

a quantitative system to evaluate the relationships between

entities. A distributed, node-centric approach to reputation

management considers a node’s behavior feedback and gives a

reputation index that nodes can use to determine the trustworthiness of their peers before establishing trust relationships.

This work further assessed risk using a Bayesian approach

to evaluate trust. Interestingly, this work used reputation as

a weight to evaluate direct observations, which is a different

approach from most existing works. Extensions of the scheme

to handle network dynamics would be useful.

Yunfang [53] proposed an integrated mechanism of policy

proof and reputation evolution into trust management for

decision-making on access control with the goal of providing

rm/objective security as well as social/subjective security.

However, this work is based on the assumption that trust

is completely transitive, and it is not clear how a more

realistic transitivity model can be incorporated into the trust

management system.

E. Key Management

Virendra et al. [99] proposed a trust-based security architecture for key management in MANETs. This architecture

aims to establish keys between nodes based on their trust

relationships, and to build secure distributed control using trust

as a metric. In their self-organizing trust-based architecture,

nodes are organized into trust-based clusters called PhysicalLogical Trust Domains (PLTDs), a group of trusted nodes

sharing a group key. Nodes can belong to multiple PLTDs.

The unique part of this work is that it considers the trust level

of each node in a physical as well as a logical sense, e.g.,

it considers both one-hop nodes as well as previously trusted

nodes that are not currently one-hop neighboring nodes. The

signicant merit of this work is in formalizing a trust metric

reecting trust decay over time and updating trust as dynamics

of the network change. However, establishing pair-wise keys

based on pair-wise trust may not be feasible in terms of

scalability and in the presence of high network dynamics in a

large MANET.

Hadjichristo et al. [63] presented a key management

framework that provides redundancy and robustness in the

establishment of Security Association (SA) between pairs of

nodes. Their proposed key management system (KMS) adopts

a modied hierarchical Public Key Infrastructure (PKI) model

where nodes can dynamically take management roles. The

scheme is designed to provide high service availability based

on trust-based SA among nodes. However, trust relationships

are derived solely from certicate chains. Adams et al. [44]

also extended their prior work [63] with a node-centric reputation management approach that considers feedback about

a node’s behavior in generating a reputation index to determine the trustworthiness of its peers before establishing
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IPSec security associations. They considered the decay of trust

over time using a three-window weighted average. They also

derived reputation values from past experiences and current

observations and introduced a rehabilitation mechanism to

give a second chance to bad nodes. However, no details

were given on the type of information that should be directly

observed to derive reputation.

Li et al. [100] demonstrated an on-demand, fully localized, and hop-by-hop public key management protocol for

MANETs. In this protocol each node generates its own

public/private key pairs, issues its certicate to neighboring

nodes, keeps received certicates in its certicate repository,

and provides authentication service by adapting to the dynamic

network topology, without reliance on any centralized server.

However, only certicate chains are used to derive trust.

Chang and Kou [101] proposed a Markov chain trust model

to obtain the trust values (TVs) for 1-hop neighbors. They designed a trust-based hierarchical key management scheme by

selecting a certicate authority server (CA) and a backup CA

with the highest TVs. This work gives a rigorous analysis of

TVs and considers a variety of attacks. However, it computes

TVs only based on direct observations and does not consider

trust decay due to using recommendations from remote nodes.

A survey of key management techniques for network-layer

security may be found in the work by Hegland et al. [102].

In contrast to secure routing that produces an operational

MANET, authentication, intrusion detection, access control,

and key management are general trust contexts that also exist

outside the area of MANETs. In these applications, it is

useful to abstract out the properties of MANETs and consider

only the inuence of MANETs on any information/evidence

gathering, aggregation, and other computation, and design a

trust management scheme that considers inuences such as

the cost/likelihood of obtaining a piece of information in

computing trust.

F. Trust Evidence Distribution and Evaluation

Several trust management schemes have been proposed

in order to provide a general framework for trust evidence

distribution or evaluation in MANETs.

Yan et al. [64] proposed a trust evaluation based security

solution for data protection, secure routing, and other network

activities. This trust evaluation model called Personal Trusted

Bubble (PTB) considers many factors including experience

statistics, data value (the higher the value of the data, the

higher is the trust needed from other PTBs to transfer it),

intrusion black list, reference (reputation/recommendation),

personal preference, and PTB policy (related to the entire

network’s security requirements and policy). Interestingly, personal preference and PTB reect the subjective characteristic

of trust in deriving trust values. Yan et al. [64] do not

validate whether their proposed trust management is correct

or useful compared to the actual trust levels, say, based on

trustworthiness in Josang and Solhaug’s terminology. In general, validation of trust models is difcult, given the inherent

subjectivity in the trust metric, but it is also critical. Jiang

and Baras [103] proposed a trust distribution scheme called

ABED (Ant-Based trust Evidence Distribution) based on the

swarm intelligence paradigm, which is highly distributed and

adaptive to mobility. The swarm intelligence paradigm is

widely used in dynamic optimization problems (e.g., the traveling salesman problem, routing in communication networks).

The key principle in swarm interaction is called stigmergy,

indirect communication through the environment. In ABED,

”pheromones” are deposited at nodes by mobile agents called

”ants” and provide the mechanism for information exchange

and interactions. These ”ants” can identify the optimal path

toward their food, resembling trust evidence in this case.

The pheromone regulation process is known to be suitable

for dynamically changing environments such as MANETs.

However, no specic attackers are considered to prove the

robustness of the proposed scheme in the presence of attacks.

In the continuing work, Baras and Jiang [104] addressed

distributed trust computation and establishment using random

graph theory. This work uses the theory of dynamic cooperative games and identies how a phase transition from a

distrusted state to a trusted state can occur in a dynamic

MANET. This work is unique in that it describes how phase

transitions occur in MANETs and how these are related to

node mobility and network topology in the process of initial

trust establishment. Trust relationships are ternary (yes, no,

don’t care) and the emphasis is on understanding steady-state

behaviors. Incorporating continuous valued trust variables,

dynamics, and transient behaviors in this framework would

be useful.

Theodorakopoulos and Baras [50] proposed a trust evidence

evaluation scheme for MANETs. The evaluation process is

modeled as a path problem in a directed graph where vertices

represent entities and edges represent trust relations. The

authors employed the theory of Semirings to show how two

nodes can establish trust relationships without prior direct

interactions. Their case study uses the PGP web of trust

to express an example trust model based on Semirings and

shows that their scheme is robust in the presence of attackers.

However, their work assumes that trust is transitive. Further,

trust and condence values are represented as binary rather

than continuous values. Even though no centralized trusted

third party exists, their work makes use of a source node

as a trusted infrastructure, which introduces vulnerability in

MANETs.

Recently, Boukerche and Ren [105] proposed a distributed

reputation management mechanism called GRE (Generalized

Reputation Evaluation), using a comprehensive computational

reputation model. GRE seeks to prevent malicious nodes from

entering a trusted community. However, no specic attack

model was addressed.

Moloney and Weber [106] presented a trust-based security

system that generates appropriate trust levels based on the

consideration of the main characteristics of MANETs as

well as context-awareness. The scheme leverages two existing

projects at Trinity College, Dublin, called SECURE and Aithe.

SECURE is used for trust management using a trust engine

and a risk engine while Aithe collects and manages context

information forwarded from sensors. It is worthwhile to extend

this work to consider attacks that can propagate incorrect

information to generate trust levels.

Very recently, Cho et al. [107] proposed a trust management

scheme for group communication systems in MANETs. This
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work proposed a composite trust metric reecting various

aspects of a MANET node such as sociability (i.e., social

trust) and task performance capability (i.e., QoS trust), and

investigated the effect of the trust chain length used by a

node to establish acceptable trust levels through subjective

trust evaluation. They also discussed the concept of objective

trust evaluation based on global knowledge as the basis of

validating subjective trust evaluation. More work remains to

be done to ascertain feasibility.

The Appendix summarizes trust management schemes surveyed in this section. In the Appendix, the methodology

explains how trust evidence is collected and performance

metrics refer to the metrics used to evaluate various trust

management schemes.

V. FUTURE RESEARCH DISCUSSION

It is clear that sooner or later intelligence will be embedded in each node with cognitive functionality, adopting

recent ideas about cognitive networks in wireless networks

[108]. Mahmoud [108] denes a cognitive network as having

a cognitive process that is capable of perceiving current

network conditions and then planning, deciding, and acting

on those conditions. Cognitive networks are able to recon-

gure the network infrastructure based on past experiences

by adapting to continuously changing network behaviors to

improve scalability (e.g., reducing complexity), survivability

(e.g., increasing reliability), and QoS (e.g., facilitating cooperation among nodes) as proactive mechanisms [48][108].

We suggest using this concept of cognitive networks so that

nodes can adapt to changing network behaviors, such as

attacker behaviors, degree of hostility, node disconnection due

to physical environment such as terrain, energy depletion, or

voluntary disconnection for energy saving. Cognition is more

than adaptation; it incorporates learning and reasoning.

Another potentially fruitful research direction is to use

social relationships in evaluating trust among collaborators in

a group setting by employing the concept of social networks.

Golbeck et al. [37][38][39] dene a social network as a

social structure of individuals who may be related directly or

indirectly to each other in order to pursue common interests.

Yu et al. [109] and Maheswaran et al. [110] use social

networks to evaluate the trust value of a node. Examples

of social networks are strong social relationships including

colleagues or relatives, membership in the same platoon, and

loose social relationships including school alumni or friends

with common interests or membership in coalition activities.

Social trust may include friendship, honesty, privacy, and

social reputation or recommendation derived from direct or

indirect interactions for ”sociable” purposes. In MANETs,

metrics used to measure these social trust properties can be

frequency of communications, malicious or benign behaviors

(e.g., false accusation or recommendation, impersonation),

private information revealed, and quality of reputation. The

notion of social trust is being incorporated into communication

networks. Trust propagation models, some based on notions of

social networking, have been proposed in multi-agent systems

[114] [115] [116].

An important and interesting research direction is to construct a composite trust metric based on social trust and

other trust components representing quality-of-service (QoS)

to successfully perform tasks to meet both performance and

trust requirements. We have seen some work in the literature

moving in this direction. Cho et al. considered honesty and

intimacy (for social trust), and unselshness and energy (for

QoS trust) for trust evaluation [107]. Kohlas et al. [111] used

honesty, competency, reliability, and maliciousness and their

corresponding negations as trust components to dene trust

relationships. Yin et al. [112] computed composite reputation

values of peers based on evidences from various domains

such as customers’ reputation scores or ranks in commercial

sites or the certied roles in certain organizations with different weights indicating the importance and robustness of

the reputation computation processes. Boursas and Hommel

[113] considered QoS aspects such as the visual quality in

multimedia and commitment in interactions to calculate node

trust levels in large distributed systems. More work remains

to be done to understand the best combination of social trust

versus QoS trust components used to construct the composite

trust metric, as well as the best weights associated with social

trust and QoS trust, especially when given application context

information for critical mission executions in MANETs.

Not much work has been done in trust management for

mobile vehicular systems. A trust architecture for vehicular

networks is proposed in [117] that incorporates a policy

control model, a proactive trust model, and a social network

based system, and takes into account dynamics. When the

environment is volatile, associating trust with data becomes

even more challenging; a solution is provided in [117] and a

case study is discussed in the context of vehicular networks.

The overall qualities of trust in decision making depends on complex interactions between the information, social/cognitive, and communications networks. Trust metrics

might be separately dened in each of the networks, but

the key issue is to elucidate the mapping of qualitative and

quantitative metrics across the networks, to dene an endto-end notion of composite trust, to determine the attributes

(presumably many others than trust) in the different networks

that affect this composite metric, and identify those that can

be controlled and those that cannot [118], especially for trust

management in a coalition environment [119].

We suggest that the following design concepts be considered

for building MANET trust management systems:

• A trust metric must reect the unique properties of trust

in MANETs, including possibly imperfect transitivity,

asymmetry, subjectivity, non-binary nature, decay over

time and space, dynamicity, and context-dependency.

• A trust metric must incorporate adequate trust components (e.g., social trust and QoS trust) capable of reecting mission difculty (e.g., high risk upon task failure),

changing network environments (e.g., lack of bandwidth,

increasingly hostile environment as attackers’ strength

increases, high communication load), and conditions of

participating nodes (e.g., low energy, compromised status).

• A trust management design must support cognitive functionality for each node to achieve adaptability to changing

network conditions and MANET environments including
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576 IEEE COMMUNICATIONS SURVEYS & TUTORIALS, VOL. 13, NO. 4, FOURTH QUARTER 2011

node density, node mobility patterns, scheduling algorithms, and trafc patterns.

• A trust management system should be situation specic

or situation aware [120][121][122]. Situational awareness

includes mission contexts and requirements in terms of

security, performance and reliability. Depending on the

required levels of security, performance and/or reliability,

a different level of trust can be adopted reecting mission

contexts and situations.

• A trust metric must adequately reect tradeoffs in altruism versus selshness, trust versus reliability, availability,

survivability, or security so as to contribute to improved

system performance. In addition, since gathering information from spatially remote areas will consume more

resources (e.g., time or energy) but improve decision

making, one should investigate the tradeoff between

resource consumption and decision making accuracy and

timeliness. One may utilize aggregation technique to

reduce resource consumption in obtaining information

from distant nodes.

• A trust management design must allow optimal settings

to be identied under various network and environmental

conditions so as to maximize the overall trust of the system for successful mission executions. Equally important

is an understanding of sensitivity to deviations from the

optimal settings.

• There has been no comparison of trust management

schemes versus conventional security schemes in terms

of metrics of interest in MANETs. One example could

be the comparison of trust management schemes to

cryptographic schemes in detecting misbehaving nodes.

• Local trust is easy to understand and compute, since it

only involves tracking behaviors of neighboring nodes.

Local trust is easy to defend from malicious attacks.

Global trust is harder to compute and update; Eigentrust

[123] is an example of a global trust metric. But a

non-local denition of trust is subject to subversion and

manipulation by colluding nodes. Zhang et al. [124]

provide a robust version of the Eigentrust algorithm. A

critical question is: is trust inherently local? How can a

global trust metric be computed and distributed reliably?

• Recently, social trust derived from social networks has

received considerable attention for establishing trust in

various applications. MANET designers may also want

to take into account social trust.

• The survey has focused on a trust value associated with

individual nodes. But often we may be interested in

associating trust with data or with a group of nodes or

entities. Many of the concepts discussed here will extend

naturally.

VI. CONCLUDING REMARKS

Trust is a multidimensional, complex, and contextdependent concept. Although trust-based decision making is

in our everyday life, trust establishment and management

in MANETs face challenges due to the severe resource

constraints, the open nature of the wireless medium, the

complex dependence between the communications, social and

application networks, and, hence, the complex dependency

of any trust metric on features, parameters, and interactions

within and amongst these networks.

In this paper, we surveyed and analyzed existing trust

management schemes in MANETs to provide MANET trust

network protocol designers with multiple perspectives on the

concept of trust, an understanding of trust properties that

should be observed in developing trust metrics for evaluating

trust, and insights on how a trust metric can be customized

to meet the requirements and goals of the targeted system. A

composite trust metric that captures aspects of communications and social networks, and corresponding trust measurement, trust distribution, and trust management schemes are

interesting research directions. For dynamic networks, such

as military MANETs, these schemes should have desirable

attributes such as ability to adapt to environmental dynamics,

scalability, reliability, and recongurability.

ACKNOWLEDGMENT

This project is supported in part by an appointment to the

U.S. Army Research Laboratory Postdoctoral Fellowship Program administered by the Oak Ridge Associated Universities

through a contract with the U.S. Army Research Laboratory.

The authors appreciate the many discussions with and critical

insights provided by our internal ARL staff including Elizabeth Bowman, Kevin Chan, Natalie Ivanic, and Brian Rivera.

The authors also give special thanks to Dakshi Agrawal and

Mudhakar Srivatsa from IBM T. J. Watson Research Center

for their valuable comments.

APPENDIX

A SURVEY ON EXISTING TRUST MANAGEMENT SCHEMES

IN MANETS

See Tables I-VI.

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CHO et al.: A SURVEY ON TRUST MANAGEMENT FOR MOBILE AD HOC NETWORKS 577

TABLE I

Authors, Year,

Ref. no

Purpose Methodology Attacks consideredPerformance

metrics

Trust property Trust management model

Marti (2000)

[67]

Secure routing Direct observation

Reputation

Black hole

False accusation

Throughput

Overhead

Detection accuracy

Dynamicity Watchdog and Pathrater

Trust revocation introducing

redemption

Underlying routing protocol:

DSR

Buchegger &

Boudec (2002)

[68]

Secure routing Direct observation

Reputation

Various

malicious packet

forwarding

DoS

N/A Weighted transitivity

Dynamicity

An extension of DSR using

a hybrid scheme of selective

altruism and utilitarianism

Buchegger

& Boudec

(2002)[69]

Secure routing Direct observation

Reputation

Forward

defection (e.g.,

route diversion)

Throughput

Goodput

Dropped packets

Overhead

Utilitya

Weighted transitivity

Dynamicity

CONFIDANT

Bayesian Model

Incentive mechanism

No punishment against misbehaving nodes

An extension of DSR

Paul & Westhoff

(2002)[70]

Secure routing Direct observation

Reputation

Masquerading

Packet modication

N/A N/A Contextaware Inference

Mechanism

An extension of DSR

Michiardi &

Molva (2002)

[71]

Secure routing Direct observation

Reputation

Selsh nodes

False information

propagation

DoS attack

N/A Dynamicity

Subjectivity

Extension of [67]

Functional trust concept introduced to combine subjective direct observation plus

indirect information.

He et al.

(2004)[72]

Secure routing Direct observation

Reputation

Packet dropping

Selsh nodes

Impersonation

False information

propagation

Throughput

Overhead

Weighted transitivitySecure and Objective

Reputation-based Incentive

(SORI) based on DSR

Incentive mechanism

Nekkanti & Lee

(2004)[73]

Secure routing N/A General outside

attackers

Average

endtoend delay

Packet delivery

ratio

Trust value

ranges from

0 to 10 as an

integer

Trust-based adaptive AODV

Li et al. (2004)

[74]

Secure routing Direct observation

Recommendation

General

malicious nodes

N/A Dynamicity

Asymmetry

Weighted transitivityb

An extension of AODV using subjective logic

a“Utility” metric refers to the ratio of the number of transmissions originated at the node itself to the number forwarded as an intermediate node on

behalf of other nodes [69]. This metric can be represented as A/(A+B) where A is the number of packets transmitted by a node for itself and B is the

number of packets transmitted for others. b“Weighted transitivity” means that trust is transitive only with a weight reecting decaying of trust depending on condence level, credibility, reliability,

time, reputation, distance, etc.

[11] S. Ruhomaa, L. Kutvonen, and E. Koutrouli, “Reputation Management

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TABLE II

Authors, Year,

Ref. no

Purpose Methodology Attacks consideredPerformance

metrics

Trust property Trust management model

Pisinou et al.

(2004) [75]

Secure routing Direct observation Black hole

Route injection

Selsh nodes

Route overhead

Algorithm

running time

Number of

routes selected

Rout errors sent

Transitivity

Dynamicity

An extension of AODV

called Trust-embedded

AODV

Buchegger &

Boudec (2004)

[76]

Secure routing Direct observation

Reputation

False information

propagation

Mean detection

time for misbehaving nodes

False alarm

Dynamicity Bayesian Model

Trust revocation introducing

redemption

Ghosh (2005)

[77]

Secure routing Reputation

Direct observation

Black hole

Gray hole

False accusation

DoS

Overhead

Routes selected

Route errors

Weighted transitivityTrust-embedded AODV

(TAODV)

Incentive mechanism

Wang et al.

(2005) [78]

Secure routing Direct observation

Reputation

False accusation

False information

Detection rate

False alarm

N/A Specication-based approach

as an extension of AODV

Zouridaki et al.

(2005 ) [79]

(2006) [80]

Secure routing Direct observation

[79][80]

Reputation by

secondhand

information [80]

Packet dropping

Packet misrouting

Packet injection

Added in [2006]

False accusation

propagation

Malicious node

colluding

Replay

Duplicate packet

forwarding

Condence level

over trust value

Trustworthiness

Opinion values

about other

nodes

Weighted transitivity

Dynamicity

Herms

Bayesian Model

Trust revocation introducing

Window scheme to ush out

stale trust information

Pirzada et al.

(2006)

[81]

Secure routing Direct observation Packet modication

Black hole

Gray hole

Packet loss

Packet

forwarded

Throughput

Overhead

Latency

Path optimality

Detection

probability

Dynamicity Trust-based reactive routing

protocols (DSR, AODV,

TORA)

Effortreturnbased trust

model

Sun et al. (2006)

[46]

Secure routing Direct observation

on packet dropping

rate

Recommendation

False recommendation

Newcomer attack

Sybil attack

Trust level

Packet dropping

ratio

Trust is a continuous value

Subjectivity

Asymmetry

Weighted transitivity

Information theory-based

modeling: Entropy-based

trust model and Probabilitybased trust model

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TABLE III

Authors, Year,

Ref. no

Purpose Methodology Attacks consideredPerformance

metrics

Trust property Trust management model

Abusalah et al.

(2006)

[81]

Secure routing Direct observation

on forwarding

packets

Recommendation

Evidences

Packet dropping Routing

overhead

Route discovery

time

N/A TARP (trustaware routing

protocol)

Sen et al. (2006)

[83]

Secure routing Direct observation

Evidences

False accusation

Packet dropping

Tampering

Packet dropping

rate

Reputation level

Dynamicity Group trust value is used

based on reputations for

neighboring nodes

Soltanali et al.

(2007)

[84]

Secure routing Reputation only

based on direct

and indirect

observation

Selsh nodes Throughput

Detection

percentage

Dynamicity

Weighted transitivity

Subjectivity

An extension of DSR with

the components of monitor,

reputation manager, opinion

manager, routing and forwarding manager

Balakrishnnan et

al. (2007) [85]

Secure routing Direct observation

Recommendation

Packet dropping

DoS (ooding)

Routing

information

modication

False accusation

False

recommendation

Packet delivery

ratio

Latency

Contextdependency

Weighted

transitivity

Trust is a

continuous

value

Fellowship model using DSR

Li et al. (2008)

[52]

Secure routing Reputation

Direct observation

Selective

misbehaving

Bad mouthing

Onoff attack

Conicting

behavior

Ratio of

trustworthiness

over reputation

for both good

and bad nodes

Weighted transitivityObjective Trust Management

Framework (OTMF)

Modied Bayesian model

Mundinger &

Boudec (2008)

[86]

Secure routing Direct observation

Recommendation

False accusation

/recommendation

Freeriding

The incorrect

reputation

threshold point

that does not

affect the system

performance

N/A A general framework to evaluate robustness of reputation

systems by analyzing them

using deviation test.

Moe et al.

(2008) [87]

Secure routing Direct observation

Recommendation

Selective packet

dropping by selfish nodes

DoS

Trustworthiness

of a node

Dynamicity Trustbased Secure Routing

(TSR)

An extension of DSR

Incentive mechanism

Reidt et al.

(2009) [88]

Secure routing N/A (assume that

information can

be derived through

observation or

other mechanisms)

N/A Communication

cost

Probability to

reach Trust

Authority (TA)

nodes

N/A A group of TA nodes are

selected based on their reliability, energy level, and connectedness with the network.

[49] Z. Liu, A. W. Joy, and R. A. Thompson, “A Dynamic Trust Model for

Mobile Ad Hoc Networks,” Proc. 10th IEEE Int’l Workshop on Future

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208

580 IEEE COMMUNICATIONS SURVEYS & TUTORIALS, VOL. 13, NO. 4, FOURTH QUARTER 2011

TABLE IV

Authors, Year,

Ref. no

Purpose Methodology Attacks consideredPerformance

metrics

Trust property Trust management model

Adnane et al.

(2009) [90]

Secure routing Direct & indirect

information

Evidences (PKI)

Fake identity

General

malicious nodes

N/A N/A Extension of OLSR

No specic trust metric

shown

Ayachi et al.

(2009) [89]

Secure routing Direct information Message

replication/

forgery/modication

N/A N/A Extension of AODV

No specic trust metric

shown

Weimerskirch et

al. (2001) [91]

Authentication Recommendation

References

Packet modication

Breach of con-

dentiality

DoS

N/A Weighted transitivityDistributed Trust Model

Verma et al.

(2001) [92]

Authentication Direct observation

Indirect

information

General inside

and outside

attackers

N/A Transitivity An extension of DSR and

ZRP

Pirzada &

McDonald

(2004) [93]

Authentication Direct observation Packet modication

Packet fabrication

Impersonation

N/A Weighted transitivity

Dynamicity

An extension of DSR

An extension of Marsh’s

trust model [125]

Davis (2004)

[47]

Authentication Evidence-based

(i.e., certicates)

False accusation N/A N/A Certicate issuance, storage,

validation, and revocation

are considered

Ngai & Lyu

(2004) [94]

Authentication Direct observation

Recommendation

False certicate Successful,

failure, and

unreachable rate

for detecting

false certicates

Trust is a continuous value

Asymmetry

Distributed trust-based

authentication

Albers et al.

(2002) [95]

Intrusion

Detection

Direct observation

for anomaly detection or misuse detection

Policybased

General

misbehaving

nodes

N/A Symmetry Local Intrusion Detection

System (LIDS)

Ahmed et al.

(2006) [96]

Intrusion

Detection

Direct observation Black hole

Packet dropping

Malicious ooding

Routing loop

Overhead

False alarm rate

N/A Leverage IDS to evaluate

trust level of other nodes

Gray et al.

(2002) [97]

Access control Policy-based (local

and global policies)

Loss of authorizationTrust level per

session

Contextdependency

Weighted

transitivity

Trust is a

continuous

value

Trustbased admission control

Luo et al. (2004)

[98]

Access control Direct observation General

misbehaving

nodes

Overhead

Delay and number of retries before ticket is received

Dynamicity URSA(Ubiquitous and RobuSt Access control)

Localized group trust model

based on threshold cryptography

Information Technology: Coding and Computing, Tiejun Huang, China,

4-6 April 2005, vol. 2, pp. 568-573.

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2000.

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IEEE Wireless Communications and Networking Conf., vol. 2, pp. 825-

830, March 2004.

[73] R. K. Nekkanti and C. Lee, “Trust-based Adaptive On Demand Ad

Hoc Routing Protocol,” Proc. 42th Annual ACM Southeast Regional

Conf., Huntsville, Alabama, 2004, pp. 88-93.

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Sky, Montana, 6-13 Mar. 2004, vol. 2, pp. 1286-1295.

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209

CHO et al.: A SURVEY ON TRUST MANAGEMENT FOR MOBILE AD HOC NETWORKS 581

TABLE V

Authors, Year,

Ref. no

Purpose Methodology Attacks consideredPerformance

metrics

Trust property Trust management model

Adams & Davis

(2005) [17]

Access control Direct observation

Reputation

General

misbehaving

nodes

N/A Conditional

transitivity1

Trust is a

continuous

value

Bayesian Model for risk assessment

Yunfang (2007)

[53]

Access control Direct observation

Reputation

Policybased

General

misbehaving

nodes

N/A Weighted transitivityIntegrated (policy-based plus

reputation-based) adaptive

trust management

Virendra et al.

(2005) [99]

Key

Management

Direct/indirect observationFalse key

generation

Compromise of

other nodes’ keys

N/A Weighted transitivity

Subjectivity

Dynamicity

Trust is a continuous value

Physical-Logical Trust Domains (PLTDs)

Hadjichristo et

al. (2005) [63]

Key

Management

Direct observation

Reputation

Stacking attack

General

malicious nodes

% of available

nodes with

capability

of service

provision

Average shortest

path

% of isolated

nodes

Success ratio to

obtain system

service

Trust ranges

from 1 to 100

Dynamicity

A modied hierarchical trust

PKI model

Adams et al.

(2005)

[17]

Key

Management

Nonrepudiation

Direct observation

Reputation

Selsh nodes

Blackmailing

Stacking attack

Trust value Contextdependency

Trust is a

continuous

number

Subjectivity

Dynamicity

Interpersonal trust model

Redemption

Simple weighted model

Li et al. (2006)

[100]

Key

Management

Direct observation Fake certicate

Tampering

Average

overhead of

the repository in

each node

Transitivity Localized Public-Key

Management (LPM)

Combining certicate chain

and communication path

based on PGP

Chang & Kuo

(2009) [101]

Key

Management

Direct observation On-Off attack

Conicting

behavior attack

Newcomer attack

Fake ID attack

Cheating attack

Collusion attack

Trust value

Packet delivery

ratio

Trust value is a

discrete value

Markov chain trust model

Yan et al. (2003)

[64]

Trust

evaluation

Direct observation

Reputation

Recommendation

Black hole

DoS

Planned metric:

security level

Subjectivity

Weighted transitivity

Personal Trusted Bubble

(PTB)

Networking Conf., Lecture Notes in Computer Science, Athens, Greece,

9-14 May 2004, vol. 3042, pp. 1446-1451.

[76] S. Buchegger and J.Y.L. Boudec, “A Robust Reputation System for P2P

and Mobile Ad-hoc Networks,” Proc. 2nd Workshop on the Economics

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Routing Solution in Mobile Ad Hoc Networks,” Mobile Networks and

Applications, vol. 10, pp. 985-995, 2005.

[78] B. Wang, S. Soltani, J. Shapiro, and P. Tab, “Local Detection of Selsh

Routing Behavior in Ad Hoc Networks,” Proc. 8th Int’l Symposium on

Parallel Architectures, Algorithms and Networks , 7-9 Dec. 2005, pp.

392-399.

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Trust Establishment Framework for Reliable Data Packet Delivery in

MANETs,” Proc. 3rd ACM Workshop on Security for Ad Hoc and

Sensor Networks, Alexandria, VA, Nov. 7, 2005.

[80] C. Zouridaki, B. L. Mark, M. Hejmo, and R. K. Thomas, “Robust

Cooperative Trust Establishment for MANETs,” Proc. 4th ACM Workshop on Security of Ad Hoc and Sensor Networks, Alexandria, VA, 30

Oct. 2006, pp. 23-34.

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Networks,” Proc. 27th Australasian Computer Science Conf., vol. 26,

pp. 47-54, 2004.

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19, no. 4, pp.78-93, 2008.

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for Mobile Ad Hoc Networks,” Int’l Symposium on Ad Hoc and

Ubiquitous Computing, 20-23 Dec. 2006. Surathkal, India, pp. 62-67.

[84] S. Soltanali, S. Pirahesh, S. Niksefat, and M. Sabaei, “An Efcient

Scheme to Motivate Cooperation in Mobile Ad Hoc Networks,” Int’l

Conf. on Networking and Services, Athens, Greece, 19-25 June 2007,

pp. 98-103.

[85] V. Balakrishnnan, V. Varadharajan, U. K. Tupakula, and P. Lucs, “Trust

and Recommendations in Mobile Ad Hoc Networks,” Int’l Conf. on

Networking and Services, Athens, Greece, 19-25 June 2007, pp. 64-69.

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65, no. 3-4, pp. 212-226, Mar. 2008.

[87] M. E. G. Moe, B. E. Helvik, and S. J. Knapskog, “TSR: Trust-based

Secure MANET Routing using HMMs,” Proc. 4th ACM Symposium

on QoS and Security for Wireless and Mobile Networks, Vancouver,

British Columbia, Canada, 27-28 Oct. 2008, pp. 83-90.

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2009 IEEE Sarnoff Symposium, March 2009.
210

582 IEEE COMMUNICATIONS SURVEYS & TUTORIALS, VOL. 13, NO. 4, FOURTH QUARTER 2011

TABLE VI

Authors, Year,

Ref. no

Purpose Methodology Attacks consideredPerformance

metrics

Trust property Trust management model

Jiang & Baras

(2004) [103]

Trust evidence

distribution

Evidence-based

(certicates)

Direct observation

General

misbehaving

nodes

Number of hops

and delay to obtain the certi-

cate

Success rate obtaining the certicate

Ant-Based adaptive trust Evidence Distribution (ABED)

based on a swarm intelligence paradigm

Baras & Jiang

(2004) [104]

Trust computationLocal direct

observation

Evidencebased

(certicates)

Recommendation

(i.e., voting)

General

misbehaving

nodes

Probability of

having at least

one secure path

between trusted

pairs

percentage of

trusted nodes in

the network

Time taken to

converge to a

steady state in

trustworthiness

of all nodes

Dynamicity Distributed trust computation

model using graph theory

and random theory

Theodorakopoulos

& Baras (2006)

[50]

Trust

evaluation

Direct observation

Recommendation

False accusation

Impersonation

Condence level

Opinions about

other nodes

Transitivity

Trust and con-

dence value is

binary

Trust evaluation model based

on Semirings theory

Boukerche &

Ren (2008)

[105]

Reputation

evaluation

Direct observation

Reputation

General

misbehaving

nodes

Query overhead

Security

overhead

Percentage of

packets

Number of

nodes

Percentage of

malicious nodes

Weighted transitivityGeneralized Reputation

Evaluation (GRE) Prototype

Groupbased trust model

Moloney & Weber (2005) [106]General

security level

identication

Direct/indirect observation

Recommendation

General

misbehaving

nodes

N/A ContextdependencyContext-aware security system for MANETs

Cho et al. (2009)

[107]

Battleeld trust

establishment

under no prior

interactions

Direct/indirect observation

Recommendation

General

misbehaving

nodes

Trust level Dynamicity

Weighted

transitivity

Contextdependency

Asymmetry

Subjectivity

Quantitative modeling technique used called hierarchical Stochastic petri nets

[89] M. A. Ayachi, C. Bidan, T. Abbes, and A. Bouhoula, “Misbehavior

Detection Using Implicit Trust Relations in the AODV Routing Protocol,” 2009 Int’l Conf. on Computational Science and Engineering,

Vancouver, Canada, vol. 2, 29-31 Aug. 2009, pp. 802-808.

[90] A. Adnane, C. Bidan, R. T. de Sousa, “Trust-based Countermeasures

for Securing OLSR Protocol,” 2009 Int’l Conf. on Computational

Science and Engineering, Vancouver, Canada, vol. 2, 28-31 Aug. 2009,

pp. 745-752.

[91] A. Weimerskirch and G. Thonet, “A Distributed Light-Weight Authentication Model for Ad-hoc Networks,” Proc. of 4th Int’l Conf. on

Information Security and Cryptology, 6-7 Dec. 2001.

[92] R. R. S. Verma, D. O’Mahony and H. Tewari, “NTM – Progressive

Trust Negotiation in Ad Hoc Networks,” Proc. 1st Joint IEI/IEE

Symposium on Telecommunications Systems Research, Dublin, Ireland,

27 Nov. 2001.

[93] A. A. Pirzada, C. McDonald, and A. Datta, “Performance Comparison of Trust-based Reactive Routing Protocols,” IEEE Trans. Mobile

Comput., vol. 5, no. 6, June 2006, pp. 695-710.

[94] E. C. H. Ngai and M. R. Lyu, “Trust and Clustering-based Authentication Services in Mobile Ad Hoc Networks,” Proc. 24th Int’l Conf.

on Distributed Computing Systems Workshops, 23-24 March 2004, pp.

582-587.

[95] P. Albers, O. Camp, J.-M. Percher, B. Jouga, L. Me, R. Puttini, “Security in Ad Hoc Networks: a General Intrusion Detection Architecture

Enhancing Trust Based Approaches”, Proc. 1st Int’l Workshop on

Wireless Information Systems, Apr. 2002, pp. 1-12.

[96] E. Ahmed, K. Samad and W. Mahmood, “Cluster-based Intrusion Detection (CBID) Architecture for Mobile Ad Hoc Networks.” AusCERT

Asia Pacic Information Technology Security Conf., Gold Coast, Australia, 21-26 May 2006.

[97] E. Gray, P. O’Connell, C. Jensen, a. Weber, J.-M. Seigneur, and C.

Yong, “Towards a Framework for Assessing Trust-based Admission

Control in Collaborative Ad Hoc Applications,” Technical Report,

TCD-CS-2002-66, Trinity College Dublin, 2002.

[98] H. Luo, J. Kong, P. Zerfos, S. Lu, and L. Zhang, “URSA: Ubiquitous

and Robust Access Control for Mobile Ad Hoc Networks,” IEEE/ACM

Trans. Netw., vol. 12, no. 6, Dec. 2004, pp. 1049-1063.

[99] M. Virendra, M. Jadliwala, M. Chandrasekaran, and S. Upadhyaya,

“Quantifying Trust in Mobile Ad-Hoc Networks,” Proc. Int’l Conf.

Integration of Knowledge Intensive Multi-Agent Systems, 18-21 April

2005, pp. 65-70.

[100] R. Li, J. Li, P. Liu, and H. H. Chen, “On Demand Public Key Management for Mobile Ad Hoc Networks,” Wiley’s Wireless Communications

and Mobile Computing, May 2006, vol. 6, no. 3, pp. 295-306.

[101] B. J. Chang and S. L. Kuo, “Markov Chain Trust Model for Trust Value

Analysis and Key Management in Distributed Multicast MANETs,”

IEEE Trans. Veh. Technol., vol. 58, no. 4, May 2009, pp. 1846-1863.

[102] A. Hegland, E. Winjum, S. F. Mjolsnes, C. Rong, O. Kure and P.

Spilling, “A Survey of Key Management in Ad Hoc Networks,” IEEE

Commun. Surveys and Tutorials, vol. 8, no. 3, pp. 48-66, 2006.

[103] T. Jiang and J. S. Baras, “Ant-based Adaptive Trust Evidence Distribution in MANET,” Proc. 2nd Int’l Conf. on Mobile Distributed

Computing Systems Workshops, Tokyo, Japan, 23-24 March 2004,

pp.588-593.
211 CHO et al.: A SURVEY ON TRUST MANAGEMENT FOR MOBILE AD HOC NETWORKS 583

[104] J. S. Baras and T. Jiang, “Cooperative Games, Phase Transition on

Graphs and Distributed Trust in MANETs,” Proc. 43th IEEE Conf. on

Decision and Control, Atlantis, Bahamas, 14-17 Dec. 2004, vol. 1, pp.

93-98.

[105] A. Boukerche and Y. Ren, “A Security Management Scheme using a

Novel Computational Reputation Model for Wireless and Mobile Ad

Hoc Networks,” Proc. Int’l Workshop on Modeling Analysis and Simulation of Wireless and Mobile Systems, Vancouver, British Columbia,

Canada, pp. 88-95, 2008.

[106] M. Moloney and S. Weber, “A Context-aware Trust-based Security

System for Ad Hoc Networks,” Proc. 1st Int’l Conf. on Security and

Privacy for Emerging Areas in Communication Networks-Workshop,

5-9 Sept. 2005, pp. 153-160.

[107] J. H. Cho, A. Swami and I.R. Chen, “Modeling and Analysis of Trust

Management for Cognitive Mission-driven Group Communication Systems in Mobile Ad Hoc Networks,” 2009 Int’l Conf. on Computational

Science and Engineering, vol. 2, Vancouver, Canada, 29-31 Aug. 2009,

pp. 641-650.

[108] Q. Mahmoud (Editor), “Cognitive Networks: Towards Self-Aware

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Defending Against Sybil Attacks via Social Networks,” IEEE/ACM

Trans. Netw., vol. 16, no. 3, June 2008, pp. 576-589.

[110] M. Maheswaran, H. C. Tang, and A. Ghunaim, “Toward a Gravitybased Trust Model for Social Networking Systems,” Proc. 27th Int’l

Conf. on Distributed Computing Systems Workshops, 22-29 June 2007,

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[111] R. Kohlas, J. Jonczy, and R. Haenni, “A Trust Evaluation Method

Based on Logic and Probability Theory,” Trust Management II: IFIP

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[112] G. Yin, D. Shi, H. Wang, and M. Guo, “RepCom: Towards Reputation

Composition over Peer-to-Peer Communities,” 2009 Int’l Conf. on

Computational Science and Engineering, vol. 2, Vancouver, Canada,

29-31 Aug. 2009, pp. 765-771.

[113] L. Boursas and W. Hommel, “Multidimensional Dynamic Trust Management for Federated Services,” 2009 Int’l Conf. on Computational

Science and Engineering, vol. 2, Vancouver, Canada, 29-31 Aug. 2009,

pp.684-689.

[114] R. Guha, R. Kumar, P. Raghavan, and A. Tomkins, “Propagation of

Trust and Distrust,” Proc. 13th Int’l Conf. on World Wide Web, New

York, NY, 19-21 May 2004, pp. 403–412.

[115] Y. Wang and M. P. Singh, “Trust Representation and Aggregation in

a Distributed Agent System,” Proc. 21st National Conf. on Articial

Intelligence (AAAI), 2006, Boston, MA, pp. 1425–1430.

[116] Y. Wang and M. P. Singh, “Formal Trust Model for Multiagent

Systems,” Proc. 20th Int’l Joint Conf. on Articial Intelligence, Jan.

2007, pp. 1551–1556.

[117] X. Hong, D. Huang, M. Gerla, and Z. Cao, “SAT: Building New

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2009, pp. 138-148.

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Dissertation, Department of Mathematics and Computer Science: University of Stirling, 1994.

Jin-Hee Cho received her BA from Ewha Womans

University, and MA from Washington University in

St. Louis, MO in 1997, and 1999 respectively. She

also received her MS, and PhD in Computer Science

from Virginia Tech in 2004, and 2008 respectively.

She received an IREAN fellowship through the

NSF IGERT program during her Ph.D. study and a

postdoctoral research fellowship through the Army

Research Laboratory/Oak Ridge Associated Universities postdoctoral fellowship program from January

2009 to June 2010. After then, she joined ARL,

Adelphi, MD as a computer scientist in the Network Science Division,

Computational and Information Sciences Directorate. Her research interests

include wireless mobile networks, mobile ad hoc networks, sensor networks,

secure group communications, group key management, network security,

intrusion detection, performance analysis, trust management, social/cognitive

networks, and network economic modeling. She is a member of the IEEE

and ACM.

Ananthram Swami received the B.Tech. degree

from the Indian Institute of Technology (IIT), Bombay; the M.S. degree from Rice University, Houston,

TX, and the Ph.D. degree from the University of

Southern California(USC), Los Angeles, all in electrical engineering.

He has held positions with Unocal Corporation,

USC, CS-3 and Malgudi Systems. He was a Statistical Consultant to the California Lottery, developed HOSAT, a MATLAB-based toolbox for nonGaussian signal processing, and has held visiting

faculty positions at INP, Toulouse, France. He is the ST for Network Science at

the U.S. Army Research Laboratory. His work is in the broad areas of signal

processing, wireless communications, sensor and mobile ad hoc networks.

He is co-editor of the book Wireless Sensor Networks: Signal Processing and

Communications Perspectives (New York: Wiley, 2007).

Dr. Swami is a member of the IEEE Signal Processing Society’s (SPS)

Technical Committee on Sensor Array and Multichannel systems and serves

on the Senior Editorial Board of the IEEE Journal on Selected Topics in Signal

Processing. He was a tutorial speaker on Networking Cognitive Radios for

Dynamic Spectrum Access at IEEE ICC 2010, co-chair of IEEE SPAWC’10,

and has served on the IEEE SPS BoG.

Ing-Ray Chen received the BS degree from the

National Taiwan University, Taipei, Taiwan, and the

MS and PhD degrees in computer science from the

University of Houston. He is a professor in the

Department of Computer Science at Virginia Tech.

His research interests include mobile computing,

wireless networks, security, fault tolerance, multimedia, trust management, real-time intelligent systems,

and performance analysis. Dr. Chen currently serves

as an editor for The Computer Journal, Wireless

Personal Communications, Wireless Communications and Mobile Computing, Security and Communication Networks, and

International Journal on Articial Intelligence Tools. He is a member of the

IEEE and ACM.

Research@

213

Temporal Complexity of the Order Parameter 

at the Phase Transition

Malgorzata Turalska, Bruce J. West, and Paolo Grigolini

Physical Review E, 83, 061142 (2011)
214 PHYSICAL REVIEW E 83, 061142 (2011)

Temporal complexity of the order parameter at the phase transition

Malgorzata Turalska,1 Bruce J. West,2 and Paolo Grigolini1

1Center for Nonlinear Science, University of North Texas, P.O. Box 311427, Denton, Texas 76203-1427, USA 2Information Science Directorate, US Army Research Office, Durham, North Carolina 27709, USA

(Received 9 February 2011; revised manuscript received 25 April 2011; published 24 June 2011)

We study a decision making model in a condition where it is equivalent to the two-dimensional Ising model,

and we show that at the onset of phase transition it generates temporal complexity, namely, nonstationary and

nonergodic fluctuations. We argue that this is a general property of criticality, thereby opening the door to

the application of the recently discovered phenomenon of complexity matching: For an efficient transfer of

information to occur, a perturbing complex network must share the same temporal complexity as the perturbed

complex network.

DOI: 10.1103/PhysRevE.83.061142 PACS number(s): 05.40.−a, 05.10.−a, 05.50.+q, 05.70.Jk

I. INTRODUCTION

Phase transitions and critical phenomena occur frequently

in nature and have been widely studied by physicists, see

for instance [1]. The Ising model [2] originally introduced to

explain ferromagnetic phase transition is well known, and the

exact solution found by Onsager [3] for the occurrence of phase

transition in the two-dimensional case is widely recognized as

an example of outstanding theoretical achievement. In the last

few years some scientists have used the Ising model to shed

light on biological and neurophysiological processes [4–7].

More precisely, the authors of [4] used the Ising model

to explain the collective behavior of biological networks

and the authors of [5–7] adopted the Ising model for the

purpose of supporting their hypothesis that the brain works

at criticality without establishing a clear distinction between

phase transition and self-organized criticality [8]. Finally,

we have to mention that the Ising model is frequently used, see

for instance [9,10], to model neurophysiological data with the

constraint of maximal entropy. Although relevant to explore

the cognition properties of the brain, this is a perspective

different from the one adopted in this paper.

Couzin [11] made the conjecture that the cooperative

interaction between the birds of a flock may mimic the

brain cognition properties, and the experimental observation

of Cavagna et al. [12] confirmed the criticality condition of

a coordinated flock of birds. It is important to emphasize that

the connection between phase transition processes and cooperative behavior of biological systems was already stressed in

the pioneering paper of Vicsek et al. [13]. It is thought [4] that

biological systems in general operate at criticality.

The phase transition condition is not confined to biological

systems. The recent work on econophysics [14,15] suggests

that the same arguments can be applied to sociological

systems. As a matter of fact, Sornette [15] proposes a model

of cooperative economical interaction, of the same type as

the Ising model, which generates at criticality a transition

from subcritical to supercritical patterns very similar to those

produced by the Ising model. As stressed by the authors

of [5], the Ising patterns at criticality correspond to the

emergence of correlation links yielding a scale-free network

statistically indistinguishable from that experimentally observed within the brain, using functional magnetic resonance

imaging.

These studies emphasize the spatial and network complexity emerging from the cooperative interaction of the

network’s units, but overlook the temporal complexity of

these networks. Herein we are attempting to fill this gap

and prove that the temporal complexity emerges at criticality.

Temporal complexity is defined as follows. The time dynamics

of complex networks is characterized by the occurrence of

significant events, which may be financial crashes [15], brain

quakes [5,6], or the changes of direction of a flock of birds

[12,13]. The time interval between two consecutive events

is given by a distribution density ψ(t), which, drastically

departing from the conventional Poisson statistics, has the

inverse-power-law form

ψ(τ ) ∝

1

τμ , (1)

with μ < 2. The occurrence of an event does not have any

memory of the occurrence of earlier events. This property is

usually denoted as renewal, but it must not be confused with the

ordinary Poisson and Markov condition: The signal generated

by these events is characterized by long-range correlation in

time, and, most importantly, it is essentially nonstationary,

thereby breaking the ergodicity that is a fundamental property

of statistical physics [16]. We refer to it as the non-Poisson

renewal condition. A significant event is interpreted as a failure

whose occurrence brings the network back to a brand new

condition. We refer to these as crucial events.

Events of this kind may be very difficult to detect. The

theoretical discussion [16], for instance, applies to the case of

dichotomous signals, where crucial events correspond to the

abrupt transition from one to the other value of the signal. In

the case of the decision making model (DMM) under study, the

single units in isolation produce perfect dichotomous signals.

They can be easily detected, but they are of a Poisson kind,

thereby strikingly departing from the condition of temporal

complexity. When the control parameter is extremely large, the

global signal undergoes abrupt transitions that however depart

again from the condition of temporal complexity, because

they are predominantly of a Poisson kind. At criticality the

global signal loses these abrupt transitions that are replaced

by smoother fluctuations around the mean value. Herein we

consider a model generating a signal whose time mean value

vanishes. We assume the crossing of the origin to be the

proxy of a significant event. Thus, we interpret the signal as a

1539-3755/2011/83(6)/061142(6) 061142-1 ©2011 American Physical Society
TURALSKA, WEST, AND GRIGOLINI PHYSICAL REVIEW E 83, 061142 (2011) 215

dichotomous fluctuation making abrupt changes from 1 to −1

and back again. We evaluate the correlation function of this

ideal dichotomous signal, denoted by the symbol �(t,t�

), and

we prove that its dependence on the times t and t� exactly fits

the prediction of the non-Poisson renewal condition.

The connection between phase transition, and its spatial

and network complexity, with temporal complexity is not well

known. In the literature there are only a few papers where this

connection is discussed. One is [17], which converts the 3D

Ising model into a critical map generating type I intermittent

dynamics and consequently the same temporal complexity

as the one under discussion herein. The emergence of nonPoisson renewal properties was also discussed in [18,19],

which is devoted to illustrating the cooperative properties of

a DMM. The foundation of these papers is sociological and

similar to that of the economical model illustrated in the work

of Sornette [15]. Thus, the authors [18,19] did not discuss the

connection of the DMM with the Ising model and adopted the

nonrealistic but often made assumption of all-to-all coupling

among the network’s units, thereby confining the emergence

of an inverse power law, namely, of temporal complexity, to

the same time scale as that of the units in isolation.

The motivation for the present research work is given by

the transfer of information from a complex network to another

complex network [20]. Rather than using entropic arguments,

as is usually done with the transfer of information, we open

the door to the concept of complexity management recently

advocated in [21]. This latter work established that a complex

network driven by crucial events is sensitive to stimuli with the

same complexity. In other words, the transfer of information

rests on temporal complexity and this explains the motivation

to prove that criticality also generates temporal complexity.

In Sec. II we discuss under which conditions the DMM [18,

19] is equivalent to the Ising model. In Sec. III we prove that

at criticality the function ψ(t) becomes an inverse power law

for over four decades. We shall also argue that with increasing

the network’s size ψ(t) may become an infinitely extended

inverse power law. Section IV shows that this is a renewal

process. Finally we devote Sec. V to concluding remarks.

II. DETAILED DESCRIPTION OF THE MODEL

We consider a system of L discrete variables located at the

nodes of a two-dimensional square lattice. Each unit si is a

stochastic oscillator and can be found in either of two states,

+1 or −1. The dynamic is introduced by choosing a single

unit on site i and updating it in an elementary time step with a

transition rate g:

g



s+1

i → s−1i



= g0 exp  K

M (M+1 − M−1)



, (2)

g



s−1

i → s+1i



= g0 exp 

− K

M (M+1 − M−1)



. (3)

Here M denotes the total number of nearest neighbors, and

M+1 and M−1 the number of nearest neighbors being in the

state +1 and −1, respectively. Single units change its states,

thereby making M+1 and M−1 fluctuate in time, while, of

course, the total number of nearest neighbors is conserved,

M+1 + M−1 = M.

-1.0

0.0

1.0

0 250 500 750 1000

si(t)

time [arb. units]

-1.0

0.0

1.0

0 250000 500000 750000

ξ(t)

time [arb. units]

τi τi+1

τi τi+1

(a)

(b)

FIG. 1. (a) Temporal evolution of a single unit si(t) and (b) of the

global order parameter ξ (t) for the decision making model realized on

a square lattice of L = 50 × 50 nodes, with g0 = 0.01 and K = 1.70.

To illustrate the concept of crucial events we mark the time intervals

τ between two consecutive events, according to their definitions

assumed in this paper. Notice different time scales on both plots.

All numerical calculation are performed on a square lattice

of either L = 50 × 50 or L = 100 × 100 nodes with periodic

boundary conditions. In a single time step a run over the whole

lattice is performed and for every unit si the transition rate of

Eqs. (2) or (3) is calculated according to which a node is given

the possibility to change its state. The single unit in isolation,

K = 0, fluctuates between states +1 and −1 with the transition

rate g = g0. When coupling constant K > 0, a unit in the state

+1 (−1) makes a transition to the state −1 (+1) faster or

slower according to whether M−1 > M+1 (M+1 > M−1) or

M−1 < M+1 (M+1 < M−1), respectively.

Next, we define the global order parameter ξ (t) = 1

L

L

i=1 si(t), which is characterized by the variability that

does not possess the dichotomous character of single units. In

Fig. 1 we show exemplary temporal evolution for the single

unit si(t) and for the global order parameter ξ (t). Note that

the amplitude of variable ξ (t) depends on the value of the

coupling constant K. When K = 0, all units in the system are

independent Poisson processes; thereby an average taken at

any moment of time over all of them is zero. Once the value

of the coupling increases, K > 0, single units are less and

less independent, resulting in a nonzero average. The quantity

KC is the critical value of the control parameter K, at which

point a phase transition to a global majority state occurs. In

numerical calculations we use the time average ξeq = �|ξ (t)|�

as a measure of this global majority. More precisely, after initial

106 time steps, the average is taken over the same number of

the consecutive time steps of the model.

Note that in the special case when M is the same for all the

nodes and g0 � 1, we find that our model generates the same

phase transition as the two-dimensional Ising model discussed

in the seminal paper of Onsager [3]. This is an expected result

insofar as the Ising model rests on the Hamiltonian

H = −J



L

i

σiσj , (4)

061142-2
TEMPORAL COMPLEXITY OF THE ORDER PARAMETER AT 216 ... PHYSICAL REVIEW E 83, 061142 (2011)

where σi and σj denote the Pauli operators with eigenvalues

±1. According to Boltzmann statistics equilibrium is described

by the density matrix

ρ = exp (−βH)

Z , (5)

with β = 1

kBT , kB being the Boltzmann constant and T the

absolute temperature. Thus, the off-diagonal elements of the

transfer matrix [22] become equivalent to the transition rates

of Eqs. (2) and (3), under the condition

kBT = J

K = MK . (6)

As examples of conditions yielding this equivalence, we

consider two cases. The former is the case of all-to-all

coupling, where M = L. The latter is considered here, and

it is the two-dimensional lattice where each node is coupled to

its 4 nearest neighbors, thereby setting M = 4.

The thermodynamical condition M = L = ∞ was discussed extensively by authors of [18,19], who showed that

under those conditions the ratios M±1

M are equivalent to the

probabilities p±1 for a node si to be in one of two allowed

states. The dynamic evolution of a single unit state is then

described by a two-state master equation

d

dt p+1 = −g



s+1

i → s−1i



p+1 + g



s−1

i → s+1i



p−1, (7)

d

dt p−1 = −g



s−1

i → s+1i



p−1 + g



s+1

i → s−1i



p+1. (8)

Solving the above for the difference in probabilities �(t) =

p+1(t) − p−1(t), which corresponds to the earlier defined

global order parameter ξ (t), one obtains

d

dt

� = 2g0[sinh(K�) − � cosh(K�)]. (9)

This equation yields two solutions, corresponding to global

majority states, for the values of coupling constant K>KC,

where KC = 1.

The solution to the latter condition ofM = 4,L = ∞can be

found in [23] and yields the condition for the global variable to

be ξeq = (1 − [sinh(K/2)]−4)

1/8. In this case the critical value

of the coupling constant is KC = 2 ln(1 + √2) = 1.7627. In

Fig. 2, we show the phase transition in the latter condition and

we find that, as expected, the numerical evaluation of ξeq (K) is

very close to the theoretical prediction of Onsager [3], thereby

confirming that the DMM is equivalent to the Ising model in

the limiting case g0 → 0.

This equivalence between the DMM and the Ising model is

merely formal, because the DMM does not have a Hamiltonian

origin and does not require the action of a thermal bath at

temperature T to work as does the Boltzmann picture. This

explains why the equivalence with the Ising model requires

that g0 vanish, so as to freeze the dynamics of the single units,

in the absence of cooperation.

When we release the condition g0 → 0, the equivalence of

the DMM with the Ising model is lost. In Fig. 3, we see that,

if the condition of g0 � 1 is abolished, the phase transition

emerges at values of K lower than KC of the theoretical

prediction. Simultaneously, further increase of g0 (g0 ≈ 0.40)

leads to a new regime, in which every unit is surrounded by

0.00

0.25

0.50

0.75

1.00

0.50 1.00 1.50 2.00 2.50

ξeq

K

FIG. 2. The phase diagram for global variable ξeq . Thin solid line

and dashed line are the theoretical predictions for the fully connected

and two-dimensional regular network, respectively. In both cases

L = ∞ and the latter case is the Onsager theoretical prediction [3]

for 2D regular lattice. The thick solid line corresponds to the global

states observed for two-dimensional regular lattice (L = 100 × 100

nodes) and g0 = 0.01. Periodic boundary conditions were applied.

nearest neighbors in the opposite state, yielding an update of

its state at every time step and generating the condition in

which the order parameter ξ (t) is exactly zero at all times. It

is worth pointing out that when a network characterized by

a large transition rate g (transition rate g ≈ 1) is studied, it

shows sensitivity to the initial random configuration of the

lattice. We observed that in a limited number of cases the

order parameter evolves to a global majority state, in which

all the nodes are in the same state rather than in condition

ξ (t) = ξeq = 0. Therefore one may consider a bifurcation in

the phase space of the model parameters in this regime.

III. ORDER PARAMETER REVERSAL TIMES

Let us now study the temporal complexity of the order

parameter ξ (t). As pointed out in Sec. I, to prove temporal

complexity of ξ (t) we have to observe significant events

and prove that they are crucial. At criticality the signal is

not dichotomous and, as we shall see hereby, there are good

reasons why it must depart from the dichotomous condition.

1.00

1.50

2.00

2.50

0.00

0.10

0.20

0.30

0.40

0.50

0.00

0.25

0.50

0.75

1.00

ξeq(K, g0)

K g0

ξeq(K, g0)

0.00

0.25

0.50

0.75

1.00

FIG. 3. The two-dimensional phase diagram of global variable

ξeq evaluated for a range of model parameters, g0 and K, on a twodimensional regular lattice of size L = 100 × 100 nodes.

061142-3
TURALSKA, WEST, AND GRIGOLINI PHYSICAL REVIEW E 83, 061142 (2011) 217

10-4

10-3

10-2

10-1

100

100 101 102 103 104 105 106

Ψ(τ)

τ

K=1.00

K=1.50

K=1.70

K=1.75

K=1.80

FIG. 4. Survival probability function �(τ ) for the order parameter ξ (t) evaluated on a two-dimensional lattice of size L = 50 × 50

for g0 = 0.01 and increasing values of coupling constant K. The

straight line corresponds to the slope of −0.50, namely to μ = 1.50,

since �(τ ) ∼ 1

τμ−1 .

We make the conjecture that the crossing of the origin, namely

the times at which ξ (t) changes sign, are the significant events

to observe. As illustrated on Fig. 1, we interpret the time

interval τ between two consecutive crossings as the time

duration of a given decision, even if this decision may rest

on a slight and fluctuating majority.

We evaluate the distribution density of decision-time durations τ , ψ(τ ), and the corresponding survival probability �(τ ),

where �(t) =  ∞

t dτψ(τ ). Although emerging from a simple

regular lattice, that is, one with no structural complexity,

the survival probability presented in Fig. 4 shows a scale-free

property that extends over more than four decades in time

for K ≈ KC. A further increase of K does not affect the

power-law region and has the effect of producing a more and

more extended exponential shoulder. The exponential shoulder

is expected to become predominant for K → ∞.

As mentioned earlier, a single unit in isolation fluctuates

between two states with the transition rate g = g0. The

corresponding survival probability function is an exponential

function �(τ ) = exp(−g0τ ). At the same time, as illustrated

on Fig. 5, a coupled unit tends to update its state with a

transition rate smaller than g0. This is a property of criticality

that is lost completely for very large values of the control

parameter K. Although it is a computational challenge to

explore the dynamics corresponding to extremely large values

of K, our numerical results suggest that whenK is very close to

the critical value, and a decision is reached by a slight majority,

the single units have a dynamics almost indistinguishable

from the Poisson dynamic that they would have in isolation.

As we increase the control parameter K and the majority

becomes larger the single units keep their Poisson dynamics

with a smaller rate. Although it is not possible because of

computer-time limitations to study the network’s dynamics

for values of K much larger than K = 1.80, it is plausible to

make the conjecture that the single units maintain their Poisson

dynamics and that these dynamics become closer and closer to

that of the global variable that will lose its power-law structure

and will become predominantly exponential. In other words,

unanimous and permanent consensus must be perceived as the

10-4

10-3

10-2

10-1

100

100 101 102 103 104 105 106

Ψ(τ)

τ

Kξ=1.50

Kξ=1.70

Kξ=1.80

Ks=1.50

Ks=1.70

Ks=1.80

exp-g0τ

FIG. 5. The survival probability function �(τ ) of the global order

parameter ξ (t) is compared with the transitions between two states

for a single unit si. Simulations were performed on a lattice of

size L = 50 × 50 for g0 = 0.01 and increasing values of coupling

constant K.

manifestation of a limiting condition of Poisson but infinitely

slow dynamics, where the behavior of the single units becomes

identical to that of the global signal.

To make this argument more compelling let us discuss the

properties of the observed exponential shoulder. Since one

observes the network in an organized phase (K>KC) it is

not surprising that one perceives the exponential signature of

an equilibrium regime. Therefore we assume that under those

conditions the global variable ξ (t) follows the dynamics of a

particle diffusing in a double potential well. In the equilibrium

condition one would expect the survival time of staying in

one well to be �(τ ) = exp(−aτ ), where a = A exp(−Q/D).

The parameter Q denotes the height of the potential barrier

separating the wells, and D stands for the diffusion coefficient.

Following [19] we expect the barrier to be a function of

the coupling constant K and the diffusion to depend on the

number of nodes L. By fitting the shoulder, once for a case

where L is kept constant and K varies, and secondly for an

opposite condition, we assess the above hypothesis. The fitting

procedure revealed that the barrier height Q is a linear function

of K (Q ∼ K) and that the diffusion coefficient D is inversely

proportional to the system size L (D ∼ 1/L). For L → ∞ the

transition rate becomes infinitely small and the exponential

shoulder becomes predominant and virtually coincident with

the Poisson dynamics of the single units.

While in the organized phase the effect of increasing L

is that of making the process exponential, although with a

virtually infinite transition time. At criticality the increase of

L has a dramatically different effect. The authors of [19],

based on this DMM in the all-to-all condition, found that at

criticality an inverse-power-law behavior emerges, which is

confined, however, to a time region with the same size as the

Poisson time τP ≈ 1/g0. However, this is not the case when

the units are the nodes of a regular two-dimensional lattice,

interacting only with the four nearest-neighbor nodes. We

have evaluated the time size of the inverse-power-law region

appearing before the exponential shoulder and we found that

its size tends to increase with increasing L as τP ∼ L−1.25.

Unfortunately, this observation is limited to values of L

061142-4
218 TEMPORAL COMPLEXITY OF THE ORDER PARAMETER AT ... PHYSICAL REVIEW E 83, 061142 (2011)

smaller than L = 10 000, since going beyond would exceed

the limits of our computer facilities. However, on the basis of

this observation we reach the conclusion that for L → ∞ the

size of the temporal complexity must become infinitely large.

Thus, criticality is a kind of complex singularity embedded in a

Poisson sea.

These intuitive arguments also explain why the signal ξ (t)

at criticality must depart from the dichotomous condition.

This is a consequence of the fact that the single units are

not rigidly bound to follow the opinion of the majority.

There is, consequently, a subtle connection between criticality,

free will, and correlation between different units. Although

the single units may change opinion, they cannot do that in a

way totally uncorrelated from the behavior of the other units,

insofar as this would be incompatible with the emergence of a

majority, as slight as it might be.

IV. TESTING RENEWAL PROPERTIES

As pointed out earlier, the signal ξ (t) at criticality is not

dichotomous. However, replacing it with a dichotomous signal

corresponding to 1 (yes) when ξ (t) > 0 and to −1 (no) when

ξ (t) < 0, thereby turning it into an ideal dichotomous signal,

is the most convenient way to reveal the emergence of crucial

events at criticality. The theory [24] (see also the earlier

work [25]) allows us to evaluate the nonstationary correlation

function �(t,t�

) in the specific case when the renewal process

rests on the waiting time distribution density ψ(t) assumed to

be an ideal inverse power law. In general, regardless of whether

this ideal condition of infinitely extended inverse power law is

realized or not, it is possible to establish whether the process

is renewal by noticing that

�(t,t�

) = �(τ,ta), (10)

where τ = t − t� and ta = t�

. As explained [26], the function

�(τ,ta) can be evaluated by using the time series {ti}, where ti

are the times at which the fluctuation ξ (t) crosses the origin.We

use a mobile window of length ta to evaluate the waiting time

distribution ψ(τ,ta) and the corresponding survival probability

�(τ,ta), by locating the origin of the window on the time

of occurrence of an event and measuring the time distance

between the end of the window and the time of occurrence of

the first event after the window end. To establish whether

the process is renewal or not we shuffle the sequence of

reversal times, thereby generating a new time series {t

s

i }.

We use the same procedure based on a mobile window to

determine �(ta,s,τ ), which corresponds to the nonstationary

correlation function in the renewal case. If the two survival

probabilities coincide, we conclude that the process is renewal.

If the renewal process is exponential, the correlation function

is stationary, and there is no aging.

In the ideal case of an infinitely extended inverse power law,

with power index μ < 2, as discussed in earlier work [16,24],

aging is perennial. In the case studied in this paper, we have

to take into account that this ideal condition would be realized

by assigning an infinitely large value to L, with the ensuing

consequence that temporal complexity would become virtually

invisible, due to the joint effect of an extremely slight majority

and of a very extended sojourn in a given decision state. To

make temporal complexity visible, we need a compromise,

10-4

10-3

10-2

10-1

100

100 101 102 103 104 105 106

Ψ(τ)

τ

t

a=0

t

a=100

t

a,s=100

t

a=1,000

t

a,s=1,000

t

a=10,000

t

a,s=10,000

FIG. 6. Testing for renewal property. Survival probability function �(τ ) of the order parameter ξ (t) evaluated on lattice of size

L = 50 × 50 and g0 = 0.01, K = 1.70 was aged respectively by

time ta = 100, ta = 1000, and ta = 10 000. Those aged survival

probabilities �(ta ,τ ) are compared with the aged renewal prediction

�(ta,s,τ ).

and we have to set a limit on the time extension of the

inverse power law. As a consequence, we obtain the very

interesting result depicted in Fig. 6, where aging, namely the

slowing down of the survival probability, becomes very large

in the inverse-power-law region, with the survival probability

remaining virtually constant, and in the correspondence of

the exponential shoulder it is virtually suppressed, in accordance with the principle that Poisson statistics annihilates

aging.

V. CONCLUSIONS

This paper establishes that at the onset of phase transition,

in addition to the spatial and network complexity shown by a

number of earlier investigations [4–7,9,12,14,15], we also have

temporal complexity. This significant result reveals a path for

the transport of information from one complex network to

another. As pointed out in Sec. I, there is a general agreement

that complex systems are a set of many units interacting at

the onset of phase transition. The present analysis proves that

these complex networks are characterized also by temporal

complexity, and consequently a perturbing complex network

is expected to exert its influence on another complex network

via the recently discovered complexity management process

[21,31]. Although the present theoretical predictions are based

on the assumption that ψ(τ ) is an ideal inverse power law and

the real complex networks, as shown herein, reach this ideal

condition only when their size is infinitely large, the result

in Fig. 5 indicates that the time region generating ergodicity

breakdown may become so extended as to make a complex

network virtually insensitive to stimuli that do not share the

same extended nonergodic condition.

In the last few years the attention of investigators has

been moving from chaos synchronization of two nonlinear

oscillators [27] to the synchronization dynamics of many units

in large-scale networks [28], called inner synchronization [29]

for convenience. The cooperation induced phase transition

discussed herein can to some extent be thought of as a form of

061142-5
TURALSKA, WEST, AND GRIGOLINI PHYSICAL REVIEW E 83, 061142 (2011) 219

inner synchronization. As pointed out [29] the challenge that

the researchers in the field of complexity have now to address

is the phenomenon of outer synchronization [30].

In this light, we conclude that the results presented here

provide a way of addressing outer synchronization, redirecting

the attention of researchers from the details of the topology

of the complex networks to the analysis of the temporal

complexity that inner synchronization may generate. In fact the

principle of complex management [21,31] rests on temporal

complexity, regardless of the way it is generated.

ACKNOWLEDGMENTS

M.G. and P.G. acknowledge financial support from ARO

and Welch through Grants No. W911NF-05-1-0205 and No.

B-1577, respectively.

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061142-6

Research@

221

Noncoherent Physical-Layer 

Network Coding with FSK Modulation: 

Relay Receiver Design Issues

Matthew C. Valenti, Don Torrieri, and Terry Ferrett

IEEE Transactions on Communications, 59(9), 2595-2604 (2011)
222

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 9, SEPTEMBER 2011 2595

Noncoherent Physical-Layer Network Coding with

FSK Modulation: Relay Receiver Design Issues

Matthew C. Valenti, Senior Member, IEEE, Don Torrieri, Senior Member, IEEE,

and Terry Ferrett, Student Member, IEEE

Abstract—A channel-coded physical-layer network coding

strategy is rened for practical operation. The system uses

frequency-shift keying (FSK) modulation and operates noncoherently, providing advantages over coherent operation: there

are no requirements for perfect power control, phase synchronism, or estimates of carrier-phase offset. In contrast with

analog network coding, which relays received analog signals plus

noise, the system relays digital network codewords, obtained by

digital demodulation and channel decoding at the relay. The

emphasis of this paper is on the relay receiver formulation.

Closed-form expressions are derived that provide bitwise loglikelihood ratios, which may be passed through a standard

error-correction decoder. The role of fading-amplitude estimates

is investigated, and an effective fading-amplitude estimator is

developed. Simulation results are presented for a Rayleigh blockfading channel, and the inuence of block length is explored.

An example realization of the proposed system demonstrates a

32.4% throughput improvement compared to a similar system

that performs network coding at the link layer. By properly

selecting the rates of the channel codes, this benet may be

achieved without requiring an increase in transmit power.

Index Terms—Network coding, two-way relay channel,

frequency-shift keying, noncoherent reception, channel estimation.

I. INTRODUCTION

I

N the two-way relay channel (TWRC), a pair of source

terminals exchange information through an intermediate

relay without a direct link between the sources [1]. The

exchange can occur in two, three, or four orthogonal time

slots, depending on how the information is encoded [2]. With

a traditional transmission scheduling scheme, the exchange

requires four slots. In each of the rst two slots, one of the

terminals transmits a packet to the relay, while in each of

the last two slots, the relay transmits a packet to each of the

terminals. By using network coding [3], the number of slots

can be reduced. With link-layer network coding (LNC), the

third and fourth slots are combined into one slot by having

the relay add (modulo-2) the packets that it receives from the

Paper approved by G. Bauch, the Editor for MIMO, Coding and Relaying

of the IEEE Communications Society. Manuscript received January 12, 2011;

revised March 25, 2011.

Portions of this paper were presented at the IEEE Military Communication

Conference (MILCOM), San Jose, CA, Oct. 2010.

M. C. Valenti’s contribution was sponsored by the National Science

Foundation under Award No. CNS-0750821, and by the United States Army

Research Laboratory under Contract W911NF-10-0109.

M. C. Valenti and T. Ferrett are with West Virginia University, Morgantown,

WV (e-mail: {valenti, terry.ferrett}@ieee.org).

D. Torrieri is with the US Army Research Laboratory, Adelphi, MD (email: dtorr@arl.army.mil).

Digital Object Identier 10.1109/TCOMM.2011.063011.110030

Time Slot 3

Time Slot 1

Time Slot 2

(a) (b)

N1 R N2

ΓR(u) ΓR(u)

N1 N2

N1 N2 N1 N2

N2

R

R

R

N1 R

ΓS(u1)

ΓR(u)

ΓS(u1) ΓS(u2)

Γ ΓR(u) S(u2)

Fig. 1. (a) Link-layer network coding, and (b) Physical-layer network coding.

two terminals. During the third step, the relay sends the sum

of the two packets, and each terminal is able to recover the

information from the other terminal by subtracting (or adding,

modulo-2) its own packet from the received signal. With

physical-layer network coding (PNC), the rst two slots are

combined by having the two terminals transmit their packets at

the same time [2]. The relay receives a combination of both

modulated packets during the rst slot, which it broadcasts

(after appropriate processing) to the two terminals during the

second slot. PNC-based strategies capable of supporting more

than just two source terminals over the TWRC may be found

in [4].

The transmission schedules for LNC and PNC are illustrated

in Fig. 1. The source terminals NN1 and NN2 transmit messages

u1 and u2, respectively, where each message is a packet

containing many information bits. The messages are (channel)

encoded and modulated by the function ΓSS(⋅). In the case of

LNC, the two messages are sent in orthogonal time slots, while

in the case of PNC, they are sent to the relay at the same time

over a multiple-access channel (MAC). For both LNC and

PNC, the relay broadcasts the encoded and modulated signal

ΓRR(u) in the nal time slot, where u is the network codeword

and ΓRR(⋅) is the function used by the relay to encode and

modulate the network codeword. Using the received version

of ΓRR(u) and knowledge of its own message, each terminal

is able to estimate the message sent by the other terminal.

There are several options for implementing PNC. The relay

may simply amplify and forward the signal received from the

end nodes, without performing demodulation and decoding.

This PNC scheme is referred to as analog network coding

(ANC) in [5] and PNC over an innite eld (PNCI) in [6].

Another option is for the relay to perform demodulation and

decoding in an effort to estimate the network codeword, which

is remodulated and broadcast to the terminals. This scheme

is simply called PNC in [2] and PNC over a nite eld

(PNCF) in [6], but in this paper we refer to it as digital

network coding (DNC) to distinguish it from ANC. Under

0090-6778/11$25.00 ⃝c 2011 IEEE
223

2596 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 9, SEPTEMBER 2011

many channel conditions, DNC offers enhanced performance

over ANC. This is because the decoding operation at the relay

helps DNC to remove noise from the MAC phase, while the

noise is amplied by the relay when ANC is used. However,

ANC avoids the computational complexity of demodulation

and decoding at the relay.

Symbol timing is a critical consideration in systems employing PNC. Synchronization of the clocks and packet transmissions at the two source nodes can be achieved by network

timing updates. These updates are routine in networks with

scheduling mechanisms, such as cellular networks. When the

propagation times of the signals from the sources differ, the

symbols arrive at the relay misaligned. The timing offset is

ττ = Δdd/cc, where cc is the speed of light, and Δdd is the

difference in link distances from the sources to the relay. For

insignicant delay, we need ττ ττ ττss/2, where ss denotes

the symbol period. This constraint limits the symbol rate. As

an example, assume Δdd = 300 meters. Then, ss >> 2 μμμμ is

required, and the symbol rate is limited to 250 kilosymbols/s.

An alternative is to delay the transmission of the node closer

to the relay by ττ. However, this requires tracking the distances

between the sources and the relay.

A common assumption made in the PNC literature is

that the signals are coherently demodulated and that perfect

channel-state information (CSI) is available at the receivers.

For instance, decode-and-forward relaying has been considered for binary phase-shift keying [7] and minimum-shift

keying [8] modulations, but in both cases the relay must

perform coherent reception. An amplify-and-forward protocol

is considered in [9], which allows the decision to be deferred

by the relay to the end-node, though detection is still coherent.

When two signals arrive concurrently at a common receiver,

neither coherent detection nor the cophasing of the two

signals (so that they arrive with a constant phase offset) is

practical. The latter would require preambles that detract from

the overall throughput, stable phases, and small frequency

mismatches. To solve this problem, frequency-shift keying

(FSK) was proposed for DNC systems in [10] and [11].

A key benet of using FSK modulation is that it permits

noncoherent reception, which eliminates the need for phase

synchronization. An alternative to noncoherent FSK is to use

differential modulation, which has been explored in [12].

In PNC systems, it is desirable to protect the data with

a channel code. The combination of channel coding and

physical-layer network coding is considered in [13] In [11],

we investigate the use of a binary turbo code in a noncoherent

DNC system. When using a binary turbo code in a DNC

system, the relay demodulator must be able to produce bitwise

log-likelihood ratios (LLRs) that are introduced to the input

of the channel decoder.

Channel estimation is an important issue, especially when

a channel code is used. A training-based channel estimation

scheme for PNC at the relay assuming amplify-and-forward

operation is considered in [14]. The relay estimates channel

parameters from training symbols and adapts its broadcast

power in order to maximize the signal-to-noise ratio at the end

nodes. Estimation of both channel gains in the two-way relay

channel at the end nodes, rather than the relay, is considered

in [15]. Novel channel estimators are presented which provide

better performance than common techniques such as leastsquare and linear-minimum-mean-squared error estimation. In

[16], we propose a blind channel estimator for the relay of the

noncoherent DNC system.

In this paper, we investigate receiver-design issues related

to the use of noncoherent FSK in DNC systems. While

noncoherent FSK has been previously proposed for DNC

sytems in [10], we make the following specic contributions:

1) We provide closed-form expressions for the relay receiver decision rule with different types of CSI. This

is in contrast with [10], which resorted to numerical

methods to solve the decision rule (see the comment

below equation (8) in [10]).

2) We consider the use of a turbo code for additional

data protection. This requires that the relay receiver be

formulated so that it produces bitwise LLRs, which may

be passed through a standard turbo decoder.

3) We provide results for Rayleigh block-fading channels.

The results in [10] were only for a phase-fading channel.

4) We propose a channel estimator that is capable of

determining the fading amplitudes of the channels from

the two terminals to the relay. The estimator does not

require pilot symbols.

The remainder of this paper is organized as follows.

Section II presents the system model used throughout the

paper. Section III derives the relay receiver, while Section

IV discusses channel-estimation issues. Section V provides

simulation results, and Section VI concludes the paper.

II. SYSTEM MODEL

The discrete-time system model shown in Fig. 2 gives

an overview of the processing at all three nodes. Terminal

NNii, ii ∈ {1, 2}, generates a length-KK information sequence,

uii = [uuiii1, ..., uuiiiii]. The two terminals channel-encode

and modulate their information sequences using the function

ΓSS(⋅), which is common to both nodes. A rate-rr1 turbo code

is used, and the resulting length LLSS = KK 1 turbo codeword

generated by NNii is denoted by bii = [bbiii1, ...bbiiiiiSS ] (not shown

in the diagram). The signal transmitted by node NNii during

signaling interval kk ss ≤ tt ≤ (kk + 1) ss is

ii(tt) = √2Eii

ss

cos [2ππ

(

ffccii +

bbiiiii

ss

)

(tt − kk ss)

]

(1)

where Eii is the transmit energy, ffccii is the carrier frequency

of node NNii (in practice, the carrier frequencies of the two

nodes are not necessarily the same), and ss is the symbol period. Note that (1) is continuous-phase frequency-shift keying

(CPFSK) with a unity modulation index, which is orthogonal

under noncoherent demodulation and has a continuous phase

transition from one symbol to the next [17]. The orthogonallymodulated signal ii(tt) may be represented in discrete time by

the 2 × LLSS matrix Xii = [xiii1, ..., xiiiiiSS ] with kkttt column

xiiiii =

{

[10]TT if bbiiiii = 0

[01]TT if bbiiiii = 1. (2)

For the DNC system, the signals are transmitted simultaneously by the two source nodes over a MAC channel. The

relay receives the noisy electromagnetic sum of interfered and
224

VALENTI et al.: NONCOHERENT PHYSICAL-LAYER NETWORK CODING WITH FSK MODULATION: RELAY RECEIVER DESIGN ISSUES 2597

Relay

Node 2

Node 1

MAC

Channel

Broadcast

Channel

ΓS (·)

ΓS (·)

Γ−1

R (·)

Γ−1

R (·)

Γ−1

S (·) ΓR(·)

u1

u2 u ̃1

u ̃2

u ̃

uˆ

Y uX

X1 Z1

X2 Z2

Fig. 2. Discrete-time system model.

faded signals, Y, and applies the demodulation and channeldecoding function Γ−1

SS (⋅). The demodulation operation yields

a soft estimate of the network-and-channel-coded message

b = b1 ⊕ b2 (not shown), while the channel-decoding operation yields a hard-decision on the network-coded message

u = u1 ⊕u2. With the LNC system, the two sources transmit

during orthogonal time slots. The received versions of X1 and

X2 are demodulated independently to provide soft estimates

of b1 and b2. These soft estimates are combined and turbo

decoded to yield a hard estimate of u. The key distinction

between DNC and LNC is that with the DNC system, the

estimate of b is obtained directly from Y, while with LNC it

is found by independently demodulating the two source signals

and then combining them.

During the broadcast phase, the relay encodes and modulates u using the function ΓRR(⋅), which may be different

than the function ΓSS(⋅) used by the sources. The channel

code applied by the relay is a rate-rr2 turbo code, yielding

a length LLRR = KKK 2 turbo codeword. The code rates rr1

and rr2 used by the sources and relays, respectively, do not

need to be the same. In the simulation results, we contemplate

using a stronger code for the MAC phase than the broadcast

phase, i.e. rr1 < rr2. The relay broadcasts its encoded and

modulated signal, which may be represented in discrete-time

by the 2×LLRR matrix X. The signal traverses two independent

fading channels, and the end nodes receive independently

faded versions of X: Z1 at NN1 and Z2 at NN2. The end

nodes demodulate and decode their received signals using

the function Γ−1

RR (⋅), and form estimates of u. Let uˆ denote

the estimate at NN1 and u ̃ denote the estimate at NN2. Next,

estimates of the transmitted information messages are formed,

u ̃2 = uˆ ⊕ u1 at NN1 and u ̃1 = u ̃ ⊕ u2 at NN2. Since the

links in the broadcast phase are conventional point-to-point

links, specic details of the receiver formulation will not be

presented here. A detailed exposition of receiver design for

turbo-coded CPFSK systems in block fading channels can be

found in [18].

All of the channels in the system are modeled as blockfading channels. A block is dened as a set of NN symbols

that all experience the same fading amplitude. The duration

of each block corresponds roughly to the channel coherence

time. Ideally both sources transmit with the same carrier

frequency ffcc1 = ffcc2 . However, due to instabilities in each

source node’s oscillator and different Doppler shifts due to

independent motion, it is not feasible to assume that these

two frequencies are the same at the relay receiver. At best,

the relay receiver could lock onto one of the two frequencies,

in which case the received phase of the other signal would

drift from one symbol to the next. To model this behavior,

we let the phase shift within a block vary independently from

symbol to symbol.

The signal matrix Xii transmitted by node NNii may be

partitioned into NNbb = LLSS/NN blocks according to

Xii =

[

X(1)

ii ... X(NNbb)ii

]

(3)

where each block X(l)

ii , 1 ≤ l ≤ NNbb, is a 2 × NN matrix, and

NNbb is assumed to be an integer. The channel associated with

block X(l)

ii is represented by the NN × NN diagonal matrix

H(l)

ii = αα(l)ii × diag(exp{jjjj(l)iii1 }, ..., exp{jjjj(l)

iii }) (4)

where αα(l)

ii is a real-valued fading amplitude and (l)

iiiii is the

phase shift of the kkttt symbol. The { (l)

iiiii} are independent and

identically distributed over the interval [0, 2ππ). The {αα(l)

ii } are

normalized so that Eii represents the average energy of terminal

NNii received by the relay. The lttt block at the sampled output

of the relay receiver’s matched-lters is then

Y(l) = X(l)

1 H(l)1 + X(l)2 H(l)2 + N(l) (5)

where N(l) is a 2 × NN noise matrix whose elements are i.i.d.

circularly-symmetric complex Gaussian random variables with

zero mean and variance NN0.

III. RELAY RECEIVER

At the relay, each block Y(l) of the channel observation

matrix Y is passed to a channel estimator, which computes

estimates of the αα(l)

1 and αα(l)2 . A full description of the estimator is given in Section IV. The fading-amplitude estimates

and channel observations are used to obtain soft estimates of

the network-and-channel-coded sequence b. The demodulator

operates on a symbol-by-symbol basis, and therefore we

may focus on a single signaling interval by dropping the

dependence on the symbol interval kk and the block index l.

Let bb1 and bb2 be the turbo-coded bits transmitted by terminals

NN1 and NN2, and let bb = bb1⊕bb2 be the corresponding networkcoded bit. The relay demodulator computes the LLR

Λ(bb) = log PP(bb = 1∣y)

PP(bb = 0∣y) = log PP(bb1 ⊕ bb2 = 1∣y)PP(bb1 ⊕ bb2 = 0∣y) (6)

where y is the corresponding column of Y. The event {bb1 ⊕

bb2 = 1} is equivalent to the union of the events {bb1 = 0, bb2 =

1} and {bb1 = 1, bb2 = 0}. Similarly, the event {bb1 ⊕ bb2 = 0}
225

2598 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 9, SEPTEMBER 2011

is equivalent to the union of the events {bb1 = 0, bb2 = 0} and

{bb1 = 1, bb2 = 1}. It follows that

Λ(bb) = log PP ({bb1 = 0, bb2 = 1}∪{bb1 = 1, bb2 = 0}∣y)

PP ({bb1 = 0, bb2 = 0}∪{bb1 = 1, bb2 = 1}∣y)

= log PP ({bb1 = 0, bb2 = 1}∣y) + PP ({bb1 = 1, bb2 = 0}∣y)

PP ({bb1 = 0, bb2 = 0}∣y) + PP ({bb1 = 1, bb2 = 1}∣y)

(7)

where the second line follows from the rst because the events

are mutually exclusive.

A. LNC Receiver

In the LNC system, the LLR’s of bb1 and bb2 are rst

computed independently during the orthogonal time slots and

are then combined according to the rules of LLR arithmetic.

The LLR of the signal sent from node NNii to the relay is

Λ(bbii) = log PP(bbii = 1∣y)

PP(bbii = 0∣y) (8)

where y is the signal received during the time slot that node

NNii transmits. When the fading amplitudes ααii, ii = 1, 2, are

known, but the phases θθii, ii = 1, 2, are not known, then (8) is

found using [19]

Λ(bbii) = log II0

(2ααii∣yy2∣

NN0

)

− log II0

(2ααii∣yy1∣

NN0

)

(9)

where II0(⋅) is the zeroth-order Bessel function of the rst

kind and yy1 and yy2 are the components of y. If the fading

amplitudes are not known, but have Rayleigh distributions,

then (8) is found using [19]

Λ(bbii) = (Eii/NN0)

2

1 + Eii/NN0

{

∣yy2∣

2 − ∣yy1∣2}

. (10)

Once the individual LLR’s from each end node are found

using (9) or (10), the LLR of the LNC system’s network

codeword can then be found from (7) and the independence

of bb1 and bb2 when y is given:

Λ(bb) = log eeΛ(bb1) + eeΛ(bb2)

1 + eeΛ(bb1)+Λ(bb2)

= max ∗ [Λ(bb1),Λ(bb2)] − max ∗ [0,Λ(bb1) + Λ(bb2)] (11)

where max ∗[xx ] = log(eexx + eeyy).

B. PNC Receiver

In the PNC system, it is not sensible to compute Λ(bb1) and

Λ(bb2) separately. Instead, use (7) and assume that the four

events are equally likely along with Bayes’ rule to obtain

Λ(bb)=log [pp (y∣{bb1 = 0, bb2 = 1}) + pp (y∣{bb1 = 1, bb2 = 0})]

− log [pp (y∣{bb1 = 0, bb2 = 0}) + pp (y∣{bb1 = 1, bb2 = 1})] .

(12)

The computation of each pp (y∣{bb1, bb2}) by the PNC relay

receiver given various levels of channel state information is

the subject of the remainder of this section.

1) Coherent PNC Receiver: When the fading amplitudes

and phases are known, pp (y∣{bb1, bb2}) is conditionally Gaussian. The mean is a two-dimensional complex vector whose

value depends on the values of {bb1, bb2} and the complex fading

coefcients {h1, h2}, which are the corresponding entries of

the H matrix. Let m[bb1, bb2] be the mean of y for the given

values of bb1 and bb2. When bb1 ∕= bb2, the two terminals transmit

different frequencies and

m[0, 1] = [ h1 h2

]TT

m[1, 0] = [ h2 h1

]TT

. (13)

When bb1 = bb2, the two terminals transmit the same frequency

and

m[0, 0] = [ (h1 + h2) 0 ]TT

m[1, 1] = [ 0 (h1 + h2) ]TT. (14)

Since there is a one-to-one correspondence between the event

{bb1, bb2} and the mean vector m[bb1, bb2], it is equivalent to write

pp (y∣{bb1, bb2}) as pp (y∣m[bb1, bb2]), where

pp (y∣m[bb1, bb2]) = ( 1

ππ 0

)2

exp {− 1

NN0

∥y − m[bb1, bb2]∥2

}

.

(15)

The coherent receiver computes each of the pp (y∣{bb1, bb2})

required by (12) by substituting the corresponding m[bb1, bb2]

dened by (13) and (14) into (15).

2) Noncoherent PNC Receiver with CSI: Suppose that

the receiver does not know the phases of the elements of

the complex-valued m[bb1, bb2] vectors, but does know the

magnitudes of the elements. The knowledge of the magnitudes

constitutes a type of channel-state information (CSI). Dene

μμ[bb1, bb2] to be the two-dimensional real vector whose elements

are the magnitudes of the elements of the complex vector

m[bb1, bb2]. When bb1 ∕= bb2, both frequencies are used, and

μμ[0, 1] = [ ∣h1∣ ∣h2∣

]TT = [ αα1 αα2

]TT

μμ[1, 0] = [ ∣h2∣ ∣h1∣

]TT = [ αα2 αα1

]TT

. (16)

When bb1 = bb2, only one frequency is used, and

μμ[0, 0] = [ ∣h1 + h2∣ 0 ]TT = [ αα 0 ]TT

μμ[1, 1] = [ 0 ∣h1 + h2∣

]TT = [ 0 αα ]TT (17)

where αα = ∣h1 + h2∣ = √αα2

1 + αα22 + 2αα1αα2 cos(θθ2 − θθ1).

The pdf of y conditioned on μμ[bb1, bb2] may be found by

marginalizing over the unknown phases

pp (y∣μμ[bb1, bb2]) = ∫ 2ππ

0

∫ 2ππ

0

pp(φφ1, φφ2)pp (y∣m[bb1, bb2]) dd 1dd 2.

(18)

where φφ1 and φφ2 are the phases of the rst and second

elements of m[bb1, bb2], respectively.

Assume that the ααii are Rayleigh distributed so that the

hii are circularly-symmetric zero-mean complex Gaussian∗.

∗The receiver derived in this subsection is valid even for non-Rayleigh fading, provided that the received phases over the two channels are independent

and uniform over (0, 2ππ).
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VALENTI et al.: NONCOHERENT PHYSICAL-LAYER NETWORK CODING WITH FSK MODULATION: RELAY RECEIVER DESIGN ISSUES 2599

When bb1 ∕ bb2 each element of m[bb1, bb2] is a circularlysymmetric zero-mean complex Gaussian and therefore has

uniform phase. On the other hand, when bb1 = bb2, one element

is h1+h2, which is the sum of two circularly-symmetric zeromean complex Gaussians, while the other element is zero.

Since the sum of two circularly-symmetric complex Gaussians

is also a circularly-symmetric complex Gaussian, it follows

that h1 + h2 is a zero mean circularly-symmetric complex

Gaussian and therefore its phase is uniform. Since the other

element is zero, its phase is irrelevant and may be set to any

arbitrary distribution, which is most conveniently chosen to

be uniform. Thus, it follows that φφ1 and φφ2 are i.i.d. uniform.

Therefore, the pdf conditioned on the magnitudes is

pp (y∣μμ[bb1, bb2])= 1

ππππ0

∫ 2ππ

0

exp {−∣yy1 − μμ1[bb1, bb2]eejjjj1 ∣

2

0

}

dd 1

×

1

ππππ0

∫ 2ππ

0

exp {−∣yy2 − μμ2[bb1, bb2]eejjjj2 ∣

2

0

}

dd 2 (19)

where μμkk[bb1, bb2] is the kkttt element of μμ[bb1, bb2] and

1

2ππ

∫ 2ππ

0

exp {−∣yykk − μμkk[bb1, bb2]eejjjjkk ∣

2

0

}

dd kk

= exp {−∣yykk∣

2 + (μμkk[bb1, bb2])2

0

}

II0

(2∣yykk∣μμkk[bb1, bb2]

0

)

.

(20)

Substituting (20) into (19),

pp (y∣μμ[bb1, bb2]) = ββ

∏

2

kk=1

exp {−(μμkk[bb1, bb2])2

0

}

×II0

(2∣yykk∣μμkk[bb1, bb2]

0

)

(21)

where

ββ =

( 2

0

)2

exp {−

(∣yy1∣

2 + ∣yy2∣2

0

)} (22)

which is common to all four {bb1, bb2} and will therefore cancel

in the LLR (12).

For each event {bb1, bb2}, substitute the pp (y∣μμ[bb1, bb2]) given

in (21) with the μμ[bb1, bb2] given by (16) and (17) as the

corresponding pp (y∣{bb1, bb2}) in (12). This results in

Λ(bb) = log [ee−αα2

1/NN0 II0

(2αα1∣yy1∣

0

)

ee−αα2

2/NN0 II0

(2αα2∣yy2∣

0

)

+ee−αα2

2/NN0 II0

(2αα2∣yy1∣

0

)

ee−αα2

1/NN0 II0

(2αα1∣yy2∣

0

)]

− log [ee−αα2/NN0 II0

(2αα∣yy1∣

0

)

+ ee−αα2/NN0 II0

(2αα∣yy2∣

0

)] .

(23)

As discussed in Section IV, it is possible to accurately estimate αα1 and αα2 in the considered block fading environment,

provided the blocks are sufciently long. However, it is not

generally feasible to precisely estimate αα because the phases

θθ1 and θθ2 are varying on a symbol-by-symbol basis. Since

EE[cos(θθ2 − θθ1)] = 0, a reasonable approximation when an

estimate of αα is not available is to use

αα ≈

√

αα2

1 + αα22. (24)

3) Noncoherent PNC Receiver without CSI: Suppose that

besides not knowing the phases θθ1, θθ2, the relay receiver does

not know the magnitude vector μμ[bb1, bb2]. Then, the relay must

operate without any channel state information except for the

average energies E1, E2 and the noise variance 0. When the

magnitudes μμ[bb1, bb2] are not known, then the conditional pdf

is found by marginalizing (21) over the unknown magnitudes

pp (y∣{bb1, bb2}) = ∫ ∞

0

∫ ∞

0

pp(μμ1, μμ2)pp (y∣μμ[bb1, bb2]) dd 1dd 2.

(25)

where μμ1 and μμ2 are the magnitudes of the rst and second

elements of μμ[bb1, bb2], respectively.

According to (16), when bb1 ∕ bb2, one of the μμkk = αα1 while

the other μμkk = αα2. Since αα1 and αα2 are independent and each

ααii is Rayleigh with energy Eii, it follows that the joint pdf of

μμ1 and μμ2 when (bb1, bb2) = (0, 1) is

pp(μμ1, μμ2) = (2μμ1

E1

exp {−μμ1

E1

}) (2μμ2

E2

exp {−μμ2

E2

})

(26)

for μμ1, μμ2 ≥ 0, and when (bb1, bb2) = (1, 0) it is

pp(μμ1, μμ2) = (2μμ1

E2

exp {−μμ1

E2

}) (2μμ2

E1

exp {−μμ2

E1

})

(27)

for μμ1, μμ2 ≥ 0. Substituting (26) and (21) into (25) yields

pp (y∣{bb1 = 0, bb2 = 1}) = ∣yy1∣

2

NN2

0

E1 + 0

+ ∣yy2∣

2

NN2

0

E2 + 0

+ log [( 1

E1E2

) ( 1

E1

+

1

0

) ( 1

E2

+

1

0

)]−1

. (28)

Similarly, substituting (27) and (21) into (25) yields

pp (y∣{bb1 = 1, bb2 = 0}) = ∣yy1∣

2

NN2

0

E2 + 0

+ ∣yy2∣

2

NN2

0

E1 + 0

+ log [( 1

E1E2

) ( 1

E1

+

1

0

) ( 1

E2

+

1

0

)]−1

. (29)

As indicated by (17), when bb1 = bb2, one of the μμkk = αα

while the other μμkk = 0. As discussed below (18), in a

Rayleigh-fading environment, h1 and h2 are independent,

complex-valued, circularly-symmetric Gaussian variables, and

therefore h = h1 + h2 is also a complex-valued, circularlysymmetric Gaussian variable. It follows that αα = ∣h∣ is

Rayleigh with energy E1 + E2, and the pdf of the nonzero

μμkk is

pp(μμkk) = 2μμkk

E1 + E2

exp {− μμkk

E1 + E2

}

, μμkk ≥ 0. (30)

For the μμkk = 0, its pdf may be represented by an impulse at

the origin, i.e. pp(μμkk) = δδ(μμkk). Substituting these pdfs with

the appropriate μμ[bb1, bb2] into (25) yields

pp (y∣{bb1, bb2}) = log [( 1

E1 + E2

) ( 1

E1 + E2

+

1

0

)]−1

+ ∣yyii∣

2

NN2

0

E1+E2 + 0

(31)
227

2600 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 9, SEPTEMBER 2011

where ii = 1 when (bb1, bb2) = (0, 0) and ii = 2 when (bb1, bb2) =

(1, 1).

Substituting (28) and (29) for the two bb1 ∕ bb2 and (31) for

the two bb1 = bb2 into (12) yields

Λ(bb) = log [

ξξ1ξξ2

ξξξξ0

]

+ log [exp {−∣yy1∣

2

ξξ − ∣yy2∣

2

0

}

+ exp {−∣yy1∣

2

0

− ∣yy2∣

2

ξξ

}]

− log [exp {−∣yy1∣

2

ξξ1

− ∣yy2∣

2

ξξ2

}

+ exp {−∣yy1∣

2

ξξ2

− ∣yy2∣

2

ξξ1

}]

(32)

where ξξ1 = E1 + 0, ξξ2 = E2 + 0, and ξξ = E1 + E2 + 0.

IV. CHANNEL ESTIMATOR

The goal of the channel estimator is to estimate the values

of the fading amplitudes αα1 and αα2 for a particular fading

block. Let the fading amplitudes of a block be represented

by the pair {AA AA}, where AA ≥ . Thus, AA = max{αα1, αα2}

and = min{αα1, αα2}. Note that in (23), exchanging αα1 and

αα2 does not change the nal expression. Therefore (23) is

commutative in αα1 and αα2, and may be written as

Λ(bb) = max ∗

[

FF

(2AA∣yy1∣

0

)

+ FF

(2 ∣yy2∣

0

)

,

FF

(2 ∣yy1∣

0

)

+ FF

(2AA∣yy2∣

0

)]

− max ∗

[

FF

(

2

√

AA2 + 2∣yy1∣

0

)

, FF (

2

√

AA2 + 2∣yy2∣

0

)]

(33)

where the approximation αα ≈ √αα2

1 + αα22 has been used and

FF(xx) = log[II0(xx)], which may be efciently and accurately

computed through the following piecewise polynomial t:

FF(xx) = log[II0(xx)] ≈ ⎧

⎨

⎩

0.22594xx2 + 0.012495xx − 0.0011272 0 < xx ≤ 1

0.12454xx2 + 0.21758xx − 0.10782 1 < xx ≤ 2

0.028787xx2 + 0.63126xx − 0.56413 2 < xx ≤ 5

0.003012xx2 + 0.88523xx − 1.2115 5 < xx ≤ 15

0.00053203xx2 + 0.95304xx − 1.6829 15 < xx ≤ 30

0.00013134xx2 + 0.97674xx − 2.0388 30 < xx ≤ 60

0.9943xx − 2.6446 60 < xx ≤ 120

0.99722xx − 3.0039 120 < xx ≤ 500

0.99916xx − 3.6114 xx x 500.

(34)

A. Fading Amplitude Estimator

To estimate AA and , rst add the two elements of each yii

to obtain

rrii = yyiii1 + yyiii2 = hiii1 + hiii2 + nniii1 + nniii2 � �� � ννii

(35)

where ννii is circularly-symmetric complex Gaussian noise with

variance 2 0, and hiiiii is the channel coefcient between

terminal NN , kk = {1, 2}, and the relay during the ii

ttt signaling

interval. The signal rrii is the noisy sum of two complex fading

coefcients, and therefore the fading-amplitude estimation

algorithm proposed by Hamkins in [20] may be used. To

determine the values of AA and , a system of two equations

with two unknowns is required. The rst equation, found by

taking the expected value of ∣rrii∣

2 under the assumption that

the fading amplitudes are xed for the block in question, is

EE [∣rrii∣

2] = EE [

αα2

1 + αα22 + 2αα1αα2 cos(θθiii2 − θθiii1)

]

= EE [αα2

1 + αα22

]

= αα2

1 + αα22 = AA2 + 2. (36)

The second equation is found by conditioning on the event

{∣rr∣

2 > AA2+ 2}, which is equivalent to {cos(θθiii2−θθiii1) > 0}

and has expected value [20]

EE

[

∣rr∣

2

�

�

�∣rr∣

2 > AA2 + 2

]

= AA2 + 2 +

4AAAA

ππ . (37)

Solving (36) and (37) for AA and yields

AA = 1

2

(√

XX + ππ

2

(YY − XX) + √

XX + ππ

2

(XX − YY )

)

= 1

2

(√

XX + ππ

2

(YY − XX) −

√

XX + ππ

2 (XX − YY )

)

(38)

where XX = EE [∣rr∣

2]

and YY = EE

[

∣rr∣

2

�

�

�∣rr∣

2 > AA2 + 2

]

.

Since the expected values required for (38) are not known,

they may be estimated by using the corresponding statistical

averages,

XXˆ = 1 ∑

NN

ii=1

∣rrii∣

2

YYˆ = 2 ∑

ii:∣rrii∣2>XXˆ

∣rrii∣

2 (39)

where is the size of the fading block and the factor 2/

used to compute YYˆ assumes that ∣rrii∣

2 > XXˆ for approximately 2 symbols. If this assumption is not true, then

the multiplication by 2/ can be replaced with a division

by the number of samples that satisfy ∣rrii∣

2 > XXˆ. As an

alternative to summing over the ∣rrii∣

2 > XXˆ , Hamkins proposes

summing over those ∣rrii∣

2 greater than the median value of

{∣rr1∣

2, ..., ∣rrNN ∣2} [20].

The estimator works by computing estimates XXˆ and YYˆ using

(39) and the {rr1, ..., rrNN } for the block. These estimates are

used in place of XX and YY in (38), which yields estimates AAˆ

and ˆ of AA and . These estimates are then used in place of

AA and in (33).

B. Transmission-Case Detection

According to (35), the two elements of yii are always added

together. When bb1 = bb2, only one tone is used, and the noise

can be reduced if the receiver processes only the tone used and

ignores the other tone. This requires that the receiver be able

to detect whether the rst tone, the second tone, or both tones

were used, which may be implemented using a variation of the

“no-CSI” receiver described in subsection III-B3. In [16], we

contemplate an estimator that uses such a transmission-case

detector. However, we found that the performances with and

without the transmission-case detector were virtually identical

and do not consider it further in this paper. At best, proper use

of the transmission-case detector reduces the noise variance
228

VALENTI et al.: NONCOHERENT PHYSICAL-LAYER NETWORK CODING WITH FSK MODULATION: RELAY RECEIVER DESIGN ISSUES 2601

from 2NN0 to NN0 during the symbol intervals that both nodes

transmit the same tone. As will be seen in the numerical

results, the estimator is resilient enough against noise that this

reduction in noise variance is not meaningful and does not

justify the additional complexity.

C. Amplitude Estimation for Single-Transmitter Links

During the broadcast phase, there is only a single transmission, and the dual-amplitude estimator described in subsection IV-A is not necessary. Similarly, the estimator is not

needed by the LNC system during the MAC phase since the

two transmissions are over orthogonal channels. To estimate

the fading amplitudes for the links involving only a single

transmitter and receiver, the simple averaging technique given

by (29) in [21] is used, which is described as follows. Consider

the ii

ttt signaling interval during the lttt fading block. Given

transmission of tone kk, in the absence of noise, the kkttt

matched-lter output at the receiver is yykkkkk = ααααjjjjii , and has

magnitude ∣yykkkkk∣ = αα. All other matched-lter outputs in the

ii

ttt signaling interval are 0. An estimate could be formed by

taking the maximum ∣yykkkkk∣ over any column of Yl. In the

presence of noise, an estimate of αα can be formed by averaging

across all columns of the fading block

ααˆ = 1

NN

∑

NN

=1

max

kk ∣yykkkkk∣. (40)

V. SIMULATION STUDY

This section presents simulated performance results for the

relay receiver described in Section III. The simulated link

model is as described in Section II, with specic simulation

parameters given in the following subsections. The goal of

the simulations is to compare the performance of comparable

DNC and LNC systems and to assess the robustness of the

channel estimator proposed in IV. Because the relay-broadcast

phase of the DNC and LNC systems operate in exactly the

same manner and have the same performance, we only focus

on the performance of the MAC phase.

A. Uncoded Performance with Perfect Channel Estimates

We initially consider a system that does not use an outer

error-correcting code, and thus b = u , ii = 1, 2. We compare

the performance of the LNC and DNC systems. With the LNC

system, the two nodes transmit their messages in orthogonal

time slots and the relay receiver rst generates the individual

LLR’s during each time slot using either (9) or (10), and

then the two LLR’s are combined using (11). When there

is no outer error-correcting code, performance using (9) is

approximately the same as that using (10). A bit error is

declared at the relay whenever a hard decision using (11)

results in an erroneous decision on the corresponding bit of the

network codeword b. Such an error will usually occur if one

of the two bits bb1, bb2 is received incorrectly, and therefore the

error rate of the LNC system is approximately PPbb ≈ 2pp(1−pp)

where pp is the bit error rate of noncoherent binary FSK

modulation [17].

With the DNC system, the two nodes transmit simultaneously, and the relay receiver computes the LLR using (23)

0 10 20 30 40 50 10−4

10−3

10−2

10−1

100

Eb/N0(dB)

BER

DNC: No CSI

DNC: known α1, α2

DNC: known α1, α2, α

LNC

Fig. 3. Bit error rate at the relay in Rayleigh fading when DNC and LNC is

used and E2 = E1. Depending on the amount of channel state information that

is available, the PNC system will use one of three different relay receivers.

when the magnitudes μμ[bb1, bb2] are known or (32) when they

are not. A hard decision is made on the LLR and a bit

error is declared if the estimate of the corresponding network

codeword bit bb is incorrect. We assume that the channel

estimates are perfect, and since there is no error-correction

coding, the size of the fading block is irrelevant provided that

the channel coherence time is not exceeded.

Initially, we set the average received energy to be the same

over both channels, i.e. E2 = E1 = Ess = Ebb. Fig. 3 shows the

performance of the LNC and DNC systems in Rayleigh fading

with equal energy signals. As anticipated, the LNC system

offers the best performance, which is approximately 3 dB

worse than a standard binary CPFSK system with noncoherent

detection (the loss relative to conventional CPFSK is due to the

fact that both bits must usually be received correctly). Three

curves for the DNC system are shown in Fig. 3, corresponding

to receivers that exploit different amounts of available channel

state information. The best performance is achieved using a

receiver implemented with (23), which requires knowledge

of αα1, αα2, and αα. The performance of the DNC system

implemented with (23) is only about 0.25 dB worse than that

of the LNC system. The worst performance is achieved using

a receiver implemented using (32), which does not require

knowledge of the fading amplitudes. The loss due to using

(32) instead of (23) is about 10 dB, indicating that estimating

the fading amplitudes at the relay is necessary.

While it may be feasible to estimate αα1 and αα2, estimating

αα may prove to be more difcult because it will depend on

not only the individual fading amplitudes, but also on the

phase difference between the two channels. Since the phase

difference might change more quickly than the individual

amplitudes, it might not be practical to estimate αα. If that

is the case, then the approximation given by (24) can be used

in place of the actual value of αα. The performance using this

technique is also shown in Fig. 3 and shows a loss of about 3

dB with respect to the known-μμ[bb1, bb2] system, which requires

knowledge of αα.

The performance of DNC is sensitive to the balance of
229

2602 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 9, SEPTEMBER 2011

20 25 30 35 40 45 50 55 10−4

10−3

10−2

10−1

Eb/N0(dB)

BER

DNC: No CSI

DNC: known α1, α2

DNC: known α1, α2, α

Fig. 4. Bit error rate at the relay in Rayleigh fading of DNC with three

different receivers and either E2 = E1 (solid line) or E2 = 4E1 (dashed

line).

power received over the two channels. Performance is best

when E1 = E2. In order to evaluate how robust the DNC relay

receivers are to an imbalance of power, the simulations were

repeated with E2 = 4E1, while keeping Ebb = Ess = (E1+E2)/2.

These results are shown in Fig. 4 for the three receiver

formulations that were considered in the previous gure.

When the power is imbalanced in this way, there is a loss

of about 2 dB. However, the loss is the same for all three

receiver implementations, suggesting that they are robust to

an imbalance of power.

B. Uncoded Performance with Channel Estimation

We now consider the inuence of channel estimation, but

still assume that the system does not use error-correction

coding. In the simulations, the information frames generated

at the end nodes contain KK = 2048 bits per frame. The fading

blocks are length NN = {8, 32, 128} symbols per block. The

DNC relay implements (32) and then makes a hard decision

on each information bit.

The bit error-rate performance of the uncoded system is

shown in Fig. 5. The performance is shown with the estimator

using the three block sizes NN = {8, 32, 128} as well as for

the case of perfect estimates of αα1 and αα2. A narrow range

of error rates is shown to better highlight the differences in

performance. In general, smaller fading blocks lead to a less

accurate estimation of the fading amplitudes, as the number

of samples available for estimation decreases. Moving from

block size NN = 128 to 32 worsens performance by roughly

0.25 dB, and from NN = 32 to 8 by 0.75 dB.

C. Performance with an Outer Turbo Code

Now consider a system that uses an outer turbo code.

The terminals each encode length KK = 1229 information

sequences into length LL = 2048 codewords, using a rate

rr1 ≈ 0.6 UMTS turbo code [22]. The relay performs turbo

decoding using the codeword LLR’s computed by (32). The

28 29 30 31 32 33 34 35 10−3

10−2

Eb/N0(dB)

BER

N=128

N=32

N=8

Perfect Estimation

Fig. 5. Inuence of fading-block length NN on uncoded DNC error-rate

performance at the relay. In addition to curves for three values of NN, a curve

is shown indicating the performance with perfect fading-amplitude knowledge.

17 18 19 20 21 22 23 24 10−4

10−3

10−2

10−1

100

Eb/N0(dB)

BER

N=128

N=64

N=32

N=16

N=8

Fig. 6. Inuence of fading-block length NN on turbo-coded DNC errorrate performance at the relay. Two curves are shown for each value of

NN = {8, 16, 32, 64, 128}. Solid curves denote perfect fading-amplitude

knowledge. Dashed curves denote estimated fading amplitudes.

fading-block lengths simulated are NN = {8, 16, 32, 64, 128}

symbols per block.

The error performance of the coded system is shown in

Fig. 6, both with perfect channel estimates and with estimated

fading amplitudes. A good tradeoff between diversity and

estimation accuracy is achieved for block sizes NN = 16

and NN = 32, which exhibit the best performance of all

systems that must estimate the fading amplitudes. For NN N 16

performance degrades due to the lack of enough observations

per block for accurate channel estimates, while for NN N 32

performance degrades due to the reduction in time diversity.

Fig. 7 shows the SNR required to reach an error rate of

10−4 at the relay as a function of the block length NN. In each

case, information is coded with the same (2048, 1229) turbo

code used for Fig. 6. Curves for three systems are shown: The

noncoherent receiver with known {αα1, αα2}, the noncoherent
230

VALENTI et al.: NONCOHERENT PHYSICAL-LAYER NETWORK CODING WITH FSK MODULATION: RELAY RECEIVER DESIGN ISSUES 2603

0 20 40 60 80 100 120 140 22

23

24

25

26

27

28

29

30

31

32

33

Eb/N0(dB)

Symbols Per Fading Block

α1, α2 known

α1, α2 estimated

No CSI

Fig. 7. Signal-to-noise ratio required to reach a bit error rate of 10−4 at the

relay as a function of fading-block length. The performance of three systems

is shown: The noncoherent receiver with known {αα1, αα2}, the noncoherent

receiver with estimated {αα1, αα2}, and the noncoherent receiver that does not

use CSI. All systems use a Turbo code with rate 1229/2048.

12 14 16 18 20 22 10−4

10−3

10−2

10−1

100

Eb/N0(dB)

BER

N=128

N=64

N=32

N=16

N=8

Fig. 8. Comparison of error-rate performance between the turbo-coded DNC

and LNC systems at the relay. The solid lines denote DNC, while the dashed

lines denote LNC.

receiver with estimated {αα1, αα2}, and the noncoherent receiver

that does not use CSI. When {αα1, αα2} are not estimated,

performance improves with decreasing NN because of the

increased number of blocks per codeword, which increases

the time diversity. However, when {αα1, αα2} are estimated,

the performance gets worse when the block size is smaller

than NN = 16. The loss of time diversity as the block size

increases is a common problem for any system operating over

a slow-fading channel, and the system proposed in this paper

is no exception. The performance gap between the knownCSI and no-CSI receiver formulations widens with increasing

block length.

An error-rate performance comparison between DNC and

LNC is shown in Fig. 8. Both systems use the same

(2048, 1229) turbo code. The LNC system outperforms the

18 20 22 24 26 28 30 10−4

10−3

10−2

10−1

100

Eb/N0(dB)

BER

LNC, rate=4500/5056

DNC, rate=4500/5056

DNC, rate=4500/6400

Fig. 9. Comparison of the performance of turbo-coded DNC and LNC at

the relay with block size NN = 32. For the DNC system, two code rates are

shown, with the lower rate code offering comparable performance to the LNC

system.

DNC system by margins ranging between 4 and 6 dB.

While the LNC system is more energy efcient than the

DNC system when the same-rate turbo code is used, the

throughput of the LNC system is worse than that of the DNC

system because the two terminals must transmit in orthogonal

time slots. The loss in energy efciency from using DNC versus LNC can be recovered by having the source terminals use

a lower-rate turbo code. Consider the performance comparison

shown in Fig. 9 for block size NN = 32. At EEbb/NN0 ≈ 24 dB,

DNC using a rate rr1 = 4500/6400 code matches the errorrate performance of LNC using a rate rr1 = 4500/5056 code.

Because the two terminals transmit at the same time, the endto-end throughput of DNC is higher than that of LNC, even

though the DNC terminals transmit to the relay with a lowerrate channel code.

To illustrate the throughput improvement of DNC over

LNC, consider the following transmission schedule for the

two systems. Assume the source terminals use rate rr1 =

4500/6400 in DNC, and rr1 = 4500/5056 in LNC. Assume

operation at EEbb/NN0 = 24 dB, yielding approximately equal

relay error-rate performance. Further, assume that both systems use code rate rr2 = 4500/5056 for relay broadcast,

yielding approximately equal end-to-end performance. DNC

requires 6400 channel uses for transmission to the relay versus

2 × 5056 = 10112 for LNC. Both systems require 5056

channel uses for relay broadcast. The throughput for DNC

is thus TT (DDDDDD) = 9000/(6400 + 5056) = 9000/11, 456 bits

per channel use, and for LNC TT (LL ) = 9000/(3× 5056) =

9000/15, 168 bits per channel use. The percentage throughput

increase of DNC over LNC is thus (TT (DDDDDD)

/TT (LL ) − 1) ×

100 ≈ 32.4%.

VI. CONCLUSION

A throughput-improving technique for relaying in the twoway relay network, digital network coding, is rened for practical operation. The system operates noncoherently, providing

advantages over coherent operation: there are no requirements
231 2604 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 9, SEPTEMBER 2011

for perfect power control, phase synchronism, or estimates of

carrier-phase offset.

A computationally simple technique for estimating fading

amplitudes at the relay is implemented. Error-rate performance

in the noncoherent Rayleigh block-fading channel at several

block sizes is presented. The system is simulated with and

without an outer error-correcting code. The coded error-rate

performance of the system using estimation differs from that

with ideal estimates by margins between 0.7 − 1.5 dB.

When the same-rate turbo code is used, digital network

coding has a higher throughput but lower energy-efciency

than link-layer network coding . The energy loss of DNC can

be recovered by using a lower-rate turbo code during the MAC

phase. Even when the loss of spectral efciency due to the

lower-rate turbo code is taken into account, the DNC system

is able to achieve a higher throughput than LNC at the same

energy-efciency. In the particular example presented in this

paper, the DNC system is capable of achieving throughputs

that are 32.4% larger than that of the equivalent LNC system,

while operating at the same energy efciency.

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for interference and fading channels," IEEE Trans. Wireless Commun.,

vol. 56, pp. 1343-1351, Aug. 2008.

[22] European Telecommunications Standards Institute, “Universal mobile

telecommunications system (UMTS): multiplexing and channel coding

(FDD)," 3GPP TS 25.212 version 7.4.0, June 2006.

Matthew C. Valenti is a Professor in Lane Department of Computer Science and Electrical Engineering at West Virginia University. He holds

B.S. and Ph.D. degrees in Electrical Engineering

from Virginia Tech and a M.S. in Electrical Engineering from the Johns Hopkins University. From

1992 to 1995 he was an electronics engineer at

the U.S. Naval Research Laboratory. He serves as

an associate editor for IEEE TRANSACTIONS ON

WIRELESS COMMUNICATIONS and as co-chair of

the Technical Program Committee for Globecom2013, and has served as an editor for IEEE TRANSACTIONS ON VEHICULAR

TECHNOLOGY and as track or symposium co-chair for the Fall 2007 VTC,

ICC-2009, Milcom-2010, and ICC-2011. His research interests are in the areas

of communication theory, error correction coding, applied information theory,

wireless networks, simulation, and grid computing. His research is funded by

the NSF and DoD.

Don Torrieri is a research engineer and Fellow

of the US Army Research Laboratory. His primary

research interests are communication systems, adaptive arrays, and signal processing. He received the

Ph.D. degree from the University of Maryland. He is

the author of many articles and several books including Principles of Spread-Spectrum Communication

Systems, 2nd ed. (Springer, 2011). He teaches graduate courses at Johns Hopkins University and has

taught many short courses. In 2004, he received the

Military Communications Conference achievement

award for sustained contributions to the eld.

Terry Ferrett is a research assistant at West Virginia

University, Morgantown, WV completing his Ph.D.

degree in electrical engineering. He received the

B.S. degrees in electrical engineering and computer

engineering in 2005 and the M.S. degree in electrical

engineering in 2008 from West Virginia University.

He is the architect of a cluster computing resource

utilized by electrical engineering students at West

Virginia University to conduct communication theory research. His research interests are network coding, digital receiver design, the information theory of

relay channels, cluster and grid computing, and software project management.

Research@

233

Transmission of Information Between 

Complex Systems: 1/ f Resonance

Gerardo Aquino, Mauro Bologna, Bruce J. West, and Paolo Grigolini

Physical Review E, 83, 051130 (2011)
234 PHYSICAL REVIEW E 83, 051130 (2011)

Transmission of information between complex systems: 1/ f resonance

Gerardo Aquino,1,* Mauro Bologna,2 Bruce J. West,3 and Paolo Grigolini4

1Faculty of Natural Sciences, Imperial College London, SW7 2AZ London, United Kingdom 2Instituto de Alta Investigacion, Universidad de Tarapac  ́ a-Casilla 7-D Arica, Chile  ́ 3Physics Department, Duke University, North Carolina, USA 4Center for Nonlinear Science, University of North Texas, P.O. Box 311427, Denton, Texas 76203-1427, USA

(Received 21 February 2011; published 31 May 2011)

We study the transport of information between two complex systems with similar properties. Both systems

generate non-Poisson renewal fluctuations with a power-law spectrum 1/f 3−μ, the case μ = 2 corresponding to

ideal 1/f noise. We denote by μS and μP the power-law indexes of the system of interest S and the perturbing

system P, respectively. By adopting a generalized fluctuation-dissipation theorem (FDT) we show that the ideal

condition of 1/f noise for both systems corresponds to maximal information transport. We prove that to make

the system S respond when μS < 2 we have to set the condition μP < 2. In the latter case, if μP < μS , the

system S inherits the relaxation properties of the perturbing system. In the case where μP > 2, no response

and no information transmission occurs in the long-time limit. We consider two possible generalizations of

the fluctuation dissipation theorem and show that both lead to maximal information transport in the condition

of 1/f noise.

DOI: 10.1103/PhysRevE.83.051130 PACS number(s): 05.40.−a, 87.18.Tt, 87.19.lm, 89.70.Hj

I. INTRODUCTION

Linear response theory (LRT) [1] is one of the basic ways

of obtaining information from fluctuations in nonequilibrium

statistical physics [2,3] that is currently adopted to address new

phenomena such as glassy systems [2] and granular matter [3].

An even more challenging issue is the application of LRT

to complex processes such as physiological processes and

especially the understanding of brain dynamics.

It is becoming more widely accepted that the brain operates

at criticality [4,5] and that the critical condition of a phase

transition has manifestations that extend beyond the conventional condition of temperature driven systems [6]. Frantsuzov

et al. [7] adopted a model of cooperatively interacting units to

propose a solution to the long-standing mystery of the origin

of the power-law distribution of the blinking times in colloidal

quantum dot fluorescence. On the other hand, this form of

intermittence is characterized by the condition of renewal

aging [8] and by the consequent ergodicity breakdown [9] that

makes it impossible to use conventional LRT. We refer to these

systems as complex systems. It is now very well understood

[10] that the breakdown of the ergodic condition is caused by

the occurrence of crucial events. It is interesting to remark

that the same perspective applies to complex networks, when

the cooperation-induced phase transition produces temporal

complexity and crucial events as illustrated in in the work of

Ref. [11]. These events are renewal, namely, the time interval

between two consecutive events does not have any relation

whatsoever with the earlier or later time intervals between

two consecutive events. Yet, if a Gibbs ensemble is suitably

prepared initially, namely, in all the systems of the ensemble an

event occurs at the time origin, then the rate of event production

turns out to be time dependent rather than constant as in the

ordinary Poisson case. The authors of Refs. [12,13] applied

this theoretical perspective to the liquid crystal dynamics and

*g.aquino@imperial.ac.uk

experimentally realized a true cascade of renewal events. The

time interval between two consecutive crucial events is given

by a waiting times probability density function (PDF) ψ(τ )

with the following asymptotic form

ψ(τ ) = (μ − 1)

T μ−1

(τ + T )

μ−1 , (1)

and the power-law index μ fulfilling the inequality

1 <μ< 3. (2)

It is important to explain the origin of the special form of

Eq. (1). First of all we want to stress that according to a point

of view in the field of complexity, only the asymptotic time

behavior matters, namely, ψ(τ ) ∝ 1/τμ. The adoption of this

widely shared point of view, as we shall see hereby, would

prevent us from establishing a correct accordance with the

experiments on the response of complex systems to external

perturbations. Thus, the choice of Eq. (1) is dictated by the

need for defining a border between the asymptotic time regime

(τ � T ) and the microscopic time regime (τ  T ). Given the

neurophysiology interest of this paper and especially the focus

on brain dynamics, we refer the interested reader to the work of

Ref. [14], where the waiting times PDF of Eq. (1) is obtained by

means of a Fechner transformation [14] from the conventional

Poisson distribution.

The rate of cascade of renewal events tends to a vanishing

value as 1/t 2−μ when μ < 2 and to a constant value as 1/tμ−2

when μ > 2. It is evident that in both cases the time duration

of the out-of-equilibrium condition is infinite, thereby raising

the challenging task of going beyond conventional LRT to

describe the dynamics.

Conventional LRT is given by the following expression:

σ(t) = �ξS(t)� = �

 t

0

dsχ(t,s)ξP (s), (3)

where ξS(t) is the fluctuation produced by the system of

interest S. The symbol �ξS(t)� denotes the Gibbs average

1539-3755/2011/83(5)/051130(12) 051130-1 ©2011 American Physical Society
AQUINO, BOLOGNA, WEST, AND GRIGOLINI PHYSICAL REVIEW E 83, 051130 (2011) 235

over the fluctuations. In the absence of perturbation this

average is assumed to vanish. The variable ξP (t) denotes the

time-dependent perturbation and � its intensity. LRT predicts

the response of S on the basis of the unperturbed correlation

function of ξS. In fact, the function χ(t,s), called the linear

response function, is related to the correlation function of the

fluctuation ξS, whose quadratic mean value is assumed for

simplicity to be normalized to unity

�S(t,s) ≡ �ξS(t)ξS(s)�, (4)

by the following expression

χ(t,s) = d

ds

�S(t,s). (5)

Note that the traditional LRT refers to the stationary case

�S(t,s) = �S(t − s), (6)

and as a consequence

d

ds

�S(t,s) = − d

dt

�S(t,s). (7)

This condition is not fulfilled by complex systems. For these

latter systems the choices of linear response functions

χ(t,s) = d

ds

�S(t,s), (8)

and

χ(t,s) = − d

dt

�S(t,s), (9)

are not equivalent.

The authors of Refs. [15–18] have discussed the foundation

of both choices and have established that the new form of LRT

is determined by the physical way through which perturbation

determines a bias. For the sake of simplicity these authors have

made the assumption that ξS(t) is a dichotomous signal. Using

the jargon of turbulence theory they called the time intervals

between two consecutive crucial eventslaminar regions. At the

moment of a crucial event occurrence, unperturbed dynamics

are realized by the random selection of either the positive,

ξS = 1, or the negative value, ξS = −1. In other words, they

assume that the occurrence of a crucial event generates the

tossing of a fair coin which determines the sign of the next

laminar region. Consequently, the external perturbation can

generate a bias in two different ways. The first way rests on

affecting the fairness of the coin tossing process. If ξP (t) > 0

(ξP (t) < 0) the choice of the positive (negative) sign is more

probable than the choice of the negative (positive) sign. This

prescription leads to the choice of Eq. (8) and is denoted as

phenomenological LRT.

The experiments done by the authors of Refs. [12,13] show

that nature prefers Eq. (9), the dynamic LRT. What is the

theoretical argument in favor of the dynamical theory? To

afford a convincing answer to this important question, let us

go back to the special form of Eq. (1). We note that we do

not know the Hamiltonian of our complex system, and we

do not even know if a satisfactory discussion of the complex

dynamics can be made using a Hamiltonian formalism. Let

us assume that Eq. (1) is a reliable representation for the

distribution length of the laminar region. In this case, a

reasonable conjecture is that the external perturbation affects

either μ or T , or both parameters defining the form of Eq. (1).

We know that μ defines the system’s complexity and emerges

from the cooperative interaction among interacting units. A

weak external perturbation is not expected to change the

system’s complexity. It is therefore reasonable to assume that

the external perturbation affects T , by enlarging (reducing)

its value if ξP and ξS have the same (opposite) sign. This

assumption leads to the choice of Eq. (9), as shown earlier

[15–18].

It is important to notice that the response of a complex

system of the same nature as the one discussed in this paper

has been studied by many authors [19–28]. These authors did

not establish a connection between their results and the LRT

of Refs. [15–18] and in some cases they made the misleading

conjecture that their results establish the “death of LRT.”

Actually, these theoretical treatments are asymptotic in time

and the only possible connection with LRT is through the

adoption of the phenomenological theory of Eq. (8), as the

readers can establish by a careful reading of Ref. [28].

We are now in a position to define the main purpose of

this paper. We draw the attention of the readers to the recent

results of the authors Ref. [29]. This paper addresses the

important issue of studying the response of a complex system

to a complex external perturbation with the surprising result

that a complex system does not respond to stimuli that are not

complex (i.e., that have a stationary Fourier spectrum). It is

important to stress that the authors of Ref. [29] focused on

the correlation between ξS(t) and ξP (t) in the long-time limit.

This is an ideal condition that has the effect of restraining the

definition of complexity to the systems with μ  2. In fact, in

the long-time limit a system with μ > 2 reaches the normal

condition of a constant rate of event production, thereby

recovering the ordinary Poisson condition. The condition

μ = 2 is of fundamental importance for brain function. In

fact, recent work [30,31] established that the brain works

with μ = 2, which, in turn, is known [32] to correspond to

making the brain action become the source of ideal 1/f noise.

The results of the authors of Ref. [29] may therefore have

important applications to design the most convenient stimuli

to drive complex networks, and especially brain dynamics, via

what was defined as “complexity management” [29]. However,

an apparent weakness found in Ref. [29] is that these results

are derived from the adoption of the phenomenological LRT,

thereby raising doubt that those complex systems, which

have been proven to obey the dynamical LRT [12,13], may

not obey the principle of complexity management (CM)

established in Ref. [29]. Herein we prove that the more

realistic dynamical LRT generates CM. In addition to this

main purpose, the present paper affords technical details on

the theory developed in Ref. [29] that, due to space limitations,

were not conveniently illustrated.

II. FDT FOR THE NONERGODIC RENEWAL REGIME:

PHENOMENOLOGICAL AND DYNAMICAL

APPROACH

The authors of Refs. [15,16] discovered a form of FDT that

applies to systems or networks whose dynamics are dominated

by non-Poisson renewal events. In the stationary case this

051130-2
236 TRANSMISSION OF INFORMATION BETWEEN COMPLEX ... PHYSICAL REVIEW E 83, 051130 (2011)

FDT becomes indistinguishable from the ordinary theoretical

prediction [1].

Herein we investigate the consequences of the adoption

of either the “phenomenological” choice Eq. (8) or the

“dynamical” choice (9), in the special case where both ξP (t)

and ξS(t) are event-dominated processes and show that the

transmission of information from P to S is determined by the

dialogue between the critical events of ξS(t) and the critical

events of ξP (t). Specifically, this discussion is devoted to

studying the transport of information from P to S, using

both forms of generalized FDT (gFDT). Note that there is

no limitation on the form of ξP provided that the coupling is

weak enough as to be compatible with the emergence of the

linear response form of Eq. (3).

To discuss the problem of the transmission of information

fromP to S, it is convenient to imagine the ideal case of a Gibbs

ensemble of composite systems S + P. Thus, in this paper we

imagine that P generates a fluctuating signal ξP (t) and that

for any signal ξP (t) there exists a response ξS(t). For each

perturbing signal we have to make infinitely many experiments

and average over all possible responses. For simplicity we take

both signals ξS(t) and ξP (t) to be dichotomous and fluctuating

between values ±1. It is important to remark that Eq. (3), for

the response of the system S to the perturbation P, is valid

when the system is prepared at time t = 0 and placed at the

beginning of a laminar phase, and the interaction with the

perturbation P is turned on at the same time.

In the general case of a dicothomous renewal process ξ (t),

generated with a waiting-times PDF ψ(t), the probability

density that fixed a time t�

, the first next event is observed

at time t>t� is given [33] by

ψ(t,t�

) = ψ(t) +∞

n=1

 t�

0

R(t

��)ψ(t − t��)dt��, (10)

with

R(t) = ∞

n=1

ψn(t), (11)

where ψn(t) denotes the n-times convolution of ψ(t). R(t) is

therefore the probability density of having an event occurring

exactly at time t.

It can be shown as well, see Ref. [33], that the autocorrelation function of the process is related to ψ(t,t�

) by

�ξ (t)ξ (t

�

)� =  ∞

t

dxψ(x,t�) = �(t,t�), (12)

and therefore coincides with the survival probability �(t,t�

)

for the first event (i.e., the probability that, for fixed t�

, no event

is observed until time t>t�). We assume that the fluctuation

ξS(t) generated by S is a dicothomous renewal process defined

by the probability density

ψS(t) = (μS − 1)

TS

μS−1

(t + TS)μs

. (13)

We therefore name, respectively, ψS(t,t�

), RS(t), and �S(t,t�)

the functions obtained by replacing in Eqs. (10), (11), and (12)

ψ(t) with ψS(t).

Let us consider now the Gibbs ensemble of composite

systems S + P, and evaluate the average �ξS(t)�SP . Note that

the average is over the separate statistics of the two systems S

and P

�ξS(t)�SP = ��ξS(t)�S�P . (14)

We select all the responses to the same perturbation, characterized by a given ξP (t), we evaluate their average, denoted by

�ξS(t)�S, and finally we construct the average over all possible

perturbations denoted by �...�P so as to obtain the final result

denoted by �...�SP . In conclusion with this procedure we obtain

�σ(t)� = ��ξS(t)�� = �

 t

0

dt�χ(t,t�)�ξP (t

�

)�, (15)

where, for notational convenience, we drop the subscripts, but

we understand the averages in the sense described above.

If necessary, the signal ξP (t) must share the same properties

as ξS(t) and for simplicity they are both assumed to be dichotomous signals with random renewal fluctuations between

the values +1 and −1. ξP (t) is therefore a non-Poissonian

dichotomic fluctuation with the following waiting-time PDF:

ψP (t) = (μP − 1)

T μP −1

(t + TP )μP

. (16)

It is therefore convenient to define the additional functions

ψP (t,t�

), RP (t), and �P (t,t�) obtained, analogously as for S,

from Eqs. (10), (11), and (12) after replacing ψ(t) with the

waiting-time distribution ψP (t).

The spectrum of this type of fluctuating signal, in the

absence of perturbation, as calculated in Refs. [32,34], is

S(f ) ∝ Lμ−2

f μ−3, (17)

valid for μ < 2, remarkably, even though a stationary autocorrelation function cannot be defined in this case. In the case

μ > 2, S(f ) = A/f 3−μ, with A independent of L, the length

of the sequence under study.

At this point it should be clear to the reader that to get

the important results of this paper on the transmission of the

statistical properties of P to S we must use Eq. (15). This

leads us to give a prescription to define �ξP (t)�. We assume

that the perturbation P is prepared at t = 0 at the beginning of

a laminar phase (i.e., with the analogous prescription adopted

for the system S). This assumption allows us to replace �ξP (t�

)�

with �P (t�

) in Eq. (15).

III. PHENOMENOLOGICAL APPROACH

In this section we study the response of a complex system

producing non-Poissonian renewal fluctuations to a perturbing

signal with similar properties within the phenomenological

approach. We analyze both the average response and the

input-output correlation [i.e., the correlation between the

perturbing fluctuating signal (input) and the signal produced

by the system of interest S (output)]. In the phenomenological

approach, the waiting times between the events generating the

dichotomic fluctuations remain unchanged by the perturbation.

The external perturbation introduces a bias so that when

an event occurs the probability that the dichotomic variable

changes or keeps its value is slightly different. The function

χ(t,s) in this approach is given by [15,16]

χ(t,t�

) = d�S(t,t�

)

dt� = RS(t

�

)�S(t − t

�

). (18)

051130-3
AQUINO, BOLOGNA, WEST, AND GRIGOLINI PHYSICAL REVIEW E 83, 051130 (2011) 237

The function �S(t,t�) is the autocorrelation function of ξS(t),

namely, the survival probability of age t�, and RS(t) for the

case of the discrete signals considered here, is the rate at which

events are produced by S prepared at t = 0 [i.e., the bits per

second encoded in ξS(t)]. This rate is time independent only

in the Poisson case. In the non-Poisson case it depends on

time, thereby making �S(t,t�

) nonstationary. The brand new

survival probability �S(t) = �S(t,t�

=0) is given by [15,16,18]

�S (t) = (1 + t/TS)

1−μS , (19)

from which the corresponding waiting-times PDF ψS(t) =

−d�S(t)/dt is derived. In the range of parameters 1 < μS < 3

considered here, it is known [10] that

RS(t) ≈ −sin πμS

TS

(TS/t)

2−μS , for 1 < μS < 2, (20)

RS(t) ≈ 1

τS

[1 + (TS/t)

μS−2

], for 2 < μS < 3, (21)

with τS = TS/(μS − 2) the mean value of ψS(t).

When μS < 2 the experimental preparation of S induces

a sequence of events whose rate RS tends to vanish for t →

∞, yielding a perennial out-of-equilibrium condition, and an

explanation of the death of linear response [18–24,28] as well.

In fact, the response to a harmonic perturbation of frequency

f is proportional to 1/(f t)

2−μS [18]. In the case 2 < μS < 3,

on the contrary, the preparation-induced cascade of events, in

the limit t → ∞, becomes stationary and virtually identical to

that of a Poisson process. The theoretical analysis of this paper

is done in the asymptotic time regime. Thus, we refer to the

case 2 <μ< 3 as stationary, in contrast to the nonstationary

case μ  2 of perennial transition. Similarly to the rate of

events RS(t) the spectral intensity per unit time tends to vanish

for μ < 2 as an effect of increasing L [see Eq. (17)]. The

ideal 1/f -noise condition, corresponding to μ = 2, generates

instead a logarithmic decrease of the spectral intensity with

time, and consequently a spectrum virtually independent of L.

A. Average response to perturbation

As previously mentioned, the nonstationary LRT (NSLRT)

of Eq. (3) rests on the preparation of S at time t = 0. We apply

the same preparation condition to P, thereby generating the

cascades RS(t) and RP (t) described by Eqs. (20) and (21), with

the appropriate indexing. Under this condition the relaxation

of �ξP (t)� becomes identical to the survival probability �P (t).

Assuming the condition of Eq. (18) we have the following

expression for the average response

�σ(t)� = ε

 t

0

RS(t

�

)�S(t − t

�

)�P (t

�

)dt�. (22)

The preparation of both S and P makes the average over many

realizations of the response σ(t) to a given stimulus P vanish

for t → ∞.

Stationary case: 2 < μS < 3.

In this regime a finite time scale for the fluctuation ξS exists

and RS(t) reaches the constant value 1/τS. The inverse powerlaw relaxation of �S(t) allows us to approximate Eq. (22) by

replacing RS(t�

) with its value for t� � t, that is,

�σ(t)� � �

 t

0

dt��S(t − t

�

)�P (t

�

)/τS, (23)

which becomes exact for t → ∞. The asymptotic behavior of

Eq. (23) is easily obtained in the Laplace domain

�σˆ (s)� � �

1

τS

1 − ψˆ S(s)

s

1 − ψˆP (s)

s , (24)

which can be studied in the limit of small s. In fact, since [35]

ψˆ (s) � 1 − τ s + �(μ − 2)sμ−1, μ> 2, (25)

and

ψˆ (s) � 1 + �(μ − 2)sμ−1, μ< 2, (26)

it follows that for 1 < μP < 2 and 2 < μP < μS, the timeasymptotic behavior is �σ(t)� ∼ t 1−μP , which is proportional

to �ξP (t)� for large t, meaning that the system S “inherits” the

relaxation properties of the perturbation P.

For 2 < μS < μP , instead, the asymptotic dominant term

is �σ(t)� ∼ t 1−μS , which is proportional to the ordinary

unperturbed relaxation to equilibrium �ξS(t)�, when an initial

bias for ξS(t) is introduced. We see therefore that for μP < μS,

when ξP (t) is slower than ξS(t), the perturbation P imposes

on the system S its own relaxation properties thereby allowing

one to “manage” the complexity of a system or network by

using an appropriate stimulus.

Nonstationary case: 1 < μS  2.

In this regime, the system S violates the finite-time

scale condition necessary for stationary dynamics and, in

fact, RS(t) ∝ tμS−2, see Eq. (20). With such replacement in

Eq. (22), a convolution form appears which can easily studied

via a Laplace transformation. In the Laplace domain (see

Appendix A for details on coefficients)

�σˆ (s � 0)� � aS sμS−2 + aP sμP −2

, (27)

which implies that if μP > 2 or if 1 < μS < μP then �σ(t)� ∼

t 1−μS . When 1 < μP < μS, we have �σ(t)� ∼ t 1−μP . Also in

this case the perturbation P forces onto S its own relaxation

properties to equilibrium.

B. Input-output correlation function

We study the cross-correlation (or input-output correlation)

function between the system S and the stimulus P : C(t) ≡

��ξS (t) ξP (t)��, which is also used as an indicator of aperiodic

stochastic resonance [36]. Multiplying both sides of Eq. (3) by

ξP (t) and averaging over the fluctuations of the perturbation

P we obtain

�(t) ≡ C(t)/ε =

 t

0

dt�RS(t

�

)�S(t − t

�

)�P (t,t�). (28)

Note that both Eqs. (22) and (28) depend on the survival

probability of P, but in the former such survival probability

depends on the single time t� whereas in the latter it depends

on both t� and t. We limit ourselves to report the results for

the asymptotic value �∞ of �(t). When ξS(t) and ξP (t) are

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238 TRANSMISSION OF INFORMATION BETWEEN COMPLEX ... PHYSICAL REVIEW E 83, 051130 (2011)

not stationary, that is, when 1 < μS  2 and 1 < μP  2,

Eq. (28), in the limit t → ∞, gives

�∞ = ζ (μS,μP ) ≡ �(μS + μP − 2)

× 3F2 [{μP − 1,μP − 1,μP + μS − 2},{μP ,μP },1]

�(2 − μP )�(μP )2�(μS − 1) ,

(29)

where 3F2 is the generalized hypergeometric function. For

more details see Appendix A. In the case 2 < μP < 3, �∞ is

simply zero.

In the case 2 < μS < 3, inserting into Eq. (28) expression

(21) for RS(t), leads to

�(t) =

 t

0

dt� �S(t − t�

)

τS

�P (t,t�) = � ̃ S(0)�P (t,t)

− � ̃ S(t)�P (t,0) −

 t

0

dt�

� ̃ S(t − t

�

)

d�P (t,t�

)

dt� , (30)

where � ̃ S(t) is given by Eq. (19) after replacing μS with

μS − 1. Equation (30) is exact for t � τS and for 1 < μP  2

it leads to �∞ = 1 since both the second and third terms

disappear for t → ∞ and the first is trivially 1 in the same

limit. For Eq. (30) 2 < μP < 3 it yields

�∞ = 1 − (μP − 2)T μP −2

P T μS−2S �T 4−μS−μP

×B



�T /TP

�T /TS

,μS + μP − 4,

2 − μP

3 − μS



�T →0 =

× μS − 2

μP + μS − 4

, (31)

where �T = |TS − TP |, B[x,a,b] is the incomplete Beta

function and the upper (lower) choice of the parameters refers

to the case TS > (<)TP . The final expression in Eq. (31)

corresponds to the case TS = TP . The results are summarized

in Table I. For illustrative purposes, we supplement Table I with

Fig. 1, showing the three-dimensional (3D) plot of the crosscorrelation function �∞ in the same parameter range: Square

II and square III correspond to the condition of minimal and

maximal correlation, respectively. Intuitively it is so because of

the difference of time scales between S and P in such regions.

In III fluctuations ξS(t) and ξP (t) have a finite and an infinite

time scale, respectively, thereby allowing ξS(t) to adapt to the

stimulus-induced bias so as to yield maximal correlation. In

II the role of the time scales is inverted, the bias induced by

P on the longer (diverging) time scale of the process ξS(t)

is asymptotically averaged out due to the many intervening

switching events of ξP (t), producing no correlation. The

vertex μS = μP = 2, representing a 1/f -noise generating

system under the stimulus of a 1/f -noise perturbation, marks

TABLE I. Summary of the asymptotic values of the crosscorrelation function �(t) in the phenomenological case.

μS↓ μP→ 1 < μP  2 2 < μP < 3

1<μS2 �∞ = ζ (μS ,μP )

* I �∞ = 0 II

2<μS<3 �∞ = 1 III �∞ = μS−2

μS+μP −4 IV

*See Eq. (29).

III

II

I

IV

1

1.5

2

2.5

3

S

1

1.5

2

2.5

3

P

0.5

1

μ

μ

Φ∞

FIG. 1. (Color online) The asymptotic limit of �(t) is displayed

for μS ,μP ∈]1,3[. The vertex μS = μP = 2 marks the transition to a

condition of maximal input-output cross correlation.

the abrupt transition from vanishing (square II) to maximal

correlation (III).

IV. DYNAMICAL APPROACH

In this section we extend the analysis of the previous section

to the dynamical approach. Within such approach we derive

both the average response and the input-output correlation

function.

A. Average response to perturbation

Starting from the property

�σ(t)� = ��ξS(t)��, (32)

obtained by averaging over the fluctuations of both S and P

and using the dynamical condition of Eq. (9), one obtains

χ(t,t�

) = − d

dt

�S(t,t�

) = ψS(t,t�). (33)

Equation (3) then becomes

�σ(t)� = �

 t

0

dt�ψS(t,t�)�ξP (t

�

)�. (34)

For simplicity we prepare the perturbation ξP (t) at t = 0.

To observe the influence of P on S we select all the systems

of the Gibbs ensemble so that ξP (0) = 1. In this case �ξP (t�

)�

is given by the survival probability �P (t) [33]. Thus, Eq. (34)

yields

�σ(t)� = �

 t

0

dt�ψS(t,t�)�P (t

�

), (35)

where �P (t�

) is the survival probability for the process ξP (t)

�P (t) =

 ∞

t

dxψP (x) =

 TP

t + TP

μP −1

. (36)

This slow decay corresponds to the probability that no

perturbation event occurs up to time t. The system S evolves

in time so as to reach a steady value that corresponds to

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AQUINO, BOLOGNA, WEST, AND GRIGOLINI PHYSICAL REVIEW E 83, 051130 (2011) 239

a constant perturbation abruptly applied at t = 0. However,

during this process a perturbation event occurs that has the

effect of suddenly changing the external field. Thus, σ(t) does

not reach a steady value, but after reaching a maximum value

will decay. Under the specific conditions discussed in this

section, the time asymptotic decay of �σ(t)� in Eq. (35) has

the same power-law index as that of survival probability �P (t).

We interpret this phenomenon as the transmission into S of the

statistics of P.

We explore again the whole range of parameters 1 < μS <

3 and 1 < μP < 3, respectively, for the system S and the

perturbation P, depending on the values of the power-law

indexes μS, μP characterizing their waiting-times PDFs. The

value μ = 2 marks the transition from a finite to an infinite

mean time (i.e., the transition to a nonergodic, nonstationary

condition. In fact, while for μ > 2 a mean time exists, a finite

time scale can be defined and a stationary condition is reached,

for μ < 2 such condition is never achieved, not even in the

infinite mean time.

Stationary case: 2 < μS < 3.

In this regime the waiting-times PDF ψS(t) has a finite

mean value τS. Therefore a finite time scale tC exists such that

for t>t� > tC the following approximation corresponding to

reaching a stationary condition, is valid [33]

ψS(t,t�

) �

1

τS

 ∞

t

dxψS(x − t

�

) = �S(t − t�

)

τS

, (37)

where τS = TS/(μS − 2) is the mean value of ψS(t). Equation (37) is exact for tc → ∞. We can also use Eq. (3) for

the response in the case when the interaction is turned on at

a later time tC that we assume to be so large as to satisfy

the approximation Eq. (37) using the following procedure.

We introduce into Eq. (3) an effective perturbation which is

turned on at time t = 0 but is zero until tC [i.e., ξ eff

P (t) =

θ(t − tC)ξP (t − tC)]. In this scheme we can assume that the

“real” process ξP (t) is prepared at time t = tC in a brand new

condition. Time tC therefore corresponds to the age of the

system S. The average response �σ(t)� of S in this case reads

�σ(t)� = ε

 t

0

dt�ψS(t,t�)�θ(t

� − tC)ξP (t� − tC)�

= ε

 t

tC

dt�ψS(t,t�)�P (t

� − tC), (38)

and the approximation (37) can be used to replace ψS(t,t�

).

With the substitution τ = t� − tC and after renaming t − tC as

t back again, the average response of the system S of age tC

reads

�σ(t)�

ε

|tC =

 t

0

dτ

�S(t − τ )

τS

�P (τ ) = � ̃ S(0)�P (t)

− � ̃ S(t)�P (0) −

 t

0

dt�

� ̃ S(t − t

�

)

d

dt�

�P (t

�

),

(39)

where � ̃ S(t) is defined as

� ̃ S(t) = (1 + t/TS)

μS−2 . (40)

Equation (39) is exact for tC → ∞ and coincides with the

expression (23) obtained in the phenomenological case in the

same regime. Therefore, in the limit t � TS,TP , the same

considerations apply, that is, if 1 < μP < 2 and 2 < μS < μP ,

the time-asymptotic behavior is �σ(t)� ∼ t 1−μP , which is

proportional to �ξP (t)� for large t, therefore the system S

always “inherits” the relaxation properties of the perturbation.

If 2 < μS < μP , the asymptotic dominant term is �σ(t)� ∼

t 1−μS , which is proportional to the ordinary unperturbed

relaxation to equilibrium �ξS(t)�.

Nonstationary case: 1 < μS < 2.

Let us make the assumption that, although at time t = 0

half of the S elements of the Gibbs ensemble are in the

state ξS = +1 and half in the state ξS = −1, all of them

are at the beginning of their sojourn in the corresponding

states. This is an out-of-equilibrium condition, corresponding

to preparing the system at t = 0. The calculations are detailed

in Appendix B. Using the dynamic theory we obtain

�σ(t)�

� ≈ k1(μS,μP )tμP −1 +

k2(μS,μP )

tμP +1−μS

, (41)

where the first coefficient is given by

k1(μS,μP ) = TP μP −1 sin(πμS)�(2 − μS)�(1 − μP + μS)

π�(3 − μP ) ,

(42)

and the second coefficient is determined to be

k2(μS,μP ) = [sin(πμS)�(3 − μS)�(1 − μP + μS)�(2μS − 3) − (2 − μS)�(2μS − μP − 1)]

TP

1−μP TSμS−2π(μP − 2)(2 − μS)�(3 − μS)�(μS − μP )�(2μS − 3) . (43)

In this range of parameters the dominant term is always the

first term in Eq. (41) which, if μP < μS, is also slower than

the unperturbed relaxation to equilibrium of ξS. In the latter

range, therefore, the system S relaxes to equilibrium inheriting

the same properties of the perturbation P.

The result in Eq. (41) is of special interest since it discriminates between the two approaches in the nonstationary regime.

It is this difference that allowed to determine that liquid crystals

[13] follow the prediction of the dynamical approach. In fact,

the phenomenological approach disregards the influence of the

perturbation on the occurrence time of the S events [19,28],

while the dynamical theory does not, thereby affording a

criterion for information transport that we judge to be a more

appropriate representation of the communication among complex systems with μ < 2. However, the equivalence between

the phenomenological and the dynamic theories in the case

when the system S is infinitely aged (i.e., for μS > 2) indicates

that the generalization of FDT given by Eq. (9), namely, the

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240 TRANSMISSION OF INFORMATION BETWEEN COMPLEX ... PHYSICAL REVIEW E 83, 051130 (2011)

dynamical theory, becomes active only when the system S

is in a far from equilibrium condition and begins drifting

toward equilibrium. Although equilibrium is never reached

when μ < 2, the correlation function �S(t,t�

) tends to recover

the property �S(t,t�

) = � ̃ S(t − t�) that makes the phenomenological theory formally equivalent to the dynamical theory.

B. Input-output correlation function

Herein we study the asymptotic limit of the input-output

correlation function

�∞ ≡ lim

t→∞ C(t)/�, (44)

within the dynamical approach. The input-output correlation

function is again defined by the average over the fluctuations

in both the system S and perturbation P

C(t) ≡ �ξS(t)ξP (t)�. (45)

The asymptotic limit of C(t) is independent of the way the

system and the perturbation are prepared, so we can use the

prescription leading to Eq. (3) obtained assuming that both S

and P are prepared at time t = 0. We therefore use the same

arguments as those yielding Eq. (28) and, adapting them to the

dynamic theory, we obtain

�(t) = C(t)/� =

 t

0

dt�ψS(t,t�)�P (t,t�). (46)

Nonstationary case I: μS < 2, μP < 2.

We use Eq. (46) with the general expressions for ψS(t,t�

)

and �P (t,t�) as obtained through Eqs. (10) and (12), respectively. In this case, taking the limit t → ∞ yields (see

Appendix B for details)

ζD = lim

t→∞ �(t) = −sin πμP

π

�(μP + μS − 1)

(μP − 1)�(μP + 1)�(μS − 1)

× F [{μP − 1,μP − 1,μP + μS − 1},{μP ,μP + 1},1] .

(47)

Nonstationary case II: μS < 2, μP > 2.

In this case we can assume that, when the interaction is

turned on, the perturbation P has already reached a stationary

condition, so that in Eq. (46) ψS(t,t�

) is given again by Eq. (10)

but �P (t,t�

) = � ̃ P (t − t�), with

� ̃ P (t) = (1 + t/TP )

μP −2

, (48)

and this expression has to be directly inserted into Eq. (46).

The power-law index μP − 2 reflect the stationary condition realized with the preparation of the perturbation P at a

time tP = −∞. In this case, we obtain

lim

t→∞ �(t) = 0. (49)

In the time asymptotic limit, the system S turns out to be

independent of P in spite of the fact that at t = 0 we switch

on the S-P interaction.

Stationary case I: μS > 2, μP > 2.

We again use Eq. (46) and assume that, when the interaction

is turned on, the perturbation P has already reached a

stationary condition so that in Eq. (46) �P (t,t�

) = � ̃ P (t − t�)

with � ̃ P (t) given by Eq. (48). For μS > 2 a finite mean time

τS of ψ(t) exists, therefore a finite time scale tC ∝ τS exists

such that for t>tC Eq. (37) can be used again to approximate

ψS(t,t�). With such substitutions, the expression for

the correlation function becomes asymptotically equal to the

correlation obtained in the phenomenological approach in the

same regime [i.e., Eq. (30), leading to the final result Eq. (31)].

Stationary case II: μS > 2, μP < 2.

In this case again a finite time scale tC can be found such

that the approximation in Eq. (37) is valid. Thus, again Eq.

(46) can be rewritten

�(t) �

 tC

0

dt�ψS(t,t�)�P (t,t�)

+

 t

tC

dt�

 d

dt�

� ̃ S(t − t

�

)



�P (t,t�). (50)

This case is therefore equivalent to what is obtained for

the phenomenological case in the same range of parameters.

Again, in the asymptotic limit t → ∞, the first term in Eq. (50)

vanishes and after integrating the second term by parts, one

obtains

�(t) � � ̃ S(0)�P (t,t) − � ̃ S(t − tC)�P (t,tC)

−

 t

tC

dt�

� ̃ S(t − t

�

)

d�P (t,t�

)

dt� . (51)

In the asymptotic limit the second term on the right-hand

side trivially vanishes and also the third term side can be shown

to vanish (see Appendix B). The only remaining contribution

in Eq. (51) is given by the first term, which is exactly one. It

follows

�∞ = lim

t→∞ C(t)/� = 1. (52)

Let us make here a few remarks on these results. Of course

in the absence of coupling, � = 0, which means that C(t)

always vanishes. We switch the interaction on at t = 0. Thus,

we always have the zero cross-correlation initial condition

C(0) = 0.

As an effect of switching on the interaction at t = 0 we

realize the condition C(t) �= 0, for t > 0. However, we find

that there exist special conditions (region II in Fig. 2) for

which the cross correlation again goes to zero, asymptotically

�∞ = lim

t→∞ C(t)/� = 0. (53)

This indicates that only in such conditions, the system S, after

a transient, recovers the condition of statistical independence

of the perturbation ξP (t). The fluctuations ξS with μS < 2 turn

out to be statistically independent of ξP (t) only when μP > 2.

The environmental perturbation with μP < 2, on the other

hand, is characterized by the remarkable property of forcing

the system S to respond, regardless of the value of μS.

Table II and the plot in Fig. 2 summarize such results

and show the same qualitative properties observed for the

correlation function in the phenomenological case, with the

correlation functions for the two approaches being identical

in squares I, II, and IV. The condition of ideal 1/f noise (i.e.,

μS = μP = 2) marks the transition from a condition of zero

to maximal correlation.

051130-7
AQUINO, BOLOGNA, WEST, AND GRIGOLINI PHYSICAL REVIEW E 83, 051130 (2011) 241

TABLE II. Summary of the asymptotic values of the crosscorrelation function �(t) in the dynamical case.

μS↓ μP→ 1 < μP  2 2 < μP < 3

1<μS2 �∞ = ζD(μS ,μP )

* I �∞ = 0 II

2<μS<3 �∞ = 1 III �∞ = μS−2

μS+μP −4 IV

*See Eq. (47).

V. FINITE RESPONSE AND 1/ f RESONANCE

Herein we proceed to demonstrate that the intensity of

the response σ(t) to a single realization of the stimulus does

not decay if �∞ �= 0. This demonstration, therefore, is valid

for both the dynamical and the phenomenological approach,

leading to a general result. Let us define with pt(ξ i

S) the

probability that at time t the variable ξS takes the value

i = ±1 and with pt(ξ i

S|ξ j

P ) the conditional probability for

the occurrence, at time t, of a value ξS = i = ±1, given

the occurrence of a value ξP = j = ±1. By definition, the

nonvanishing �∞ yields

C(t) ≡ 

i,j=±1

ij pt



ξ i

S|ξ j

P



pt



ξ j

P



→ ε�∞. (54)

We note that for t → ∞, on a time scale such that �ξP (t)�,

which decays as �ξP (0)�t 1−μP , is a second-order quantity

O(ε2) we have that

pt



ξ j

P



= 1/2 + O(ε2

), (55)

and

�(t) = �∞ + O(ε2

). (56)

Thus is due to Eq. (54) and to the symmetry of the considered

dichotomous processes

pt



ξ i

S



ξ j

P



→ 1

2 + i jε

�∞

2 . (57)

III

II

I IV

1

1.5

2

2.5

3

S

1

1.5

2

2.5

3

P

0.5

1

μ μ

Φ∞

FIG. 2. (Color online) The asymptotic limit of �(t) is displayed

for the different regimes of parameters for 1 < μS < 3 and 1 < μP <

3, in the dynamical approach.

In the same long-time scale, Eq. (57) yields

�σ(t)�± ≡ 

i

pt



ξ i

S



ξ ±1

P



i � ±ε�∞, (58)

where the subscript ± indicates the value of ξP at time t.

Summing Eq. (58) over the two values of ξP gives a total

average null response, as expected. But if the magnitude |σ(t)|

of the response to a single instance of the input ξP (t) is

considered instead, its total average is

�|σ(t)|� =

1

2



±

�|σ(t)|�±  1

2



±

|�σ(t)�±| � ε�∞, (59)

where an equality holds if terms of order O(ε2) are neglected.

Thus when �∞ > 0, the response σ(t) to a single instance of

the input ξP (t) does not die out and remains proportional to

the stimulus intensity, no matter how large t becomes. Square

III in both Figs. 1 and 2 is the plateau region of maximal cross

correlation and response. Claims regarding the death of linear

response, as in Refs. [19–28], are therefore appropriate only

in relation to the vanishing correlation of square II. The total

average response �σ(t)� always tends to vanish for t → ∞ for

reasons that do not imply a lack of response except in the case

of square II.

The reason for the striking difference between the response

to a harmonic perturbation and the response to a nonergodic

stimulus is intimately related to the emergence of 1/f noise

and to its spectrum described by Eq. (17), which assigns the

weight S(f )/L = 1/(f L)

3−μP to the spectral component of

frequency f of a nonergodic stimulus. As a consequence, the

stimulus generates, in time, lower and lower frequencies f so

as to keep 1/(f L)

2−μS (i.e., the response intensity to frequency

f [18]) finite, thereby yielding Eq. (59). The death of linear

response [19–28] is caused by the fact that stimuli with fixed

frequencies cannot cope with the decreasing frequency of the

cascade of events of Eq. (20).

We have afforded a compelling proof that the intensity of the

single realizations of σ(t), with μS < 2, does not decay if the

perturbation ξP (t) falls in the same complexity basin (μP < 2).

This is the phenomenon of complexity management which

allows to define the right stimulus to obtain a response from

a system or network producing nonergodic fluctuations. Now

we argue that 1/f stimuli generate the maximum information

transport by looking at the mutual information

I (t) = 

i,j

pt



ξ j

P



pt



ξ i

S



ξ j

P



log[pt



ξ i

S



ξ j

P

pt

ξ i

S



]. (60)

Using Eqs. (55), (56), and (57) it follows that

I (t → ∞) � ε2

�2

∞, (61)

and the information transmission rate is obtained by multiplying I (t) by the input rate [37], given by RP (t). IfμP > 2, Fig. 1

shows that �∞ < 1. Although square III in Figs. 1 and 2, indicates that all stimuli with μP  2 induce maximal correlation,

μP < 2 corresponds to a stimulus with decaying events rate

(input bits/s) RP (t). So even if a response is produced in this

regime, the rate of information vanishes in time.

Only at the crucial condition μP = 2, of ideal 1/f noise,

does this algebraic decay become logarithmic, and, consequently, a steady and maximal information transmission rate is

051130-8
242 TRANSMISSION OF INFORMATION BETWEEN COMPLEX ... PHYSICAL REVIEW E 83, 051130 (2011)

achieved. This is the phenomenon that we call 1/f resonance.

The above considerations are valid for both the dynamical

and phenomenological approaches, therefore we consider the

condition of maximal information transmission achieved in

the ideal 1/f -noise condition, a fundamental property of

1/f -noise renewal processes.

VI. CONCLUDING REMARKS

The growing interest for the dynamics of complex networks

is shifting the attention of the researchers from the synchronization of two stochastic units [38] to the synchronization of

a large number of units [39], an interesting phenomenon that

is closely related to the very popular model of Kuramoto [40].

The single units of the processes of chaos synchronization are

chaotic and do surprisingly synchronize while maintaining

the erratic dynamics that they have in isolation. Although

the single units of the Kuramoto model are regular, it is

becoming increasingly evident that the emergence of a global

synchronization is a condition independent of whether the

single units are regular or stochastic. The single units of the

work of Refs. [11,41] are Poisson processes and if one of them

drove the other, they would obey the principle of aperiodic

stochastic resonance [42]. If the two units are bidirectionally

coupled they are expected to undergo a condition of perfect

synchronization if the coupling is sufficiently intense. When

the number of interacting units is very large a phase transition

occurs from the noncooperative to the cooperative behavior

[11,41]. It is important to stress that at criticality no permanent

consensus is reached, and the mean value of the global field

vanishes. Yet, this condition is strikingly different from the

noncooperative condition. The whole network remains in the

“yes” (“no”) state for an extended time before making a

transition to the “no” (“yes”) state.

It is surprising that the phase-transition literature seems to

have overlooked, with only a few exceptions [11,41,43], that

the transitions from the “yes” (“no”) to the “no” (“yes”) state

occurring at criticality are the “crucial” events defined in Sec.I.

In other words, the time interval between two consecutive

transitions is derived from a PDF that has the asymptotic time

structure of Eq. (1) with a power index μ fitting the inequality

condition of Eq. (2). Some authors [11,41] argued that μ = 1.5

and others, [7], releasing the condition that all the units share

the same Poisson rate, generate a global condition with crucial

events characterized byμ < 2, but significantly departing from

the value μ = 1.5. Note that the theoretical arguments of

Ref. [44], yield the misleading impression that the crucial

value of μ is a consequence of ordinary statistical physics.

According to some authors [45,46] the Kuramoto phenomenon can be defined as inner synchronization. A network

of cooperating units located on the nodes of a complex network

may reach inner synchronization with different values of the

control parameter, depending on the network topology [47].

This is a subject of increasing interest with attractive applications to the dynamics of the human brain [45]. If we adopt

this perspective, we can address the problem of information

transmission from one to another complex network as a process

of outer synchronization. This is an interesting issue, but the

conditions to fulfill to realize outer synchronization are not yet

clear [45].

An important result of this article is the discovery of a

promising road to settle the problem of information transmission from one to another complex network. In fact, if the

inner synchronization corresponds to a criticality condition

and criticality generates crucial events with a power-law index

μ fulfilling the inequality of Eq. (2), then a complex network

at criticality is a generator of 1/f noise, with S(f ) ∝ 1/f 3−μ.

Thus the problem of information transmission from one to

another complex network, in this regime, becomes equivalent

to the phenomenon of 1/f resonance illustrated in this article.

This is essentially the main result of earlier work [29]. The

main conclusions of Ref. [29] are illustrated by Fig. 1, which

is obtained using the phenomenological LRT.

What are the limits of this earlier result? The experiments

[12,13] yield support to the dynamical rather than the phenomenological LRT, thereby generating doubt that the results

of Ref. [27], although very attractive, may not completely

reflect reality. It is important to stress that phenomenological

LRT is a natural consequence of adopting the asymptotic

time perspective replacing the waiting-times PDF ψ(τ ) of

Eq. (1) with ψ(τ ) ∝ 1/τμ. This way of proceeding, although

generating the elegant mathematics of fractional derivatives,

has as an ultimate effect the misleading discovery of the death

of linear response [19–24,28]. We do not adopt the asymptotic

time perspective but the special form of Eq. (1). This is not

a unique way of connecting the long-time to the short-time

regime. However, whatever form we adopt we are convinced

that there will be a parameter playing the same microscopc

role of the parameter T of Eq. (1). It is reasonable to assume

that an external perturbation may perturb either T or μ, or both

of them. However, the perturbation of μ is incompatible with

the assumption of a weak stimulus. In fact, μ is a consequence

of the cooperation among the units of the network, and a

perturbation may affect μ only if its strength is large enough

to influence the interaction among the units of the network.

Thus, an external weak perturbation can only have an effect on

T , thereby making the dynamical LRT become the proper way

to study the response of a complex network to a weak external

stimulus, in accordance with the experimental results [12,13].

For these reasons, we can conclude that Fig. 2 is the original,

and important, result of this paper. We hope that it may open the

road to the dynamical solution of the problem of information

transmission [48] from one to another complex network, a

research topic that is still in its infancy.

ACKNOWLEDGMENTS

PG acknowledges financial support from ARO and

Welch through Grants No. W911NF-05-1-0205 and No. B1577, respectively. MB acknowledges financial support from

FONDECYT project No. 1110231.

APPENDIX A

In this Appendix we record more details about the derivation of both the average response �σ(t)� and the input-output

correlation function �(t) in the case of the phenomenological

approach.

When we adopt the phenomenological theory, we obtain

for the average response to external perturbation, in the

nonstationary case [cf. Eq. (27)] the following asymptotic

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AQUINO, BOLOGNA, WEST, AND GRIGOLINI PHYSICAL REVIEW E 83, 051130 (2011) 243

expression:

�σ(t)�

� = k1(μS,μP )tμP −1 +

k2(μS,μP )

tμS−1 , (A1)

where the coefficient of the first term is

k1(μS,μP ) = �(μP − μS)

�(1 − μP )�(μS)

− �(1 − μP + μS)

�(2 − μP )�(μS)

, (A2)

and the coefficient of the second term

k2(μS,μP ) = �(μP − μS)

�(1 − μS)�(μP )

. (A3)

Note the logarithmic corrections corresponding to μS = μP ,

with

�σ(t)�

� ≈ sin(πμS)π

log t

tμS−1 +

A(μS)

tμS−1 , (A4)

and

A(μS) = sin(πμS)

π

 1

μS − 1 − 2γ − 2ψL(μS)



− cos(πμS),

(A5)

where γ is the Euler’s constant and ψL(z) the logarithm

derivative of the � function. These predictions are qualitatively

equivalent to the dynamical theory predictions with the

assumption that the system S has been prepared in the very

distant past.

The general expression of the correlation function in the

phenomenological approach is given by Eq. (28).

Nonstationary case I: μS < 2,μP < 2.

In this range the asymptotic approximation for the function

R(t) defined in Eq. (11) for both S and P is

R(t) �

sin πμ

π

t

μ−2

. (A6)

The power-law properties of �S(t), in the long-time limit,

shift the dominant contribution to the integral of Eq. (28) to

the range t� ∼ t, this allows to adopt the approximation (A6)

inside the integral. It follows

�(t) �

sin πμS

π

sin πμP

π

×

 t

0

τμS−2�S(t − τ )dτ  τ

0

xμP −2�P (t − x)dx.

(A7)

Using a generalized Newton binomial expansion of the powerlaw form of the functions �P (t) we obtain the following

expression:

�(t) �

sin πμS

π

sin πμP

π

∞

n=0



1 − μP

n

 (−1)n

(t + TP )n+μP −1

 t

0

t

�n+μP +μS−3

�S(t − t

�

)dt�, (A8)

while the convolution with �S(t) in the integral, leads to

�(t) �

sin πμS

π

sin πμP

π

∞

n=0



1 − μP

n

 (−1)n

(t + TP )n+μP −1

× tn+μP −1

n + μP − 1

�(n + μP + μS − 2)�(2 − μS)

�(n + μP ) ,

(A9)

where we have used the fact that for t large the main

contribution to the integral comes from the range of values

x  τ  t and therefore the asymptotic approximations for

RS(t) an RP (t) are justified. The limit for t → ∞ cancels

the time dependence and leaves a sum which leads to the

expression of Eq. (47) in the text.

Nonstationary case II: μS < 2,μP > 2.

In this case the function RP (t) tends asymptotically to a

constant value and therefore it easy to see, following analogous

procedure to the previous case, that the correlation tends to

zero.

Stationary case I: μS > 2,μP < 2.

As mentioned in the text, from Eq. (30) it is enough to show

that the third term on the right-hand side is zero. Such terms

read as

sin πμP

π

 t

0

� ̃ S(t − t

�

)dt�t

�μP −2

�P (t − t

�

). (A10)

Using a generalized binomial expansion for both � ̃ S(t) and

�P (t) we get

�(t) �

sin πμP

π



n,m



1 − μP

n

1 − μS

m

 (−1)n

(t + TP )n+μP −1

× (−1)m

(t + TS)m+μS−2

 t

0

t

�n+m+μP −2

dt�. (A11)

Carrying out the integration shows that the leading term in t

vanishes for t → ∞ Such a demonstration is valid also for the

dynamical case in the same range of parameters [see Eqs. (51)

and (52)].

APPENDIX B

This Appendix is devoted to detailed calculations involved

the evaluation of the asymptotic limits of the function �(t)

and �σ(t)� in the dynamic approach.

Herein only the parameter range μS < 2,μP < 2 is considered, the other ranges being analyzed in detail in the main text

and in Appendix A.

�(t) is given by the following integral:

�(t) =

 t

0

ψS(t,t�)�P (t,t�)dt�, (B1)

where ψS(t,t�

) is defined in Eq. (10) and �P (t,t�) in Eq. (12),

each with the appropriate labeling. �(t) can be decomposed

as sum of four contributions

�(t) = 

4

i=1

�i(t), (B2)

where

�1(t) = tψS(t)�P (t), (B3)

�2(t) = ψS(t)

 t

0

 t�

0

RP (t

��)�P (t − t��)dt��dt�

= ψS(t)

 t

0

(t − t

��)RP (t��)�P (t − t��)dt��, (B4)

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244 TRANSMISSION OF INFORMATION BETWEEN COMPLEX ... PHYSICAL REVIEW E 83, 051130 (2011)

�3(t) = �P (t)

 t

0

 t�

0

RS(t

��)ψS(t − t��)dt��dt�

= �P (t)

 t

0

(t − t

��)RS(t��)ψS(t − t��)dt��, (B5)

�4(t) =

 t

0

 t�

0

 t�

0

RS(t

��)ψS(t − t��)RP (τ )�P (t − τ )dt�

dt��dτ.

(B6)

Here ψS(t) and �P (t) are defined in Eqs. (13) and (36). Using

the methods of Ref. [49], Eqs. (B4), (B5) can be evaluated in

the asymptotic limit to yield

�1(t) ≈ c1

tμS+μP −2 , �2(t) ≈ c2tμS−1 , �3(t) ≈ c3tμP −1 .

Since in the limit t → ∞ the three contribute vanish, it is

not important the evaluation of the constants c1, c2, c3. To

evaluate �4(t) we first need an analytical expression for RS(t)

and RP (t). It can be shown that [49]

RS(t) ≈ −sin(πμS)

πT μS−1

S

1

t 2−μS

, (B7)

with the same expression, with the respective parameters being

valid for RP (t). Then we have

 t�

0

RS(τ )ψS(t − τ )dτ ≈ −(t + TS − t�

)

1−μS t�μS−1 sin(πμS)

π(t + TS) .

On the other hand

 t�

0

RP (τ )�P (t − τ )dτ

= T μP −1

P

(t + TP )μP −1

∞

n=0



1 − μP

n

 (−)

n

(t + TP )n

×

 t�

0

RP (τ )τ ndτ. (B8)

Using for RP (t) the same approximation of RS(t) Eq. (B2),

we have

 t�

0

RP (τ )�P (t − τ )dτ

≈ − sin πμP

π(t + TP )μP −1

∞

n=0



1 − μP

n

 (−)

n

(t + TP )n

t�n+μP −1

n + μP − 1

.

(B9)

The function �4(t) is

�4(t) ≈ sin πμP sin πμS

π2(t + TP )μP −1

tμS+μP −1

(t + TS)μS

∞

n=0



− t

t + TP

n

×



1 − μP

n

F



n + μP + μS − 1,μS − 1,n + μP + μS, t

t+TS



(n + μP − 1)(n + μP + μS − 1) ,

where F(a,b,c,z) is the hypergeometric function. Finally

�4(t) ≈ sin πμP�(μP + μS − 1)(t + TP )

1−μP tμS

(1 − μP )�(μP + 1)�(μS − 1)(t + TS)μS t 1−μP

× F



{μP − 1,μP − 1,μP + μS − 1},{μP ,μP + 1}, t

t + TS



,

where pFq ({a},{b},z) is the generalized hypergeometric function. In the limit for t → ∞

�∞ = − sin πμP�(μP + μS − 1)

π(μP − 1)�(μP+1)�(μS − 1)

× F [{μP − 1,μP −1,μP+μS − 1},{μP ,μP+1},1] .

(B10)

As far as the evaluation of the function �σ(t)� is concerned,

in the dynamic approach it is given by Eq. (35)

�σ(t)� = �

 t

0

ψS(t,t�)�P (t

�

)dt�, (B11)

which is to be evaluated in the parameter range μS < 2,μP <

2. Let PS(t) = RS(t) + δ(t), Eq. (B11) reads

�σ(t)� = �

 t

0

dt��P (t

�

)

 t�

0

dxPS(x)ψS(t − x), (B12)

which, after a transformation on the integration domain, turns

into

�σ(t)� = �

 t

0

dxψS(t − x)PS(x)

 t

x

dt��P (t

�

). (B13)

The long-time limit shifts the main weight in the integral on the

terms such that x  t. Therefore the approximation Eq. (B7),

valid also for PS(t) in the long-time limit, can be adopted.

Inserting such approximation in Eq. (B13) leads to

�σ(t)��−�

sin πμS

πT μS−1

S

 t

0

dxψS(t − x)xμS−2

 t

x

dt��P (t

�

).

(B14)

051130-11
AQUINO, BOLOGNA, WEST, AND GRIGOLINI PHYSICAL REVIEW E 83, 051130 (2011) 245

After getting rid of the integral in t� by direct integration

of �P (t�

), one is left with a simple convolution which is

easy to analyze in Laplace transform. After some straightforward algebra, extracting the two asymptotic leading terms,

leads to

�σ(t)� � k1t

1−μP + k2tμS−μP −1

, (B15)

that is, the asymptotic expression of Eq. (41), with the

coefficients given by Eqs. (42) and (43).

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Research@

247

Robust Control for Mobility and Wireless 

Communication in Cyber–Physical Systems 

with Application to Robot Teams

Jonathan Fink, Alejandro Ribeiro, and Vijay Kumar

Proceedings of the IEEE, 100(1), 164-178 (2012)
248 INVITED

PAPER

Robust Control for Mobility and

Wireless Communication in

Cyber–Physical Systems With

Application to Robot Teams

Control of an autonomous robot team with the aim of optimizing network performance

is discussed in this paper; the authors design a controller to ensure availability

of communication resources.

By Jonathan Fink, Student Member IEEE, Alejandro Ribeiro, Member IEEE, and

Vijay Kumar, Fellow IEEE

ABSTRACT | In this paper, a system architecture to provide

end-to-end network connectivity for autonomous teams of

robots is discussed. The core of the proposed system is a cyber–

physical controller whose goal is to ensure network connectivity as robots move to accomplish their assigned tasks. Due to

channel quality uncertainties inherent to wireless propagation,

we adopt a stochastic model where achievable rates are

modeled as random variables. The cyber component of the

controller determines routing variables that maximize the

probability of having a connected network for given positions.

The physical component determines feasible robot trajectories

that are restricted to safe configurations which ensure these

probabilities stay above a minimum reliability level. Local

trajectory planning algorithms are proposed for simple environments and leveraged to obtain global planning algorithms to

handle complex surroundings. The resulting integrated controllers are robust in that end-to-end communication survives

with high probability even if individual point-to-point links are

likely to fail with significant probability. Experiments demonstrate that the global planning algorithm succeeds in navigating

a complex environment while ensuring that end-to-end communication rates meet or exceed prescribed values within a

target failure tolerance.

KEYWORDS | Mobile ad hoc networks; motion planning; multirobot systems

I. INTRODUCTION

The confluence of advances in wireless communication,

sensing, computing, and control has led to the emergence

of cyber–physical systems. Cyber–physical systems go beyond the mere addition of cyber functionalities to physical

systems being typified by a tight blending of cyber and

physical components and functions. A canonical example

of this class of systems is the autonomous robot team

where a group of mobile robots cooperate to accomplish

tasks assigned by human operators. Consider, for example,

a search and rescue mission in a hazardous environment

where a team of robots is deployed to scout points of interest. Designated lead members of the team move to

specified locations while the remaining robots provide

mission support by configuring a multihop wireless network that permits relaying of information back to the

operators. Availability of wireless communications is

critical for task accomplishment because communications

are required to exchange information between robots as

well as to relay information to and from the human

Manuscript received March 30, 2011; revised June 22, 2011; accepted June 24, 2011.

Date of publication September 6, 2011; date of current version December 21, 2011.

This work was supported by the U.S. Army Research Laboratory under Grant

W911NF-08-2-0004, and the U.S. Office of Naval Research (ONR) under Grants

N00014-08-1-0696 and N00014-07-1-0829.

J. Fink was with the GRASP Laboratory, University of Pennsylvania, Philadelphia,

PA 19104 USA. He is now with the U.S. Army Research Laboratory, Adelphi,

MD 20783-1197 USA (e-mail: jonfink@grasp.upenn.edu).

A. Ribeiro is with the Electrical and Systems Engineering Department, University of

Pennsylvania, Philadelphia, PA 19104 USA (e-mail: aribeiro@seas.upenn.edu).

V. Kumar is with the GRASP Laboratory, University of Pennsylvania, Philadelphia,

PA 19104 USA (e-mail: kumar@grasp.upenn.edu).

Digital Object Identifier: 10.1109/JPROC.2011.2161427

164 Proceedings of the IEEE | Vol. 100, No. 1, January 2012 0018-9219/$26.00 2011 IEEE
249

operators. It then follows that deployment of the team

requires integration of (cyber) wireless communications

with (physical) mobility control. The cyber–physical

perspective posits that this integration is not just a matter

of adding wireless transceivers to the robots but requires a

tight conjoining of communications and mobility control.

Tight integration of communications and mobility

arises in autonomous robot teams because availability of

wireless infrastructure is unlikely in the harsh environments in which they are to be deployed. In the absence of

infrastructure, robots have to self-organize into a wireless

network capable of supporting the necessary information

exchanges. In this context, the spatial placement of the

robots affects the feasibility of successful communication

because some physical configurations cannot support required transmission rates between, say, human operators

and some lead members. We can therefore think of communications as determining a set of virtual obstacles that

restrict the movement of the robots. However, this is not a

hard limit as it is possible to reconfigure communication

variables to alter the set of physical configurations for

which communications are feasible. This yields a control

loop that begins with communication variables limiting

mobility and finishes with mobility determining communication variables. In the end, successful deployment of an

autonomous team of robots requires joint cyber–physical

controllers that determine (physical) trajectories for the

robots while ensuring (cyber) availability of communication resources [1].

An architecture diagram for the cyber–physical control

of autonomous teams of robots is shown in Fig. 1. As with

any mobility control system, there is a block performing

task specification, a second block executing the control

law, and a third block conducting actuation and state estimation. The task specification block interfaces with the

human operators and integrates robot observations and

requirements to determine specifications that it passes on

to the control block. These specifications come in the form

of a potential function �ðxÞ and basal end-to-end communication rate requirements ak

i;minVthe subindex i here denotes a source and the superindex k a destination; see

Section II. The goal of the autonomous team is to minimize

the potential �ðxÞ while ensuring that end-to-end communication rates exceed the minimum required level ak

i;min.

The control block is the core of the cyber–physical

system and the main focus of the work described in this

paper. The purpose of this block is to determine physical

control inputs x_ðtÞ that dictate the movement of the robots

and network variables AðtÞ that determine the communication of information through members of the team. The

control inputs x_ðtÞ are conducive to task completion, i.e.,

minimization of the task potential, while ensuring reliable

network operation, i.e., communication at or exceeding

rates ak

i;min when combined with network variables AðtÞ.

Information about achievable rates between different locations is necessary to determine mobility and communication variables. This information is provided by a radio

communication modeling block. Due to the difficulty of

predicting these rates accurately, the output of this block is

in the form of a statistical model of achievable rates. As we

discuss in Section III, models that provide the mean and

variance of achievable rates are reasonable in practice.

The execution block consists of individual robots

implementing the control law x_ðtÞ and routing packets

Fig. 1. System architecture. Task specification here represents a generic spatial application defined by a convex task potential function CðxÞ

while providing a stream of data to the human operator. Individual robot components consist of the low-level robot control, estimation,

and communication. We additionally assume that a subsystem is available to build an online model of radio communication in the environment.

The focus of this work is on developing concurrent methods for routing and mobility control.

Fink et al.: Robust Control for Mobility and Wireless Communication in Cyber–Physical Systems

Vol. 100, No. 1, January 2012 | Proceedings of the IEEE 165
250

according to variables AðtÞ. Robots also take observations

yiðtÞ, e.g., a video feed, that they relay to task planning and

perform position estimation x^iðtÞ that they feedback to the

control block. Using available technologies for mapping,

control, and state estimation, each robot can estimate its

position x^iðtÞ and control its velocity x_iðtÞ with respect to a

common known map of the environment. Since we do not

provide details about mapping or state estimation in this

paper, we refer the interested reader to, e.g., [2].

A. Communication-Aware Mobility Control

In the design of concurrent communications and mobility control algorithms for autonomous robot teams there

are three issues that arise naturally: 1) how to translate

robot positions into information about the point-to-point

rates at which pairs of robots can communicate with each

other; 2) how will the cyber–physical control loop in Fig. 1

utilize point-to-point connectivity information in order to

ensure end-to-end network integrity; and 3) how to deal

with uncertainty in the information about point-to-point

channels between individual pairs of robots.

Issues 1) and 2) are related to each other. A simple

answer to issue 1) is to identify the ability to communicate

with spatial proximity, possibly mixed with line-of-sight

information to account for physical obstacles to radio propagation. Using this model, point-to-point links between

robots are feasible if they are sufficiently close and within

sight of each other and infeasible otherwise. Matched to

this simple model of point-to-point connectivity, the ability to establish end-to-end connections is identified with

the connectedness of the resulting graph. The problem of

ensuring end-to-end network integrity then becomes one

of letting robots move to accomplish their task while

guaranteeing that the graph of point-to-point connections

consists of a single connected component [3]–[10].

The advantage of this approach is that network-wide

indicators of connectedness such as the second largest

eigenvalue of the graph Laplacian or the k-connectivity of

the network’s graph are computationally tractable. It is

therefore not difficult to insert connectivity constraints

into the task planner and, if there is leeway in the selection

of different spatial configurations, to find configurations

that optimize connectivity in terms of the metric under

consideration. Do notice that path planning is computationally challenging even in the absence of communication

constraints. The tractability of graph connectedness indicators simply means that the computational complexity of

the cyber–physical control loop is comparable to that of a

communication-unaware controller.

Aside from the value of simplicity, associating small

distance and line-of-sight with connectivity is not entirely

justified. Early solutions to wireless networking for systems of dynamic nodes were indeed based on the concept

that reliable point-to-point links could be detected and

established [11], [12]. However, it was quickly noticed that

binary channel models were not accurate representations

of wireless links [13]. As a consequence, attention was

devoted to methods that explicitly consider the reliability

of point-to-point links and choose routes to optimize endto-end reliability metrics such as probability of delivery

[14]–[16]. These more nuanced models of wireless connectivity can be incorporated without much difficulty into

mobility control algorithms that use indicators of graph

connectedness [17], [18]. If links are characterized by

reliability indicators, the binary connectivity graph can be

replaced by a weighted graph. Information about the

number of connected components can be similarly extracted from, for example, the second largest eigenvalue of

the weighted graph’s Laplacian.

Either based on binary or weighted graphs, connectedness is an indirect indicator of the ability to establish endto-end communications. Having a single connected

component just implies that multihop paths from sources

to destinations exist but does not determine whether the

network formed by the robots is able to support desired

communication rates. It is possible that some links in the

connectivity graph can support rates that are too low, or

that the resulting configuration contains bottlenecks that

limit information flow. In the cyber–physical control loop

depicted in Fig. 1, graph connectedness does not solve for

the network variables AðtÞ. Instead, it restricts movement

so that a necessary but not sufficient condition for the

existence of feasible network variables AðtÞ is not violated.

In fact, one could argue that these controllers are not

cyber–physical because the integration of wireless communications with physical mobility is minimal. A true cyber–

physical design should consider end-to-end rates and

conjoin the design of mobility and network control as done

in the systems described in the following section.

B. Robust Cyber–Physical Control

An honest metric of network integrity has to rely on the

achievability of target end-to-end communication rates.

Communication rates, however, do not depend solely on

the spatial configuration but also on the manner in which

packets are routed through the network. This leads to the

problem of joint cyber–physical control of trajectories and

routing variables [1], [19]. For given spatial configuration x,

pairs of achievable point-to-point communication rates

RijðxÞ are assumed available at the radio modeling block.

The cyber part of the controller queries the modeling block

and attempts to determine routing variables AðxÞ that

support desired communication rates ak

i;min. If this attempt

is unsuccessful, spatial configuration x is marked as unallowable. If the attempt succeeds, joint configuration

ðx; AðxÞÞ is deemed feasible and, in particular, spatial

configuration x is deemed allowable. The physical part of

the controller utilizes the information on allowable configurations x to plan physical routes that achieve

minimization of the task potential �ðxÞ while ensuring

that end-to-end rates exceed basal requirements at all

points in time. See Section II.

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Ensuring continuous reliable communication is challenging, however, because significant uncertainty in

achievable rates is inherent to the deployment of an autonomous robot team. Due to shadowing and small scale

fading, even small variations in robots’ positions lead to

significant changes in channel strength [13], [20], [21]

which translate into commensurable changes in achievable

communication rates between pairs of robots. Precise

channel state information can be acquired through measurements over time, but mobility planning algorithms

necessitate access to channel quality indicators at future

positions to which the robots are yet to be deployed. Thus,

from the perspective of the system in Fig. 1, rates RijðxÞ are

random. The outputs of the radio modeling block are rate

estimates R�ijðxÞ whose variances R~ijðxÞ are typically large.

See Section III.

Since achievability of end-to-end rates cannot be guaranteed because of rate uncertainties, it is natural to redefine network survivability in terms of the probability of

end-to-end rates exceeding their basal rates ak

i;min. Packet

routing variables AðxÞ are then determined in order to

maximize this probability [1]. The resulting routes are

robust because they are chosen to minimize the probability

of failure. At this point it has to be noted that the goal of

the self-organized network is to maintain reliable end-toend communication, not point-to-point communication.

Consequently, we can think of robust routes as exploiting

spatial redundancy to minimize the effect of point-to-point

uncertainty in end-to-end communication rates. Indeed,

by splitting traffic flows between various neighboring robots, we can ensure that while failure of a particular link

may reduce end-to-end communication rates, it does not

interrupt them completely. See Section IV.

The integration of robust routing with the physical control loop is discussed in Section V. We develop local planning algorithms for simple environments (Section V-A) that

we test in simulations for teams composed of three and four

robots (Section V-B). These simulations demonstrate that

the performance of local controllers is hindered by local

minima. To overcome this problem as well as to handle

complex environments, we leverage the local controllers to

design global planning algorithms (Section V-C). Global

planners are tested with experimental results in an indoor

office space at the University of Pennsylvania (Section V-D).

The experiments demonstrate that the global planning

algorithm succeeds in navigating a complex environment

while ensuring that end-to-end communication rates meet

or exceed prescribed values within a target failure tolerance.

We close the paper with concluding remarks (Section VI).

II. COMMUNICATIONS AND MOBILITY

Consider a team of N robots and denote their positions as

xi, for i 1⁄4 1; ... ; N. The robots are kinematic and fully

controllable which allows us to consider simple mobility

models of the form x_iðtÞ 1⁄4 uiðtÞ, where uiðtÞ is the control

input to robot i. A human operator is located at the fixed

operation center that we index as i 1⁄4 0 at position x0.

Further define vectors x :1⁄4 ðx0; ... ; xNÞ 2 R2ðNþ1Þ and

x_ :1⁄4 ðx_0; ... ; x_NÞ 2 R2ðNþ1Þ to group all positions and

velocities, respectively. The task assigned to the team is

specified through a generic scalar convex task potential

function � : R2ðNþ1Þ ! R. If the potential minimum �min

is attained at x�, i.e., if �ðx�Þ 1⁄4 �min, the configuration

x� satisfies task completion. For example, if a designated

leader agent ‘ must visit a target location x‘;goal 2 R2

, we

can define �ðxÞ1⁄4kx‘ � x‘;goalk2

. The minimum �min 1⁄4

0 is attained by any configuration x� 1⁄4 ðx0; ... ; x‘; ... ;

xNÞ for which x‘ 1⁄4 x‘;goal, or equivalently by any member

of the set x� 2 fx 1⁄4 ðx0; ... ; x‘; ... ; xNÞ : x‘ 1⁄4 x‘;goalg.

Irrespective of the particular form of �ðxÞ, the control

problem is to find velocities x_ðtÞ such that at some time tf

the team configuration xðtfÞ 1⁄4 xð0Þ þ R tf

0 x_ðtÞdt satisfies

task completion in that we have �ðxðtfÞÞ 1⁄4 �min.

Mathematically, we can write this mobility control formulation as

min x_ðtÞ;t21⁄20;tf �

� xðtfÞ � �

subject to xðtÞ 1⁄4 xð0Þ þ Zt

0

x_ðuÞ du: (1)

As robots move to accomplish their task, they maintain

end-to-end data communication flows between members of

the team and/or members of the team and the operation

center. Information flows are indexed as k 1⁄4 1; ... ; K.

Flows may have multiple sources and multiple destinations. The set of destinations of the kth information flow is

denoted as destðkÞ. For agent i and flow k, the variable

ak

i;min represents the required communication rate between

agent i and any of the agents in the set of destinations

destðkÞ. For example, if the only communications of interest are from the lead robot ‘ to the operation center, there

are only K 1⁄4 1 flows. Since the flow k 1⁄4 1 is intended to

the operating center, destð1Þ 1⁄4 0 and a1

‘;min denotes the

minimum level of service for the communication from the

leader to the operating center. All other variables

ak

i;min 1⁄4 0 are null.

We model point-to-point connectivity through a rate

function RijðxÞ 1⁄4 Rijðxi; xjÞ that determines the amount of

information that agent i at position xi can send to agent j at

position xj. Since direct communication between the

source and the destination of an information flow is not

always possible, terminals self-organize into a multihop

network to relay packets for each other. Packet relaying is

determined by routing variables �k

ij which describe the

fraction of time node i spends transmitting data for flow k

to node j; see Fig. 2. Thus, the product �k

ijRijðxÞ determines

the rate of point-to-point information transmission from

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i to j. If we consider the transmission to all neighboring

terminals for which RijðxÞ > 0, the total rate at which packets leave agent i is PN

j1⁄40 �k

ijRijðxÞ. Likewise, the total rate

at which

P

i receives packets from other terminals is N

j1⁄40;j62destðkÞ �k

jiRjiðxÞ. The information rate akiðA; xÞ

available for flow k at source i is the difference between

outgoing and incoming rates

ak

iðA; xÞ 1⁄4 XN

j1⁄40

�k

ijRijðxÞ

|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}

Outgoing packets

� XN

j1⁄40;j62destðkÞ

�k

jiRjiðxÞ

|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

Incoming packets

(2)

where we define the vector A grouping all routing

variables �k

ij. Notice that the variables �kij represent time

slot shares and must therefore satisfy 0 � �k

ij � 1 for all i, j,

and k. It must also be that P

j;k �k

ij � 1 for all i to ensure

that the sum of all time shares at terminal i does not exceed

1. It is possible to, alternatively, require P

i;j;k �k

ij � 1 if we

wish for only one link to be active at any time across the

entire network.

Routing variables A and configuration-dependent rates

RijðxÞ determine the set ak

iðA; xÞ of end-to-end communication rates from each node i and flow k as per (2). The

task specification requires that end-to-end rates exceed

the minimum threshold ak

i;min. Therefore, integrity of the

communication network necessitates that

ak

iðA; xÞ � aki;min; for all i; k: (3)

Notice that ak

iðA; xÞ is a function of positions x and routing variables A. To control end-to-end connectivity, i.e., to

satisfy (3), we can resort to control of positions x, to

control of routes A, or both.

Since communication is necessary for task completion,

the mobility control problem as summarized in (1) is

redefined. The new goal is to find algorithms and control

policies that govern robot motions in order to satisfy the

task specifications in (1) and (3). Reducing �ðxÞ as per (1)

and ensuring network integrity as per (3) may be

conflicting requirements. We therefore replace (1) by a

concurrent search of trajectories xðtÞ and routes AðtÞ so

that the task potential is minimized without ever breaking

communication connectivity. Mathematically, we write

this objective as the optimization problem

min AðtÞ;x_ðtÞ;t21⁄20;tf �

� xðtfÞ � �

subject to ak

ið Þ� AðtÞ; xðtÞ aki;min

xðtÞ 1⁄4 xð0Þ þ Zt

0

x_ðuÞ du (4)

where rates ak

iðAðtÞ; xðtÞÞ are given by the expression

in (2) with A 1⁄4 AðtÞ and x 1⁄4 xðtÞ.

A drawback of the formulation in (4) is the difficulty of

ensuring that the constraints in (3) are satisfied. As seen

in (2), rates ak

iðA; xÞ depend on the link reliabilities RijðxÞ,

which are difficult to estimate. Propagation knowledge is

used by the radio communication modeling block in Fig. 1

to provide rough rate estimates R�ijðxÞ. Agents can further

measure received signal strength indicators (RSSIs) to

refine the rate estimates for the current configuration xðtÞ;

see, e.g., [22]. The challenge here is that R�ijðxÞ estimates

are needed not only for configuration xðtÞ, but for nearby

configurations to which the robots may move. The high

variability of wireless channels makes R�ijðxðtÞÞ a poor

predictor of RijðxÞ even if x is close to xðtÞ. In general,

channel estimates R�ijðxÞ at future positions x follow from

radio propagation modeling and corrections from RSSI

observations.

In formal terms, uncertainty of channel estimates

means that the variances R~ijðxÞ of channel estimates R�ijðxÞ

are typically large for possible future positions x.

Variances R~ijðxÞ can be provided by the radio modeling

block and are therefore assumed to be available at the

controller along with the mean estimates R�ijðxÞ. We seek

to redefine (4) in a manner that takes into account this

probabilistic formulation of channel rates. The important

observation here is that if point-to-point link rates become

random, so do the rates ak

iðA; xÞ of end-to-end communication flows [cf. (2)]. Consequently, it is not possible to

guarantee satisfaction of the constraints in (3). Rather, we

introduce a reliability tolerance � and require that for all i

and k

P ak

iðA; xÞ � aki;min h i � �: (5)

That is, we require that the end-to-end link between all

sources i and the destinations of all corresponding flows k

Fig. 2. Communication network. The nodes are deployed to support

end-to-end rates from node i to destination (or flow) k. Routing

variables �k

ij determine the fraction of time node i sends packets to

node j for flow k. Rij is defined as the supported rate of the wireless

channel from node i to node j.

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253

exceed their minimum required level of service with probability larger than �. As in the case of the rate requirement in (3), we can satisfy (5) by controlling A and x

separately or jointly.

In order to robustly satisfy the networking constraints

in (5), the concurrent routing and mobility problem (4) is

replaced by

min AðtÞ;x_ðtÞ;t21⁄20;tf �

� xðtfÞ � �

subject to P ak

iðA; xÞ � aki;min h i � �

xðtÞ 1⁄4 xð0Þ þ Zt

0

x_ðuÞ du: (6)

The focus of this paper is the solution of (6).

The problem formulation in (6) inherits some standard

complications from the control formulation in (1). The

problem is infinite dimensional and due to, e.g., the

presence of obstacles, not convex. The concurrent search

in (6) is further complicated by the entanglement of the

routing and mobility problems. We deal with this entanglement by fixing x and selecting A in a manner that

optimizes the reliability P1⁄2ak

iðA; xÞ � aki;min� (Section IV).

We finish with local and global searches on positions x

to minimize �ðxÞ while keeping reliabilities above the

� threshold (Section V).

III. POINT-TO-POINT RATE MODELING

With robots i and j located at positions xi and xj, we seek to

develop a probabilistic model for the supported communication rate RijðxÞ 1⁄4 Rijðxi; xjÞ between them. More specifically, we wish to model its expected value R�ijðxÞ 1⁄4

R�ijðxi; xjÞ and variance R~ijðxÞ 1⁄4 R~ijðxi; xjÞ. Our intent is to

use simple radios that do not perform rate or power

adaptationVwe use 2.4-GHz Zigbee radios in our experiments; see Section V-D. In that case, the communication

rate Rijðxi; xjÞ is a function of the packet error rate of the

channel, which in turn is a function of the signal-to-noise

ratio [23]. We therefore focus on models of the received

signal strength PRðxi; xjÞ that we cascade into models of the

packet error rate and the supported communication rate

Rijðxi; xjÞ.

Received signal power PRðxi; xjÞ is determined by three

phenomena: path loss due to the distance from the source,

shadowing due to obstacles in the propagation path, and

multipath fading that arises as a result of reflections and

refractions. Of these three phenomena, path loss and shadowing can be incorporated into a predictive model with

relative ease. Fading, however, is difficult to predetermine.

Multipath fading arises due to a spatial wave interference

pattern generated by reflections and refractions of the

electromagnetic wave which, as a consequence, inherits a

space constant in the order of the wave’s wavelength.

Thus, in order to use ray tracing models to predict the

interference pattern we need to locate all potential

reflectors and refractors with precision smaller than this

wavelength. While plausible in principle, this is intractable

in practice because wavelengths at common operating frequencies are too smallVfor example, with radios operating

at 2.4 GHz., the wavelength is about 12 cm. Given this

difficulty we adopt the following model for the received

power PR;dBmðxi; xjÞ 1⁄4 10 logðPRðxi; xjÞÞ measured in dBm:

PR;dBmðxi; xjÞ

1⁄4 L0 � 10n � log kxi � xjk � �

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

Path loss

� Wðxi; xjÞ

|fflfflfflffl{zfflfflfflffl}

Shadowing

� F

|{z}

Fading

(7)

where the term F is a zero-mean Gaussian random variable with variance �2

F modeling fading effects. The term L0

is the measured power at a reference distance d0 from the

source, n is a path loss exponent, and Wðxi; xjÞ is a nonsmooth function to model shadowing as a function of the

number of obstacles between source and destination.

Experimental data collected in an indoor office environment at the University of Pennsylvania along with the

function in (7) having parameters fit to the experimental

data are depicted in Fig. 3. The parameters in Fig. 3 are

L0 1⁄4 �51 dBm for d0 1⁄4 1 m, n 1⁄4 2:1, Wðxi; xjÞ 1⁄4 0 for

line-of-sight links and Wðxi; xjÞ 1⁄4 7.6 dB for non-line-ofsight links, and �2

F 1⁄4 32 dB2. Notice that fading can cause

Fig. 3. Experimental characterization of received power model (7) based on 1000 samples when we classify wireless channels based on

line-of-sight. (a) Line-of-sight. (b) Non-line-of-sight.

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variations in received power in the order of �10 dB. This

is verified by the standard deviation of F, which is �F �

5.7 dB. Recall that a 10-dB difference corresponds to a

change of an order of magnitude for the power measured

in linear units.

Translation of received signal strength PRðxi; xjÞ into

packet error rate peðPRðxi; xjÞÞ depends on the type of

modulation and the choice of error correcting codes.

Regardless of specifics, a generally good approximation to

the packet error rate is

pe PRðxi; xjÞ � � 1⁄4 erfc

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

k

Pðxi; xjÞ

PN0

 ! s

(8)

where PN0 is the noise power, erfcðxÞ is the complementary error function, and k is a constant that depends on

modulation and coding [23]. The translation into communication rates Rijðxi; xjÞ is now straightforward as it simply

requires multiplying the information rate of transmitted

packets R0 by the probability of successful decoding, i.e.,

Rijðxi; xjÞ 1⁄4 R01⁄21 � peðPRðxi; xjÞÞ�.

The randomness of the fading term F in (7) propagates

into randomness in the error probabilities peðPðxi; xjÞÞ and

subsequently into randomness in the rates Rijðxi; xjÞ. To

compute the mean and variance of Rijðxi; xjÞ we need to

recall that F is a Gaussian random variable. We can then

invoke the delta method [24] to approximate the mean and

variance of Rijðxi; xjÞ in terms of the mean and variance of

PRðxi; xjÞ and the derivatives of the function peðPRðxi; xjÞÞ

in (8). The complete mapping from distance to link rates is

depicted in Fig. 4. Observe that except for very short

distances smaller than 3 m there is significant variability in

achieved link rates. See [1, ch. 4] for details.

While the model we present in this section describes

the statistics of the supported rate Rijðxi; xjÞ in terms of

distances kxi � xjk, we emphasize that this is not a limitation of the proposed cyber–physical controller. In fact,

the controller is agnostic to the particulars of the radio

propagation model and relies only on the ability to query

for means and variances. It is reasonable to expect that

techniques for spatial mapping of point-to-point communication rates will outperform distance-based models when

considering unknown or highly dynamic environments.

However, as demonstrated by the experiments in

Section V-D, the robust satisfaction of communication

constraints affords the possibility to rely on coarse models

of point-to-point communication rates while still ensuring

maintenance of quality-of-service requirements.

IV. ROBUST ROUTING

The control loop in Fig. 1 embeds a cyber optimization

component and a physical reconfiguration module. The

cyber optimization component optimizes communication

routes for given positions, while the physical component

explores spatial reconfigurations that maintain network

integrity and are conducive to task achievement. The physical module relies on the cyber module for the determination of configurations that are feasible from the

perspective of maintaining target communication rates

with some desired reliability.

The major difficulty in designing the cyber component

of this control loop is the uncertainty in achievable transmission rates between nearby agents. As we discussed in

Section III, simple experiments suffice to demonstrate that

achievable rates in wireless links are difficult to predict.

Therefore, assuming that the actual channel rates RijðxÞ

coincide with their estimates R�ijðxÞ may result in a drastic

difference between predicted and actual end-to-end rates.

This mismatch subsequently leads to situations in which

the communication optimization module asserts network

integrity when actual rates do not suffice to support the

task assigned by operators.

A simple way to account for the uncertainty in RijðxÞ is

to discount R�ijðxÞ in order to reduce the likelihood of

having actual rates smaller than the assumed value. This is

definitely possible but would result in underutilization of

communication resources. A better way to account for the

uncertainty in channel rates is to recall that end-to-end

rather than point-to-point failures are relevant. With reference to Fig. 2, say that robot i decides to convey information to k by forwarding it through l. If the rate RilðxÞ

happens to be substantially smaller than estimated, communication is interrupted. If, on the contrary, robot i decides to send part of the information through l and some

other part through j, the communication between i and k

survives as long as the rate RijðxÞ is larger than estimated.

This example illustrates that it is possible to exploit spatial

redundancy through traffic splitting in order to devise robust routes that guarantee small changes in end-to-end rates

despite large variability in point-to-point rates RijðxÞ [25].

To develop robust routing algorithms, start by noticing

that computing the probability in (5), which is part of the

Fig. 4. Mapping from distances kxi � xjk between a transmitter at

location xi and a receiver at position xj to normalized link rates

Rijðxi; xjÞ=R0.

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problem formulation in (6), necessitates modeling the

probability distribution of ak

iðA; xÞ. This is difficult in

general. However, if we explicitly consider the stochastic

model of point-to-point links via their means and variances, we can compute the mean and variance of end-toend rates ak

iðA; xÞ as

a�k

iðA; xÞ :1⁄4 E akiðA; xÞ � �

1⁄4 X

j

�k

ijR�ijðxÞ � X

j62destðkÞ

�k

jiR�ijðxÞ (9)

a~k

iðA; xÞ :1⁄4 var akiðA; xÞ � �

1⁄4 X

j

�k

ij � �2

R~ijðxÞ þ X

j62destðkÞ

�k

ji � �2

R~ijðxÞ: (10)

A proxy for the probability in (5) is the difference between

ak

iðA; xÞ and its mean a�kiðA; xÞ normalized by its standard

deviation

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a~k

iðA; xÞ

q

. Indeed, a typical approximation for

the probability constraint in (5) is the condition

Pk

iðA; xÞ :1⁄4 a�k

iðA; xÞ � aki;min ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a~k

iðA; xÞ

q � ��1

ð�Þ (11)

for some function ��1

ð�Þ. If the probability distribution of

rates RijðxÞ is Gaussian, (11) is equivalent to (5) if ��1

ð�Þ is

the inverse of the normal distribution’s cumulative

distribution function. For other probability distributions,

we can apply Chebyshev’s inequality with ��1

ð�Þ 1⁄4 ffiffiffiffiffiffi1=� p

to demonstrate that (11) is a sufficient condition for

satisfying (5) though this will usually be a conservative

approximation.

For given positions x, the goal of the cyber optimization component is to find routing variables A that satisfy

(11) which is equivalent to, approximate to, or a sufficient

condition to satisfy (5). In either case, there is some indeterminacy because there is a nonunique set of variables A

that satisfy the corresponding inequality. This indeterminacy provides a degree of freedom that can be used to

increase the reliability beyond the required level. For a

given configuration x, we would like to find routes

A 1⁄4 AðxÞ that provide the maximum possible reliability.

Doing so is not conceptually difficultVit requires determination of routes AðxÞ that maximize the probability proxies Pk

iðA; xÞ in (11) in some sense. Considering that there

is a rate variable ak

i for each source–destination pair, there

are several possible choices to define this maximization.

One possibility is to maximize the sum of these probability

proxies. Another possibility is to select routes AðxÞ that

make the smallest of these terms as large as possible.

Maximizing this minimum implies that the constraints in

(11) are satisfied with significant slack and that there is

significant liberty to change the physical configuration

without violating communication constraints. This freedom of movement facilitates implementation of the physical mobility control block as we discuss in Section V.

An important observation here is that the probability

proxy constraints in (11) can be written in a manner that

defines a cone in the joint space of routing variables and

end-to-end rates [1, ch. 5]. A consequence of this property

is that the optimization problems that need to be solved to

maximize them are second-order cone program (SOCPs).

SOCPs are a particular class of convex optimization

problem that can be solved by efficient polynomial-time

algorithms. For the problems considered here, the computational complexity of these algorithms is represented as a

polynomial function of the number of agents N and the

number of flows (destinations) K, as OððK � N2Þ

3:5

Þ. In

practical implementations, the N2 term can be reduced by

eliminating links for which rate estimates R�ijðxÞ are below

a certain threshold.

Recall that the motivation for robust routing algorithms

is to reduce uncertainty in end-to-end communication

rates. We do so by taking advantage of spatial redundancy,

for which it is necessary to split traffic among various

different routes. It is fitting that we do expect to obtain this

type of solution from the maximization of the probability

proxies Pk

iðA; xÞ in (11). Indeed, increases in these terms

can be brought about by either increasing the mean

a�k

iðA; xÞ or decreasing the variance a~k

iðA; xÞ. Since the

mean is a linear function of A, traffic splitting has a minor

effect on a�k

iðA; xÞ. However, traffic splitting reduces

a~k

iðA; xÞ by a factor proportional to the splitting because

the variance is a quadratic function of AVrecall that

�ij � 1. Thus, traffic splitting tends to increase the probability proxies because it keeps a�k

iðA; xÞ more or less constant and reduces a~k

iðA; xÞ significantly.

V. CONTROL

As per (6), the objective of mobility control is to decrease

�ðxÞ while satisfying the probability proxy constraints in

(11)Vwhich are equivalent or approximations to

P1⁄2ak

iðA; xÞ � aki;min� � �. To check for the feasibility of a

cyber–physical configuration ðA; xÞ, we define the probability margin as the minimum slack in probability proxy

constraints across all flows and sources

�ðA; xÞ :1⁄4 min

i;k

a�k

iðA; xÞ � aki;min ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a~k

iðA; x0

Þ

q � ��1

ð�Þ

2

6

4

3

7

5: (12)

Notice that a necessary and sufficient condition for

feasibility of the physical configuration x is to have

�ðAðxÞ; xÞ � 0, with routing variables AðxÞ as given by

the solution of the SOCP described in Section IV. A

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256

sufficient condition for feasibility of physical configuration

x0 is the existence of a cyber configuration A for which

�ðA; x0

Þ � 0. In particular, for x0 close to x, we expect to

have �ðAðxÞ; x0

Þ � 0 since the channel statistics at x and

x0 are close.

A. Gradient Controller

In general, gradient controllers define velocities x_ðtÞ

proportional to the negative gradients �r�ðxðtÞÞ of the

task potential. Since the problem in (6) is subject to communication constraints, a local controller will be based on

potential gradients projected onto the feasible set

�ðA; xÞ � 0. The complex description of the feasible set,

however, precludes computation of projected gradients.

Instead, we consider the probability margin �ðAðxÞ; xÞ

and modify the potential �ðxÞ by adding the probability

margin constraint into the objective through a barrier

function

�ðxÞ :1⁄4 �ðxÞ � logð Þ �ð Þ AðxÞ; x : (13)

Since nonnegativity is necessary and sufficient for feasibility of physical configuration x, the potential �ðxÞ in (13)

is defined if and only if physical configuration x is feasible.

The local control law is defined to implement gradient

descent on the modified potential �ðxÞ introduced in (13),

which, in explicit terms, is given by

uðtÞ 1⁄4 �r�ð Þþ xðtÞ rx�1⁄2 � Að Þ xðtÞ ; xðtÞ

�1⁄2 � Að Þ xðtÞ ; xðtÞ : (14)

The term r�ðxðtÞÞ in (14) drives the system to satisfy the

task potential. The term rx�1⁄2AðxðtÞÞ; xðtÞ�=�1⁄2AðxðtÞÞ;

xðtÞ� serves as a barrier that drives robots away from configurations for which there is a low probability of exceeding the desired reliability in end-to-end rates.

B. Simulation Results

We implement the local controller with mobility control inputs given by (14) and communication variables

obtained from Section IV. Computing controls based on

local optimization of the network-level end-to-end rates

allows for a method of realizing team deployment while

maintaining the necessary level of network connectivity.

Fig. 5 depicts an example deployment with three robots for

a time-varying task potential

�ð Þ1⁄4 xðtÞ

x2;goal 1⁄4 ð4; 0Þ; t G 40

x2;goal 1⁄4 ð8; 0Þ; t G 60

x2;goal 1⁄4 ð6; 5Þ; t � 60.

8

<

:

(15)

Most importantly, this example demonstrates convergence of the task potential �ðxÞ while maintaining

P1⁄2ai � ai;min� > � as depicted in Fig. 6(a) and (c). However, it is also interesting to observe that when the task

potential is minimized, e.g., t � 30 s, the local control law

(14) maximizes the probability of each end-to-end rate

exceeding its minimum threshold. When the task potential

switches so that �ðxÞ is no longer minimized, the probability margin is reduced so that the primary objective,

minimization of �ðxÞ, is prioritized. Finally, Fig. 6(b) depicts the end-to-end rate of the node x2 that must remain

above a2;min 1⁄4 0:1. Remember that RijðxÞ is a stochastic

rate that affects the end-to-end rate. The envelope around

a�2 in Fig. 6(b) depicts the effect that different realizations

of communication channels RijðxÞ will have on the end-toend rate. Since the pursuit of minimization on �ðxÞ is

constrained to have a probability margin �ðAðxÞ; xÞ > 0,

the end-to-end rate exceeds its threshold in the presence of

deviations to RijðxÞ.

We perform a four robot simulation, depicted in Fig. 7,

to demonstrate how the complexity of the objective function (13) increases with more nodes. Similar to the three

robot simulation above, the task potential function is time

varying so that �ðxÞ 1⁄4 0 for t G 50 s and �ðxÞ1⁄4kx3 �

ð9; 0Þk for t � 50 s. When t G 40 and the task potential is

already minimized, control of x1 and x2 is based solely on

the maximization of log �ðAðxÞ; xÞ. From the symmetric

initial configuration, the maximization drives x1 and x2

towards a local maxima where they would be positioned at

the same point. In fact, if we consider all configurations

where x1 G x2 and both agents are on the line connecting

the access point, we find that there are two local maxima as

depicted in Fig. 8.

To further illustrate the virtual terrain of the objective

function (13) that drives local control, we observe the

convergence of �ðxðtÞÞ as depicted in Fig. 9(a). For

times 70 s G t G 90 s, convergence of �ðxðtÞÞ slows,

Fig. 5. Deployment via local control law (14) for a system with a

fixed access point, relay node x1 and lead node x2, which is

controlled by a time-varying task potential CðxðtÞÞ.

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172 Proceedings of the IEEE | Vol. 100, No. 1, January 2012
257

�ðAðxÞ; xÞ as depicted in Fig. 9(c) shows little change,

and x1 and x2 cease progress. Fortunately, perturbations in

the simulation are enough to pull the system out of the

local minima so that it can achieve a global minimum at

t 1⁄4 140 s. As the number of agents increases, the frequency of local minima in ��ðAðxÞ; xÞ becomes more and

more of an issue for local control. The addition of obstacles

into the environment adds further difficulties as it not only

affects feasible configurations x due to collision constraints, but also introduces nonsmooth components in the

underlying point-to-point communication links RijðxÞ.

C. Global Planning

Gradient-based control will drive the system towards

local minima of (13). However, as we consider larger

teams with more complicated network topologies and

complex environments with obstacles, local minima become more of an issue. We propose that a global search of

(6) is necessary in order to accomplish the high-level

situational-awareness tasks we are interested in.

To consider global search of (6), we redefine the problem to be more amenable to motion planning approaches

from the robotics literature. Let X be a bounded connected

open subset of R2N that represents the full joint state space

for the team of robots where x0 is the initial configuration

of the team. In general, the goal region will be defined as

Xg 1⁄4 fx : �ðxÞ G �min þ �g. In the telepresence application where a lead agent must visit the location x‘;g, �ðxÞ 1⁄4

kx‘ � x‘;gk2 and Xg 1⁄4 fx : kx‘ � x‘;gk G �g. The obstacle

region Xobs contains any configuration that places an

individual robot on a physical obstacle and the infeasible

Fig. 7. Snapshots from a four robot trial. The end-to-end rates and probability of meeting the problem specifications are depicted in Fig. 9.

CðxðtÞÞ 1⁄4 kx3 � ð9;0Þk for t > 50 s and CðxðtÞÞ 1⁄4 0 for t � 50 s: (a) t 1⁄4 40 s; (b) t 1⁄4 75 s; (c) t 1⁄4 140 s.

Fig. 8. Evaluation of �ðAðxÞ; xÞ, 8x 2 fx 1⁄4 ðx1 ; x2; x3Þ : x1 G x2 G x3,

x3 1⁄4 ð5;0Þ, x1 , and x2 online connecting access point and x3g.

Local maxima occur at x1 1⁄4 x2 1⁄4 1.9 m and x1 1⁄4 1.5 m, x2 1⁄4 3 m.

Fig. 6. Performance of the local control law (14) demonstrating

convergence of CðxðtÞÞ in (a), the maintenance of expected end-to-end

rate greater than the threshold of a2;min 1⁄4 0:1 in (b), and the

P1⁄2ai � �i;min� > 0:6. The envelope surrounding E1⁄2a2� in (b) depicts the

60% confidence interval for realizations of the end-to-end rate

with stochastic RijðxÞ.

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258

region represents configurations where it is infeasible to

satisfy the network constraint (11)

Xinf 1⁄4 x :

a�k

iðA; xÞ � aki;min ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a~k

iðA; xÞ

q

G ��1

ð�Þ

8

><

>:

8 A 2 �k

ij : 0 � �kij � 1;

X

j;k

�k

ij � 1

( )9

>=

>;

: (16)

The free space Xfree is then X n ðXobs [ XinfÞ. Finally, a path

in X is parameterized by a scalar s � 0 and given by

S : 1⁄20; s� ! X. A feasible path, and solution to our globalplanning problem, is then S : 1⁄20; s� ! Xfree such that

Sð0Þ 1⁄4 xinit and SðsÞ 2 Xg.

The dimensionality of our problem and the high computational cost of verifying a state as in Xfree makes deterministic search algorithms impractical. Instead we turn to

probabilistic search methods that offer good space filling

properties and efficient exploration of an unknown space

like the rapidly exploring random tree (RRT) algorithms

[26]. The basic structure of an RRT, as depicted in Fig. 10,

is to start with an initial point x0 and expand to fully

explore the workspace, adding states in a tree structure T

until a point x 2 Xg is added to the tree T . At each step

of the RRT algorithm, we pick a random state x^ 1⁄4

RANDOMSTATEðX; T Þ and select the point xmin 1⁄4

NEARESTðT ; x^Þ that is closest to x^ among those that

have already been added to T . We then attempt a virtual

drive from xmin to x^ using the subroutine x 1⁄4

EXTENDðxmin; x^Þ. The point x is the first intersection

of this virtual path with the border of the free space region

Xfree, or, if the border is not reached, the random point x^.

The point x is then added to the tree T as a branch connected to the point xmin that was closest to x^ among the

preexisting elements of the tree. The algorithm terminates

when a point x 2 Xg in the goal configuration is added to

the tree.

A common problem encountered when applying RRT

algorithms to high-dimensional state spaces is that computation of Nearest is inefficient for increasing tree sizes.

We adopt the strategy of storing the tree T in a KD-tree

data structure which stores states in Rd by recursively

subdividing based on alternating axis-aligned hyperplanes

[27]. This enables approximate nearest neighbor calculations that maintain performance even as the dimension

increases. However, there are two additional difficulties

that arise when applying standard RRT algorithms to solve

the specific high-dimensional network connectivity problem in (6): 1) the verification of feasible states as Extend is

used to expand the tree towards x^; and 2) the prohibitive

cost of uniformly exploring Xfree for our high-dimensional

problem with slow-to-compute constraints. We discuss this

two issues in the following two sections.

1) Efficient Verification of Feasible States: The

EXTENDðx1; x2Þ algorithm attempts to virtually drive

the system from x1 towards x2 by successively verifying

that points along the line connecting x1 and x2 are in Xfree.

It returns the state x as the closest state to x2 such that all

states sampled with precision �x between x1 and x2 are

in Xfree. In traditional motion planning applications,

Fig. 10. Graphical depiction of the RRT search process visualized in R2.

Fig. 9. (a) Convergence of CðxÞ, (b) end-to-end rates, and

(c) probability of success for the four robot trial depicted in Fig. 7.

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174 Proceedings of the IEEE | Vol. 100, No. 1, January 2012
259

verification that x 2 Xfree is based on an algebraic constraint or collision query with a multitude of efficient

methods for doing so [28]–[30]. While the necessary computation to determine x 62 Xobs is typically small, computation of x 62 Xinf requires a solution of the SOCP in

Section IV and can be costly for high-dimensional systems.

Consequently, we store AðxÞ for every node in T and

recompute AðxÞ only when necessary to extend new

states. By relying on the fact that an optimal robust routing

solution AðxÞ will be feasible for neighboring states, it is

often possible to extend x towards x^ without the costly

overhead of numerical optimization.

2) Biased Space Sampling: Random states x^ are chosen to

sample the space X � R2N according to a probability distribution pxðxÞ representing the belief about configuration

x being part of a feasible path SðsÞ. If nothing is known

about SðsÞ, we choose pxðxÞ uniform in the space X. In

general, at least the final configuration is known in that

SðsÞ 2 Xg. We can then bias the distribution by designing

pxðxÞ to choose configurations in Xg with higher probability. Goal biasing improves efficiency of RRT algorithms

by reducing the number of samples necessary to find a

feasible path SðsÞ in the high-dimensional space X � R2N.

In many cases of interest, however, the volume of Xg is

comparable to the volume of X and goal biasing offers little

improvement over uniform sampling. In, for example, the

telepresence application, the goal position of the leader x‘;g

is known, but the positions of the remaining robots are

free. Thus, goal biasing would reduce the exploration cost

along the components associated with x‘ but keep the cost

of exploring the remaining 2ðN � 1Þ dimensions fixed. To

further reduce exploration cost in this case we construct a

prediction X~g � Xg of the final configuration and bias

sampling towards this prediction.

Constructing a final configuration prediction X~g is task

specific. We describe here a method applicable to the telepresence application. To determine the configuration prediction X~g we determine configuration predictions X~i;g for

each robot and compute X~g as the Cartesian product of

these individual sets, i.e., X~g 1⁄4 QN

i1⁄41 X~i;g. Notice that for the

lead robot we can make X~‘;g 1⁄4 fx‘ 2 R2 : kx‘ � x‘;gk G �g.

Observe now that X � R2N is the Cartesian product

X 1⁄4 QN

i1⁄41 Xi of the N decoupled spaces Xi 2 R2 corresponding to each individual robot. If we further assume a

homogeneous team of robots, then all robots operate in the

same space Xi 1⁄4 Y, with a common set of physical obstacles Yobs, and consequently a common free space Yfree 1⁄4

Y n Yobs. It follows that the joint free space Xfree is also a

Cartesian product of N identical sets Yfree minus those

configurations for which a network cannot be established

with sufficient reliability

Xfree 1⁄4 ðYfreeÞ

N n Xinf: (17)

While infeasible network configurations are captured by

Xinf as given in (16), Xfree can otherwise be described by

the free space of individual robots.

To exploit this observation, we first determine an

obstacle-free path � : 1⁄20; s� ! R2 such that �ð0Þ 1⁄4 x0 is

the position of the operating center and �ðsÞ 2 X‘;g. The

obstacle-free path � : 1⁄20; s� ! Yfree is split into N � 1 equal

length segments �k. The ith robot is then assigned to a

segment by the function kðiÞ based on Euclidian distance

to its midpoint. Segments are enlarged to define the region

X~i;g for i 61⁄4 0; ‘. Since this is a heuristic for the goal

configuration, the only requirement on X~i;g is that

�kðiÞ : 1⁄20; s� ! X~i;g. A typical choice is

X~i;g 1⁄4 xi : mins xi � �kðiÞðsÞ �

� �� G d

~

g

n o

where d

~

g is a parameter controlling the enlarged size of

X~i;g. The predicted final configuration is then computed as

the Cartesian product X~g 1⁄4 QN

i1⁄41 X~i;g. The construction of

X~g described above is based on the heuristic that a feasible

goal configuration in an environment with obstacles will

resemble a line-of-sight communication chain. Increasing

the size of X~g with large values of d

~

g limits the implication

of this assumption.

D. Experimental Results

The randomized motion planner is able to find

feasible configurations that allow target servicing at

Fig. 11. The global planning task for five robots in a complex

environment. We depict the series of waypoints that x5 must visit and

the initial conditions for x1 ; x2; x3; x4. The inset images depict our

experimental system deployed in the environment.

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260

positions not attainable with the local control approach

from Section V-A. It can additionally provide a feasible

sequence of configurations to get to the target configuration. To test the global planner on a system with one fixed

access point and five robots, we introduce a sequence of

task potential functions �1ðxÞ; �2ðxÞ; ... ; �MðxÞ that

require the lead node x5 to visit a sequence of positions

while the remaining four robots act as relays to support

end-to-end communication with the access point of

a5;min 1⁄4 0:25. We require that this end-to-end rate be

satisfied with probability � 1⁄4 0:75. The problem is made

more complicated by the introduction of obstacles that not

only block robot motions but also degrade received signal

strength in (7) by 7.6 dBm when line-of-sight is lost.

The global planning algorithm described in Section V-C

is queried to find a feasible path �i : 1⁄20; s� ! R10 for each

task �iðxÞ in order such that �ið0Þ 1⁄4 �i�1ðsÞ and �1ð0Þ 1⁄4

xinit. The trajectory of the lead node x5 for the concatenation of paths � 1⁄4 �1j ... j�M is depicted in Fig. 11. After

solving for a feasible path, x_ desðtÞ is computed so that xðtÞ

follows �.

We conduct an experimental deployment of five robots

controlling the feasible trajectory � for the problem depicted in Fig. 11. In this experiment, each robot is

equipped with a 2.4-GHz Zigbee radio that is used to

broadcast and receive packets in order to measure instantaneous received signal strength that can be used to

estimate the supported communication rate R^ijðtÞ between

node i and j at time t. Using these measurements, in conjunction with the network routing solution AðtÞ, we can

estimate the actual supported end-to-end rate at time t for

each node i, a^iðAðtÞ; xðtÞÞ. The metric for the performance

of our approach is then that each a^iðAðtÞ; xðtÞÞ > ai;min

with probability � across the duration of the trial.

Fig. 12 depicts the predicted a�iðAðtÞ; xðtÞÞ and measured a^iðAðtÞ; xðtÞÞ end-to-end rate of each node for the

duration of the experiment. First, note that over the entire

trajectory produced with our global planning methods, the

prediction of end-to-end rates always satisfies the problem

specification that P1⁄2aiðA; xÞ � ai;min� � �. Second, when

we examine the actual end-to-end rates based on real

measurements of point-to-point wireless channels we see

that it is generally true that a^iðAðtÞ; xðtÞÞ � ai;min. In fact,

if we analyze this constraint across the duration of the

experiment, we find that the fraction of time spent below

Fig. 12. The end-to-end rates of the nodes during the global planning

experiment depicted in Fig. 11. In each plot, the shaded envelope

depicts -ai and the variations that occur with probability � 1⁄4 0:75 based

on ~ai while the dashed black line depicts the instantaneous rate a^i

.

Note that a5;min 1⁄4 0:25 while all other ai;min 1⁄4 0.

Fig. 13. Snapshots of the network configuration during the experimental deployment based on global planning to satisfy the task in Fig. 11.

The end-to-end rate performance is depicted in Fig. 12. (a) t 1⁄4 100 s; (b) t 1⁄4 278 s; (c) t 1⁄4 482 s; (d) t 1⁄4 622 s.

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261

the minimum threshold for each of the instantaneous rates

a^1; a^2; a^3; a^4; and a^5 is 9.2%, 0.8%, 0.3%, 0.6%, and 2.9%,

respectively.

Representative configurations from the experiment are

depicted in Fig. 13. In Fig. 13, at t 1⁄4 100 s, the predicted

goal state X~g assumes the shortest line-of-sight path which

is the left hallway, i.e., a similar result to the reactive

methods in our local control algorithm. As the system

transitions to Fig. 13(b), where the lead node x5 has been

tasked to a waypoint in the right hallway, the prediction for

X~g shifts to a chain of relays going through the right

hallway. This shift in the basic topology of X~g focuses on

exploration of the joint state space so that x4 moves

towards a configuration that will lower the performance of

the network over the short term. As node x5 completes the

desired loop, it utilizes x4 as a relay channel and is able to

maintain the desired end-to-end rate. It is this dramatic

shift in network topology that highlights the advantage of

our global planning approach as we are able to accomplish

continuous end-to-end rate maintenance that would not be

possible with a purely reactive method.

With regards to the running time of the randomized

global planning algorithm, we note that it is difficult to

characterize the performance of randomized search algorithms. One factor is the complexity of Xfree which is determined both by physical obstacles in the environment as

well as the constraints placed on feasible network configurations, i.e., the solution to AðxÞ. Fig. 14 depicts the running time for a benchmark environment as we increase the

dimension of the problem N by adding robots to the system. We also vary the end-to-end rate threshold amin that

must be maintained for the lead node to demonstrate how

more strict network requirements also increase complexity

of the search space.

VI. CONCLUSION

We propose a system architecture that provides end-to-end

connectivity for autonomous teams of robots as they pursue operator-assigned tasks. This architecture is composed

of a cyber component that determines the configuration of

the wireless network and a physical component that handles mobility. Because the performance of point-to-point

wireless links is difficult to predict, we adopt a stochastic

model for supported rates and develop optimal robust solutions to the wireless routing problem. The strengths of

the proposed approach are that: 1) network availability

takes into account specific thresholds on the end-to-end

rates of individual nodes; 2) considering the stochastic

nature of communication channels network availability is

further defined in probabilistic terms; 3) by specifically

searching for robust solutions, we obtain configurations

with spatial diversity and increased probability of success

in the face of difficult to predict communication channels.

Robust routing solutions are integrated with a physical

component that handles mobility either through local

control or global planning. Local controllers converge to

solutions that are local minima of the concurrent mobility

and network routing problem. The global planner performs a randomized biased search in the space of spatial

configurations in an attempt to jointly solve the mobility

and optimal network routing problems. We perform experiments to demonstrate that the global planning algorithm

succeeds in navigating a complex environment while ensuring that end-to-end communication rates meet or exceed prescribed values within a target failure tolerance.

Global planners are able to discover dramatic shifts in

communication topology that allow for continuous communication that is not possible with purely local methods. h

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ABOUT THE AUTHORS

Jonathan Fink (Student Member, IEEE) received

the B.S. degree in electrical and computer systems

engineering from Rensselaer Polytechnic Institute, Troy, NY, 2004 and the Ph.D. degree in electrical and systems engineering from the University

of Pennsylvania, Philadelphia, in 2011.

He is a Research Scientist with the U.S. Army

Research Laboratory, Adelphi, MD. His research

interests include collaboration and planning for

multirobot systems with emphasis on communication modeling and planning for network connectivity.

Dr. Fink has received the best student paper award at the 2009

Robotics: Science and Systems Conference (RSS 2009).

Alejandro Ribeiro (Member, IEEE) received the

B.Sc. degree in electrical engineering from the

Universidad de la Republica Oriental del Uruguay,

Montevideo, Uruguay, in 1998 and the M.Sc. and

Ph.D. degrees in electrical engineering from the

Department of Electrical and Computer Engineering, the University of Minnesota, Minneapolis, in

2005 and 2007, respectively.

Since 2008, he has been an Assistant Professor

at the Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia. From 1998 to 2003, he

was a member of the technical staff at Bellsouth Montevideo. His

research interests lie in the areas of communication, signal processing,

and networking. His current research focuses on wireless communications and networking, distributed signal processing, and wireless sensor

networks.

Dr. Ribeiro is a Fulbright scholar and the recipient of a National

Science Foundation CAREER award in 2010 and best student paper

awards at the 2005 and 2006 International Conference on Acoustics,

Speech, and Signal Processing (ICASSP).

Vijay Kumar (Fellow, IEEE) received the Ph.D.

degree in mechanical engineering from The Ohio

State University, Columbus, in 1987.

He is the UPS Foundation Professor and the

Deputy Dean for Education in the School of Engineering and Applied Science, University of

Pennsylvania, Philadelphia. He has been on the

Faculty in the Department of Mechanical Engineering and Applied Mechanics with a secondary

appointment in the Department of Computer and

Information Science at the University of Pennsylvania since 1987.

Dr. Kumar is a Fellow of the American Society of Mechanical Engineers

(ASME). He has served on the editorial boards of the IEEE TRANSACTIONS ON

ROBOTICS AND AUTOMATION, Journal of Franklin Institute, the IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, ASME Journal of Mechanical

Design, the ASME Journal of Mechanisms and Robotics, and the Springer

Tracts in Advanced Robotics (STAR). He is the recipient of the 1991

National Science Foundation Presidential Young Investigator award, the

Lindback Award for Distinguished Teaching, and the 1997 Freudenstein

Award for significant accomplishments in mechanisms and robotics. He

has won best paper awards at the Distributed Autonomous Robotic

Systems 2002, the International Conference on Robotics and Automation

2004, the International Conference on Robotics and Automation 2008,

the Robotics: Science and Systems 2009, and the Distributed Autonomous Robotic Systems 2010. He is a Distinguished Lecturer in the IEEE

Robotics and Automation Society and an elected member of the Robotics

and Automation Society Administrative Committee.

Fink et al.: Robust Control for Mobility and Wireless Communication in Cyber–Physical Systems

178 Proceedings of the IEEE | Vol. 100, No. 1, January 2012
263

Biographies of ARL Authors

Jin-Hee Cho joined ARL in 2009 as a computer scientist with a research focus in developing network protocols for trustworthy 

tactical cyber space using multi-disciplinary approaches and analyzing their performance and security. Dr. Cho actively 

collaborates on trust research with ARL’s partners on the Network Science Collaborative Technology Alliance (NS CTA) and 

International Technology Alliance (ITA), in addition to scientists in ARL’s Computational and Information Sciences Directorate 

(CISD) and Sensors and Electron Devices Directorate (SEDD). Wider research interests include wireless mobile networks, sensor 

networks, delay tolerant networks, secure group communications, key management, intrusion detection, trust management, 

social/cognitive networks, market-based economic modeling, game theory, and resource allocation. Dr. Cho earned a B.A. from 

Ewha Womans University, Seoul, Korea and a Masters degree in social work from Washington University in St. Louis, Missouri, 

before earning a M.S. and Ph.D. in Computer Science from Virginia Polytechnic Institute and State University. She has published 

over 40 technical articles in the areas of networking and telecommunications, and received a best paper award in 2009 at the 

Institute of Electrical and Electronics Engineers (IEEE)/International Federation for Information Processing (IFIP) International 

Symposium on Trusted Computing and Communications (TrustCom09), Vancouver, Canada. She is a member of the IEEE and 

the Association for Computing Machinery.

Amy M. Dagro is a mechanical engineer in the Soldier Protection Sciences Branch of the Weapons and Materials Research 

Directorate. Ms. Dagro earned her B.S. in biomedical engineering from Johns Hopkins University and her M.S. in mechanical 

engineering from Columbia University. She began her research in computational biomechanics with Dr. Reuben Kraft in 2010, 

and became an ARL civilian employee in 2011. Ms. Dagro leads ARL’s research in blast and ballistic effects to the brain.

Jonathan Fink is a researcher in the Asset Control and Behavior Branch, Information Sciences Division of the Computational and 

Information Sciences Directorate (CISD). He received his Ph.D. from the University of Pennsylvania in 2011, where he worked 

in the General Robotics, Automation, Sensing and Perception (GRASP) Laboratory on communication-constrained planning for 

teams of networked robots. Since joining ARL, he has contributed to the development and experimental testing of fundamental 

capabilities for autonomous ground robots including techniques for mapping and navigation in militarily relevant environments. 

Dr. Fink has also continued his graduate research into multi-agent planning that takes place at the intersection of networking 

and robotics through collaborations with the Network Sciences Division of CISD. He is a member of the Institute of Electrical 

and Electronics Engineers and has published over 20 technical articles in the field of robotics and autonomous systems.

Piotr Franaszczuk joined ARL in July 2011 as a Senior Research Scientist in Neuroscience, with interests focusing on 

computational methods of signal analysis, neural modeling, and development of large-scale biologically realistic computational 

models of brain structures. He received his M.S. and Ph.D. in physics from the University of Warsaw in 1978 and 1988, 

respectively; his doctoral thesis received an award from the Polish Ministry of Science and Education. Dr. Franaszczuk was a 

Bennett Fellow in neurophysiology and a faculty member at the University of Maryland School of Medicine. He also taught in 

the Department of Biomedical Physics, University of Warsaw. Since 1999, Dr. Franaszczuk has been a faculty member in the 

Department of Neurology at the Johns Hopkins University School of Medicine; until June 2011, he co-directed their Epilepsy 

Research Laboratory. Dr. Franaszczuk has authored or co-authored over 100 journal and conference papers and 12 book 

chapters. He has been a reviewer for over 10 scientific journals and, since 2005, has served as grants reviewer on several 

National Institutes of Health scientific review panels.

Russell Harmon is Director of the International Research Office of the U.S. Army Engineer Research and Development Center 

(ERDC-IRO), where he conducts international science and technology engagement. Prior to joining ERDC-IRO in September 

2011, Dr. Harmon was a Program Manager at the Research Development and Engineering Command-ARL Army Research 

Office, where he managed the extramural basic research program in terrestrial sciences. A geochemist who has worked at 

the NASA Manned Spacecraft Center, the Scottish Universities Research and Reactor Center, and UK Natural Environment 

Research Council Isotope Geosciences Laboratory, Dr. Harmon has held faculty positions at Michigan State University; Southern 

Methodist University; and North Carolina State University, where he is currently an adjunct faculty member in the Marine, Earth, 

and Atmospheric Sciences Department. He is a Fellow of the Geological Society of America, National Speleological Society, 

and ARL. Dr. Harmon has authored or coauthored more than 200 peer-reviewed publications and remains actively engaged 

in geochemical and geospatial research. He holds a B.A. from the University of Texas, a M.S. from the Pennsylvania State 

University, and a Ph.D. from McMaster University.
264

Kimberly A. Pollard is a research biologist with the Perceptual Sciences Branch of the Human Research and Engineering 

Directorate. She earned a B.A. in biology from Rice University and a Ph.D. in biology from UCLA; her dissertation focused on 

acoustic communication and social complexity in animal behavior. After an Oak Ridge Associated Universities postdoctoral 

fellowship at ARL from 2009-2012, Dr. Pollard joined ARL as a civilian employee. She currently studies nontraditional forms of 

communication and perception, including bone-conducted hearing, bone conduction and vibrotactile communication devices, 

and the role of olfaction in warfare.

Ananthram Swami is the Army’s Senior Research Scientist (ST) for Network Science, with interests in the study of interactions 

and co-evolution; prediction; and control of inter-dependent networks including mobile ad hoc wireless networks, sensor 

networks, and social networks. Dr. Swami earned a B.Tech. from IIT-Bombay, a M.S. from Rice University, and a Ph.D. from the 

University of Southern California, all in electrical engineering. Dr. Swami has published more than 300 journal and conference 

papers in the areas of signal processing, communication networks, and network science; holds three patents; developed a 

Matlab toolbox on non-Gaussian signal processing; and has co-led multiple tutorials on cognitive networking. He is a Fellow 

of the Institute of Electrical and Electronics Engineers and an ARL Fellow. His research is currently supported by the Defense 

Threat Reduction Agency and the Office of the Secretary of Defense. 

Don Torrieri is an ARL engineer with primary interests in wireless communication systems, spread spectrum, adaptive arrays, and 

signal processing. He received M.S. and Ph.D. degrees in physics and electrical engineering, respectively, from the University of 

Maryland; a M.S. in electrophysics from the Polytechnic University; and a B.S. in electrical engineering from the Massachusetts 

Institute of Technology. Dr. Torrieri analyzed electronic systems at the Naval Research Laboratory, where he received three 

Group Achievement awards and three Research Publication awards. Since 1977, he has worked for the Department of the 

Army, where he spent sixteen years doing classified work for the Survivability Management Office and its predecessors. He 

wrote many blue-ribbon committee and red-team reports on Army communication systems, and is the author of many books, 

journal articles, conference papers, and technical and classified reports. He taught graduate courses at both Johns Hopkins 

University and George Washington University. His awards include the Military Communications Technical Achievement Award 

and the Best Paper Award of the Institute of Electrical and Electronics Engineers Military Communications Conference. In 1989, 

he was selected as one of the original Fellows of the Army Laboratory Command, predecessor to the Army Research Laboratory.

Bruce J. West is Chief Scientist Mathematics (ST) of the U.S. Army Research Office, Information Science Directorate. His research 

focuses on the development of mathematical tools necessary to understand complex phenomena as they apply to the nascent 

discipline of Network Science. Since 2006, he has led the development of the ARL External Nonlinear Science program. Dr. West 

graduated with a Ph.D. in nuclear physics from the University of Rochester, was a founding member of the La Jolla Institute, and 

served as Chair and Professor of Physics at the University of North Texas. He earned the Department of the Army’s Research 

and Development Achievement Award for Technical Excellence, ARL Publication Award, Commander’s Award for Civilian Service, 

and the prestigious Wilks Award in statistics. He has been recognized by Cambridge Who’s Who as one of the top 101 Industry 

Experts and has twice received their Professional of the Year Award in Applied Physical Science and Mathematics. In 2012, Dr. 

West received the President’s Meritorious Rank Award. He has produced more than 350 publications, including 13 books, and 

his published research has received more than 12,000 citations with an h-factor of 53. Dr. West is a Fellow of the American 

Physical Society, the American Association for the Advancement of Science, and ARL.
Approved for public release; distribution unlimited • March 2013