An Army-Centric

System of Systems Analysis (SoSA) Definition

by Jeffrey A. Smith, Jayashree Harikumar, and Brian G. Ruth

ARL-TR-5446 February 2011

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Army Research Laboratory

White Sands Missile Range, NM 88002-5513

ARL-TR-5446 February 2011

An Army-Centric

System of Systems Analysis (SoSA) Definition

Jeffrey A. Smith and Brian G. Ruth

Survivability/Lethality Analysis Directorate, ARL

Jayashree Harikumar

New Mexico State University Physical Science Laboratory 

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14. ABSTRACT

Over the course of the last several years there has been much discussion of a concept called the “system of systems” (SoS), and 

many proposed definitions that capture features of various SoS. These numerous definitions offer the means to identify SoS; 

yet, from an analytical point of view, we feel they do not offer the clarity required to construct a well grounded framework to 

analyze these SoS. Over the course of our research to develop such a framework, we realized that there is another perspective 

from which to view the notion of a SoS. This perspective, rooted in the analysis of a military SoS, explicitly recognizes three 

existential concepts present in any military SoS that are also likely present in non-military SoS. The purpose of this report is to 

highlight these concepts, and relate them to the base of established literature regarding SoS.

15. SUBJECT TERMS

system of systems; survivability; lethality; vulnerability; assessment; analysis; complex systems

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Jeffrey A. Smith

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iii

Contents

List of Figures iv

1. Introduction 1

2. Principal Characteristics for System of Systems Definition 2

3. A Novel Perspective on SoS 3

3.1 Nested Concepts and Inter-Related Purposes..................................................................4

3.2 Dynamic and Evolving Processes ...................................................................................5

3.2.1 State/Phase Space and World Lines....................................................................7

3.2.2 Homeostasis and Homeodynamics....................................................................14

3.3 Sociotechnical Systems Theory.....................................................................................21

3.3.1 Multiscale Analysis of Complex Systems.........................................................24

4. A Concept of SoS Analysis 26

5. Conclusions 26

6. References 28

List of Symbols, Abbreviations, and Acronyms 31

Distribution List 32
iv

List of Figures

Figure 1. Phase space for a notional flying insect. .........................................................................9

Figure 2. Dynamic response for the notional flying insect within an operational context. 

(a) Notional time series data presenting insect displacement as a function of sampling 

time. (b) The same dataset displaying displacement at time t versus displacement at time 

t + t.........................................................................................................................................11

Figure 3. World line, world sheet, and world volume associated with the time-evolved states 

of an object with different degrees of positional uncertainty. ................................................13

Figure 4. Homeostatic process within a complex adaptive system. .............................................16

Figure 5. Homeodynamic process within a complex adaptive system. ........................................19

Figure 6. Range of homeodynamic efficiency (RHE) for a general organizational SoS..............21

Figure 7. The NATO C2 conceptual model..................................................................................24
1

1. Introduction

Over the course of the last several years, there has been much discussion of a concept called the 

“system of systems” (SoS), and many proposed definitions that capture features of various SoS. 

These numerous definitions offer the means to identify systems of systems; yet, from an 

analytical point of view, we feel they do not offer the clarity required to construct a well 

grounded framework to analyze these SoS. We believe that there is another perspective from 

which we can view the notion of a SoS. This perspective is rooted in the analysis of a military 

SoS. We hypothesize that there exists three basic concepts in both military SoS and non-military 

SoS. These concepts are (1) nested concepts and inter-related purposes, (2) a notion of 

dynamical processes, and (3) identification of sociotechnical concepts in all SoS. The purpose of 

this paper is to highlight these concepts, and relate them to the base of established literature 

regarding SoS.

Information is recognized as power. Technology is used to gather and disseminate information 

in the U.S. Army. We need to assess the technology and information gathered for army 

personnel. In addition, we cannot assess what technology can give us if we do not have a good 

understanding of the system(s) in which the technology will be used. Specifically we need to do 

a SoS analysis (SoSA). To do analysis we want to develop an analysis methodology. Issues that 

confront us before we can develop an analysis methodology are:

1. Multiple definitions of SoS.

2. What behavior or property of a system characterizes it as a SoS?

3. Multiple sensors (technology) gather information. How do they work together? Do they 

interfere with each other?

4. Do SoS differ with specific Commander‟s intent?

5. Emerging behavior from systems (sensors and personnel) that operate in differing 

scenarios.

6. Interaction of the emergent behavior of the systems in a SoS.

7. Impact of collaborative technologies in a SoS.

In the following sections, we discuss the principal characteristics for SoS definitions, as found in 

literature, and the three existential concepts we believe is present in both military and nonmilitary SoS. We introduce these three existential concepts to highlight that SoS has special 

characteristics that limit the potentiality of multi-methodologies, especially in the model 

formulation and system design phases (1). We hypothesize that for SoS analysis, two basic 

model types are needed: (1) a model to abstract the basic structure and behavior of SoS, and 
2

(2) a model to combine the data with simulation for guiding decisions (2). The proposed concept 

of SoSA is that if one accepts the above model formulation then it implies that the output of a 

SoS model will be probabilities of possibilities.

2. Principal Characteristics for System of Systems Definition

Literature review (3, 4) provides many definitions for SoS. A common thread among these 

definitions is five principal characteristics that most authors think a SoS should possess. These 

characteristics were first identified by Maier (5, 6) and in this section we will address these five 

principal characteristics; and the relevance and associated debate over the interpretation of these 

characteristics.

1. Operational Independence of the elements: This principal requires the component systems 

in a SoS to be independently operational. A SoS is built to fulfill specific purpose(s). In 

such a system the normal operational mode of the component systems can be subordinated 

to the central (desired) operational mode of the system. Alternately, when component 

systems are more collaborative (less directed) the central operational mode of the system 

may be compromised.

2. Managerial Independence of the elements: In addition to being independently operational, 

this principal states that component systems may be acquired and operated independently 

and can maintain an operational existence independent to the SoS. In addition to the 

constraints imposed by principal 1, evolution of the component systems places constraints 

on acquisition and operation of the component systems and in turn the management and 

operation of the SoS.

3. Evolutionary Development: The SoS is composed of systems that are bought together to 

satisfy a higher task/purpose. The existence of SoS can be evolutionary as tasks and 

purposes are added, removed or modified with usage. This principal states that a SoS does 

not appear fully formed. This implies that as the SoS evolves it is likely that the overall 

system „value‟ is no longer approximately equal to the sum of the component system 

„value.‟ The SoS could now provide a greater functionality than its component systems or 

the interactions between the component systems could reduce the combined capability of 

the overall system. A system value is determined from the desired system attributes. 

However as the overall system evolves the component system attributes that combine to 

generate the SoS utility impact SoS costs.

4. Emergent Behavior: The behavior of a SoS is not the behavior of any one component 

system. The collective behavior of the interacting systems in a SoS defines the behavior of 

the SoS. The principal purposes of the SoS are fulfilled by these system behaviors.

However, when component system attributes interact to produce the desired emergent 
3

attribute of the overall system, it is possible that some undesired emergent behavior be 

observed. This impacts system cost, design and „knowledge‟ of the system. 

5. Geographic Distribution: The geographic extent of the component systems can be large. 

The only constraint on the geographic extent is that the components can readily exchange 

information but not mass nor energy. Component systems adapt to changing environment 

and when systems are geographically distributed there is no common thread to predict the 

emergent system behavior. In addition, for a military SoS the geological extent of the 

deployed SoS is defined by purpose, available materiel and soldier skills vice the other way 

around.

In a review of literature (7), most authors agree on the five principal characteristics, explicitly 

defined by Maier, for describing SoS. A military SoS also exhibits many of the Maier defined 

characteristics either by design consequence or interaction with its environs. A network enabled 

military SoS is driven by purpose. It has to continuously evolve in order to survive and fulfill its 

mission. At the same time, each system in a military SoS has to maintain its operational and 

managerial independence in order to fulfill its sub tasks and the overall system mission. One 

observes a continuous tradeoff between independent and interdependent behavior in a military 

SoS brought about by conventional force-on-force type of engagement and effects based 

asymmetric (different complexity at different scale) warfare.

A military system is not just about its purpose but also about its structure and how it has been put 

together by whom and why. Military systems evolve and the change occurs at both the 

component system level and the SoS level. System scalability is affected when a system change 

occurs. A change in system scalability impacts system cost. A SoS put together for a certain 

purpose may be scalable to handle extra work but the initial overhead and the cost to maintain 

the updated system has to be considered. System scalability factors, such as number and type of 

component systems, play an important role when a system is not used for the scale it has been 

modeled to. In addition, a military system (SoS) may be perfect for a certain scenario but 

relatively useless in another scenario. No system can be diverse enough to respond to every 

changing uncertainty and only the known specifics of the problem can define the SoS and the 

constraints in which the system can be used. Adaptation of a military system to its purpose, 

available materiel and Soldier skill is critical, and this implies that scalability is also a sufficient 

and necessary defining characteristic of a military SoS.

3. A Novel Perspective on SoS

In the military domain, the SoS concept has been useful in describing the complexity of the 

network enabled/connected battlefield operating systems. The Department of the Army (8) relies 

on the SoS concept for the Future Combat System (FCS) strategy (9). However, these numerous 
4

definitions, from an analytical point of view, do not offer the clarity required to construct a well 

grounded framework with which to analyze these SoS. Bjelkemyr et al. (10) pose questions for 

SoSA. They suggest that the SoS elements must first be clearly understood in addition to the 

links that tie the component systems together. Analysts attempting to develop metrics to address 

SoSA must address three components: (1) what are the drivers for transformation in the SoS? 

(2) is there a measure to describe the state of well being in the SoS? and (3) how can this state of 

well being be assessed in the SoS?

In this section, we offer a novel perspective from which to view the SoS. We believe that the 

SoS lexicon as defined in literature (2, 5, 6, 11, 12) does not apply (is not complete) in the 

military domain. In a military domain there are interrelationships and consequent emerging 

behaviors between the systems in a SoS. Consequently, analysis of a SoS in a military domain 

needs to account for the ever changing and evolving systems that define the SoS. In addition, we 

believe that the mission drives the SoS and therefore a generalized treatment of the emergence 

and evolution for all types of SoS is not possible. In a military SoS, emphasis is given to 

decision makers and assets because as stated by Dahmann (13), decision making and its 

consequences are central to the function of the military SoS.

Based on literature search and discussions therein, we believe in order to develop an analytical 

framework with which to analyze a military SoS, we need to recognize the three existential 

concepts present in any military SoS. The concepts are (1) nested concepts and inter-related 

purposes, (2) dynamic behavior and evolving processes, and (3) sociotechnical system concepts.

3.1 Nested Concepts and Inter-Related Purposes

The military SoS does not exist in vacuum. It has a mission to perform that is unique to that 

time, place, and the collective involved. The mission or purpose drives the functioning of the 

military SoS. Purpose gives the warfighter a reason to perform his task(s). The commander 

communicates his mental image of the purpose, decomposed in time, to his subordinates and 

peers via inter-related and nested concepts. An important concept to remember is that nested 

concepts do not control but just constrain the subordinate commander‟s action (14), in order to 

maintain a unity of effort.

In order to make decisions, a commander uses both the current situational context and a mental 

image of the situation projected into the future. A commander then needs to communicate this 

mental image to the subordinates so that they can focus their efforts/actions to a shared common 

goal. A commander‟s vision includes the subordinates‟ requirements and responsibilities while 

providing the purpose, and the desired end state for that military unit. In current Army doctrine, 

commander‟s intent is acknowledged as the most important part of an operations order. A nested 

concept is the vehicle to communicate this mental image throughout the organization. It defines 

the commander‟s relationship with superiors, peers, and subordinates. FM 3-0 (15) and other 

literature (16–20) defines nested concepts as a “concept whereby each succeeding echelon‟s 

concept is nested in the other.” At each level of command, commanders provide intents and 
5

concepts to unite their subordinates and peers in a coordinated effort to successfully complete a 

mission. By nesting concepts and thus purposes for subordinates the commander achieves unity 

of effort in the unit. As the subordinate units within the force get smaller and more spatially 

dispersed, this shared mental image grows in importance and remains essential to maintain unity 

of effort and a cohesive organization. Neither nested concepts nor inter-related purposes tell the 

warfighter how to get from the current state to the desired end state. The concept of operations 

provides this “bridge of action” to subordinates.

The concept of operations is a statement that directs the manner in which subordinate units 

cooperate to accomplish the mission and establishes the sequence of actions the force will use to 

achieve the end state. It defines tasks as assigned to specific units, and the interdependence of 

the former as executed by the latter. The value of concept of operations is that it establishes the 

subordinate‟s relationships with the commander‟s unifying image and informs them of their 

responsibility to attain the common goal.

Ackoff and Emery (18) characterize human systems as purposeful systems whose members are 

also purposeful individuals that intentionally and collectively formulate objectives and are 

component parts of larger purposeful systems. In a military SoS, the above characterization of a 

purposeful system implies that we can consider the commander and the subordinates a 

purposeful system. For Ackoff and Emery (18), a purposeful system or individual is idealseeking if it intentionally chooses another objective that more closely approximates its ideal. An 

ideal-seeking system or individual is necessarily one that is purposeful, but not all purposeful 

entities seek ideals. The capability of seeking ideals may well be a characteristic that 

distinguishes humans from anything that they can make, including computers.

SoS are created with the intention that its emergent behavior realizes an intended mission; thus, 

they become more than the sum of its parts. Yet, when they are improperly taken apart for 

analysis, they lose this important property. In an analytical approach (to study any problem), 

explanation of the whole is deduced from the explanation of the parts. In contrast, one could use 

the synthetic approach where a part's functionality is explained in terms of its role in the larger 

system. For military SoS, we suggest the use of nested concepts and inter-related purposes, in 

conjunction with a commander‟s intent and concept of operation to decompose the SoS for 

analysis, and then properly integrate the results into a holistic analysis of the SoS.

3.2 Dynamic and Evolving Processes

The notion of dynamic processes is a critical contributing element when considering what makes 

up a military SoS. The dynamic interplay between complementary mission-supporting purposes 

as SoS component systems operationally respond to uncertainty and change (both internal and 

external) ultimately determines success or failure in achieving the intended mission objective. 

Tracking the evolving SoS trajectory through an associated operational state space (as a function 

of time) serves to illustrate this purpose-driven dynamic. But the question remains: how do we 

characterize this type of dynamic process in order to analyze it?
6

To properly answer this question, we need to identify the common characteristics among the 

decision-making processes (DMP) that manifest across a variety of SoS examples. These 

characteristics include the following (20).

• The utility function of the decision maker utilizing a SoS DMP may be difficult to quantify. 

• The relative utility of a specific decision option to each component system within the SoS 

may greatly vary.

• The utility function for a SoS (or SoS subunit) course of action (COA) may be a nonlinear 

composite of utility functions from individual constituent systems.

• The SoS COA utility function is generally context sensitive. As the environmental context 

evolves over time, so too does utility function associated with that COA.

These common characteristics imply both a natural time component of SoS decision-making and 

also a natural uncertainty associated with the information under which decisions are made. 

Regarding the latter, we can logically ascertain the following assumptions.

• Decisions are made with the information available to the decision maker at the time of the 

decision.

• The quality of the decision is only known after the realization. A correct decision could 

result in an undesired outcome but it is still the best decision.

From these observations, we can conclude that the inherent complexity within SoS DMPs tends 

to render mission outcome prediction based solely on existing a priori knowledge of a situation 

very challenging (if not often impossible).

When collectively examined, the common characteristics of SoS-related DMPs cited in the 

previous paragraph tend to suggest several different perspectives from which to analyze SoS 

dynamic processes. The first of these perspectives is the concept of synchronous behaviors, 

which are characteristics, shared among different SoS variants that range from physical, 

biological, to social systems. These behavioral types include the processes of how people reach 

consensus in group interactions and decision making, as well as collective behaviors that exhibit 

data cluster-of-clusters type of phenomena. Coordinated synchronous behaviors demonstrated 

within a SoS require the pre-existence of a durable communication network connecting the 

various constituent component systems making up the SoS.

A second perspective from which to analyze dynamic processes within a SoS are the 

complementary concepts of non-stationary and non-ergodic behaviors. A discrete time series

x(t1), x(t2), x(t3), ... (where x(ti) is the scalar or vector state of quantity x measured/sampled at the 

i

th time step ti) is non-stationary if, for some m, the joint probability distribution of x(ti), x(t i + 1), 

x(t i + 2), ..., x(t i + m - 1) is dependent upon the value of ti (21). Similarly, a discrete time series 
7

x(t1), x(t2), x(t3), ... represents a non-ergodic process if 

( )

i

x t

, the time-averaged value of the 

process, where

 

 

N

x t

x t

N

i

i

N

i

 





1

lim

is not equal to

( )

i

x t

, the ensemble average of

( )

i

x t

(i.e., the value of x sampled at the same time 

ti over the ensemble of all possible realizations of the process), where

 

  

M

x t

x t

M

j

i j

M

i

 





1

lim

and

  i j

x(t )

is the j

th realization of 

( )

i

x t

. Thus, non-ergodic processes usually reflect dynamical 

systems far from an equilibrium state (22). A SoS will often exhibit non-stationary and nonergodic behavior in large time due to nonlinearity of the composite collective system.

A third analytic perspective from which to address dynamic processes is the notion of multiple 

time scales. SoS often exhibit important behaviors that are demonstrated over multiple time 

scales, including

• a fast time scale, 

• a slow time scale, and 

• a large time scale.

These differentiated time scales generally scale proportionally with various spatial scales within 

the SoS.

• Fine-grained/microscale functions evolve according to a fast time scale.

• Coarser-grained/mesoscale functions evolve at a slower time scale.

• Finally, collective/macroscale functions evolve according to a large time scale 

characteristic of the SoS as a composite whole.

By integrating the concepts of synchronous, non-stationary, and non-ergodic behaviors and 

multiple time scales into a single analytic meta-concept with multiple perspectives, the SoS 

analyst is provided with an insightful “lens” for the study of collective dynamic and evolving 

processes exhibited by a military SoS.

3.2.1 State/Phase Space and World Lines

In the previous section of this report, we discussed the dynamical processes that tend to 

operationally manifest within military purpose-driven SoS. Generally, these processes can be 

quantitatively captured and recorded as time-evolving metrics that, over time, generate 
8

trajectories within an associated operational state space. We can thus characterize and analyze 

the emergent behavior generated by a SoS by examining these related state space histories 

(representing the dynamical evolution of a SoS collectively or of individual component systems 

as a function of time). In this section, we introduce the necessary foundational concepts that 

provide a means to realize SoS state space based analysis.

We start out with the most basic concept of all: the phase space, which is a multi-dimensional 

coordinate system whose axes represent the measurable variables of a particular system. For a 

dynamical system, the phase space must be capable of representing all possible states that the 

system might occupy as a function of time, where each possible state of the system corresponds 

to one unique point in the phase space. Generally, a phase space is finite-dimensional, consisting 

of an infinite number of points forming a smooth manifold. The phase space axes – where each 

axis reflects a specific degree of freedom (i.e., physical, functional, cognitive, operational) –

must span the configuration state space of the system under analysis. An instantaneous state of 

the system is thus represented in phase space by a point.

Figure 1 illustrates an example of a simple phase space formed by the time-dependent variables 

displacement (from a reference location) d(t), velocity v(t), and acceleration a(t) describing the 

motion of a notional insect with the capability of flight (e.g., a common house-fly), where t is the 

time at which the variables were sampled. When plotting a set of system variables in phase 

space, there is an underlying assumption that the variables are all sampled at the same rate. If 

they are not, it is difficult to make an interpretation. In order to properly record a set of system 

variables within the respective phase space, one must think of the variables sampled at each time 

step as the coordinates for a point in the space. For the example displayed in figure 1, this 

sampling process produces a set of discrete points scattered across a small region of the phase 

space (indicating the abruptly-shifting flight of a flying insect). These points describe the phase 

space trajectory: the path through a phase space traced by a dynamical system as a function of 

time; measured as a time-ordered sequence of system states. As the system evolves over time, a 

succession of such states is produced, giving rise to the phase space trajectory that serves to 

graphically capture the time evolution of the system. For purposes of informative visualization, 

the trajectory points displayed in figure 1 are shown to fade as a function of increasing age (thus 

conveying an implied graphical description of the flying insect as a function of increasing time), 

where a forced perspective depicts larger data points as being closer to the viewer. Here, the 

phase space trajectory portrays the insect as initially staying close to the d(t) = 0 reference 

location, then flying out with increased velocity and acceleration later in time.
9

Figure 1. Phase space for a notional flying insect.

Once we have established the concept of a phase space for graphically conveying the time 

history of the state of a dynamical system (or sub-state for high-dimensional systems such as 

SoS), we can next extend the formalism by providing one (or more) means to identify phase 

space patterns indicating emergent behavior on the part of the system. One such means is the 

concept of a phase space attractor: a set of states (i.e., points within a phase space) towards 

which a dynamical system evolves (regardless of initial conditions) as a function of time. In a 

dissipative dynamical system, as time progresses, the phase space trajectory will tend towards a 

mathematical limit. We call this mathematical limit the “attractor:” an equilibrium location in 

the system phase space that describes a time-independent (i.e., stationary) situation. This yields 

a phase space point (or point cluster) towards which all possible system histories tend to 

converge.

From a practical point of view, the observation of a real dynamical process demonstrated by a 

system or SoS typically does not yield all relevant associated phase space variables. Either not 

all of these state variables are known to the analyst or not all of them can be easily measured. In 

many non-laboratory real-world situations, it is often the case where only one system/SoS state 

variable x(t) can be reliably sampled and measured as a function of time. It is possible to 

reconstruct an equivalent system phase space trajectory that preserves the topological structure of 

the original multi-dimensional phase space (i.e., the phase space attractor) via the dynamic 

measurement of only one scalar state variable via the use of time-delay embedding (23). In this 

technique, a sampled time series x(t0), x(t1), x(t2), x(t3),..., x(ti),..., x(tN) based upon timeinterval-consistent measurements of a dynamical system is used to “reconstruct” the associated 

multi-dimensional phase space trajectory 

St



via the time-delayed approximation

              

T

i delay i i i i i S t  S t  x t , x t  t , x t  2t ,, x t  m 1 t

 

,
10

where 

 i S t



is the actual system state at sample time ti,   delay i S t



is the time-delayed system state 

approximation at sample time ti, m is the embedding dimension, and t is the time delay interval. 

The preservation of the original phase space topological structure is guaranteed provided m  2d

+ 1, where d is the dimension of the attractor. Recursive application of the above approximation 

thus yields the phase space trajectory

          S t  Sdelay t

1

,Sdelay t

2

,Sdelay t

3

, ,SdelaytN m1 t





   

,

where N is the length of the original sampled time series. In this fashion, we can visually 

reconstruct the topology of system/SoS attractors from the original multi-dimensional phase 

space.

To illustrate the utility of time-delay embedding for the purpose of attractor reconstruction, we 

return to the example phase space for the notional flying insect depicted in figure 1. Suppose 

that the insect displacement coordinate is measured (under controlled conditions) once every 

5 seconds (s) for a total time interval of 500 s (i.e., N = 100). Let us further suppose that a food 

sample exhibiting an appealing olfactory signal receivable by the insect via its antennae is 

positioned at a displacement 5.5 m from the insect‟s reference location. Assuming these 

operational conditions, figure 2 depicts a notional time history of the insect using two different 

data presentation formats. In figure 2(a), a set of purely notional time series data is presented 

that purports to measure insect displacement as a function of sampling time. From these data, 

the insect can be observed to progressively explore (in a semi-random fashion) regions of its 

environment further and further removed from the d = 0 reference location until settling around a 

displacement very close to 5.5 m. In figure 2(b), the same time series dataset is used to 

reconstruct the insect‟s phase space trajectory via time-delay embedding, where embedding 

dimension m = 2 and time delay t = 5 s. Within this two-dimensional phase space 

reconstruction, one can readily and easily observe the attractor (i.e., the point cluster surrounding 

d = 5.5 m) indicating a region of the environment that the insect seems to purposely adhere to 

(probably seeking nourishment from the food at that location)*. Although almost trivially

simple, this example serves to illustrate how phase space attractors can effectively convey the 

possible presence of purposeful system behavior within an operational context.

 

*

It must be noted that, in the current context, this displacement could actually indicate any point on the circle defined by 

d = 5.5 m. However, for the purpose of this example, we assume that the attractor center-of-mass displacement is identical to the

location of the insect‟s food. In a more realistic application of this analytic approach, lat/long coordinates would probably be 

used in place of scalar displacement (thus increasing the value of embedding dimension m for phase space reconstruction).
11

Figure 2. Dynamic response for the notional flying insect within an operational context. (a) Notional time series 

data presenting insect displacement as a function of sampling time. (b) The same dataset displaying 

displacement at time t versus displacement at time t + t.

Given an understanding of the nature of phase space attractors, the inquisitive analyst might next 

seek to explore exactly how a system or SoS might dynamically evolve towards such an attractor 

state as a function of phase space topology. This leads us to the related concept of a basin of 
12

attraction: the set of points within a system-specific phase space such that any initial conditions 

chosen in this set will dynamically evolve the system state towards a particular attractor. By 

allowing the intrinsic nonlinearity of a complex system (such as a SoS) to be manifested within 

an operational regime requiring detailed functional balance (to assure an achievement of 

purpose), the resulting systemic non-equilibrium can also lead to the coexistence of multiple 

attractors in phase space. This space can then be carved up into a set of attractor basins, where 

each basin corresponds to the set of states that, if the system were to start from there, would 

evolve to a particular attractor. The coexistence of multiple attractors and related basins of 

attraction within a phase space constitutes the natural mode of complex systems capable of 

demonstrating adaptive behavior and of performing regulatory tasks. We would thus expect to 

see the system staying within one basin of attraction (corresponding to resistance to change) and 

then at some point possibly switching between different attractors (corresponding to a change in 

the long-term mode of behavior) as the system strives to adapt in support of a purpose (or set of 

inter-related purposes). The existence of low-dimensional attractors associated with complex 

systems (e.g., the point cluster depicted in figure 2(b)) suggests the possibility of higher 

dimensional attracting structures within the phase space. Within the context of a military SoS, 

the challenge is to effectively exercise purposeful DMPs so that the phase space end-state 

location reflects the convergence of “Commander‟s intent” with an attractor, where all SoS 

initial conditions lie within the basin of attraction associated with this attractor. This would thus 

effectively demonstrate a purposeful navigation of the phase space by the SoS to achieve a 

desired goal (or set of goals).

When the constituent components within a dynamical SoS are exposed to some type of 

perturbative influence arising from the operational environment (e.g., noise or damage leading to 

functional degradation), the SoS is often driven away from whatever stable mode of operation it 

exercised prior to the perturbation. To compensate, the SoS will sometimes attempt to reorganize or re-structure itself without explicit pressure or involvement from other entities 

external to the SoS. This is the essence of the concept of self-organization: a process in which 

the internal organization of a system (typically an open system) increases in complexity without 

being guided or managed by an outside source (24). This change in organizational complexity 

can evolve in either time or space, maintain a stable form, or exhibit transient structural 

phenomena. During the self-organizational process, the SoS is purposefully self-motivated to 

explore its associated phase space in search of new attractors that arise when a system is driven 

away from its prior state of equilibrium. These new attractors exist within the basins of 

attraction of the SoS and are generally dynamically unstable, thus motivating the SoS to move 

purposefully along a phase space trajectory to a new attractor in pursuit of a return to operational 

equilibrium. Such a new attractor then constitutes the self-organized state of the SoS.

Once the inter-related concepts of phase space, phase space trajectories, attractors, and basins of 

attractions have been introduced, we can next focus our attention on the specific analysis of 

instances of complex systems within an operational context. This is accomplished via utilization 
13

of the system‟s world line: the phase space history of a system, synonymous with the phase 

space trajectory resultant from a specific dynamical system instance either as measured within 

the real physical world or via simulation. In general, a “world line” is the sequential path of 

measurable events (with time and state variables as dimensions) that marks the history of a 

system as it moves forward through time. In other words, a world line is a curve in phase space 

that traces out the (time) history of an object. One usually takes the proper time of an object or 

an observer as the curve parameter t as measured along the world line. Finally, given the 

inherent uncertainty often present upon measuring the state variables describing a time-evolving 

complex system, the concept of a world line may be extended into:

• a “world sheet” reflecting uncertainty within a single measured state variable, and

• a “world volume” reflecting similar uncertainty within two (or more) state variables. 

For example, the world sheet and world volume associated with a system exhibiting positional 

uncertainty is depicted in figure 3.

Figure 3. World line, world sheet, and world volume associated with 

the time-evolved states of an object with different degrees of 

positional uncertainty.

The following are several examples of system-specific world lines.

The time series data plotted in figure 2(a) presents a two-dimensional section of the world line 

for the notional food-seeking insect from our earlier example. Here, the insect‟s displacement 
14

d(t) is but one of three recordable metrics describing the dynamic status of the measured subject 

entity (the other metrics being velocity v(t) and acceleration a(t)).

In a more practical example, the electronic logbook of a submarine (a computer software 

program that systematically records time-stamped materiel states, events, and conditions –

including vessel position – via a network of distributed sensors) provides a description of the 

marine platform‟s world line. In this context, the world line allows one to calculate the speed of 

the submarine at any point in time during its deployment, given an available measure of distance 

(a so-called metric) appropriate for the curved surface of the Earth.

In a similar sense, an airplane‟s world line is captured by the associated flight data recorder 

(FDR) – popularly referred to as a “black box” – a device that records aircraft performance 

parameters as a function of time. The data recorded by the FDR can be used for purposes of 

analyzing air safety issues, material degradation, aircraft engine performance, and after-the-fact 

in-flight accident scenarios; such analysis essentially seeks to reconstruct the airplane‟s world 

line via FDR data.

For a simple phase space with three spatial and one time dimension, a curve is defined by four 

coordinate functions xa ; a = 0, 1, 2, 3 (where x0 usually denotes the time coordinate). A 

coordinate grid in this phase space is the set of curves one obtains if three out of four coordinate 

functions are set to a constant. We can extend the concept of world lines to represent the history 

of a real or simulated object within an N+1 dimensional phase space (or state space). In this 

mathematical space, there are N degrees of freedom defining scalar attributes associated with the 

object (e.g., latitude location, speed, weight, residual amount of fuel) and one degree of freedom 

associated with a time axis.

Within the context of a military SoS operating within a battle space, a world line is a quantifiable 

measure reflecting purposeful DMPs. We can thus characterize the evolving purposeful 

behavior of the SoS by measuring CI (t): the “distance” between the current phase space 

location of the SoS and the phase space location reflecting “Commander‟s intent” as a function 

of time. Successful progressive realization of this purposeful behavior by the SoS will 

demonstrate world line convergence upon the “Commander‟s intent” phase space locus via 

monotonically decreasing values of CI (t). On the other hand, monotonically increasing values 

of CI (t) will signify a disruptive challenge to the SoS‟s purpose. Finally, it would be of interest 

to investigate correlations between CI (t) for a SoS and any discernable phase space attractors 

representing consistently repeatable SoS end-states.

3.2.2 Homeostasis and Homeodynamics

To survive and remain functional within a challenging and often hostile environment, a complex 

adaptive system must maintain (within itself) a stable and sustainable operational structure. This 

property of such a system where it is able to maintain its essential state variables within limits 

acceptable to its own structure in the face of unexpected disruptive perturbations is called 
15

homeostasis (25). To maintain homeostasis, systems typically utilize a negative-feedback-based 

control structure. This involves a process of interaction that balances various influences and 

effects such that a stable system state or system behavior can be achieved and then perpetuated 

(26).

The homeostatic process existing within an adaptive system can be generalized into a negative 

feedback control block diagram as shown in figure 4. In this control circuit, which reflects the 

general structure of regulatory networks within the human physiology (27), a “controller” seeks 

to maintain the measured time-dependent system state variable X(t) at a preset steady-state stable 

value Xstable in spite of incoming disturbance events impinging upon the system from a 

surrounding environment. The major components of the homeostatic process within the adaptive 

system include a sensor, a controller and an actuator.

• The sensor measures X(t) at time t, and then subtracts the reference value Xstable from X(t)

to generate the offset value X(t).

• The controller accepts X(t) as input, and then executes a decision-making process as a 

function of X(t).

o If X(t) > 0, then the controller formulates a process to reduce the value of X(t).

o If X(t) < 0, then the controller formulates a process to increase the value of X(t).

o If X(t) = 0, then the controller chooses to maintain the current value of X(t).

Finally, the formulated control process is passed along as the controller output.

• The actuator accepts the recommended control process from the controller as input, and 

then physically implements this process in an attempt to modulate the value of X(t). 

Finally, the modulated value of X(t) exits the actuator, and the homeostatic cycle is 

repeated.

Asynchronously concurrent with this control cycle are incoming events from the surrounding 

environment that may act to negatively disturb the function of the actuator. These environmental 

disturbances, which can perturb the value of X(t) away from Xstable, thus provide the need for 

homeostatic control in the first place.
16

Figure 4. Homeostatic process within a complex adaptive system.

Homeostasis proliferates across a broad range of system types (both biological and artificial) of 

variable complexity and adaptive capabilities. The following are some common everyday 

examples.

• The ambient internal temperature of a house T(t) can be disruptively impacted by 

environmental factors such as adverse weather conditions and heat loss from doors and 

windows. In order to stabilize house temperature, T(t) is first sensed and measured at time 

t by a thermistor (as in a modern digital thermostat), and then passed to a controller. Next, 

T(t) is subtracted from the thermostat set point Tstable to compute the offset value 

T(t) = T(t) – Tstable. Here, the controller output signal is constrained to open/close the 

flow valve supplying fuel to a furnace heater (i.e., the actuator). If T(t) > 0, the controller 

signals the furnace to close the fuel valve (and thus reduce T(t)). If, on the other hand, 

T(t) < 0, the controller signals to open the valve (to increase T(t)). Finally, if T(t) = 0, 

then the current fuel flow rate is maintained.

• Within a healthy human, the blood glucose concentration C(t) is temporarily perturbed 

from a stable reference level Cstable (typically a narrow range from 90 to 110 mg glucose/dl 

blood) by vigorous exercise or the consumption of food. To counteract such disturbances 

and regulate blood glucose levels, the human body employs a very efficient homeostatic 

control process (28). A decrease in blood glucose concentration (where the offset blood 

glucose concentration C(t) = C(t) – Cstable < 0) is detected by alpha cells within the 

pancreas, which respond by secreting glucagon into the bloodstream. As glucose levels 
17

subsequently increase (to the point where C(t) > 0), the heightened blood glucose 

concentration level is next detected by beta cells (also within the pancreas), which in turn 

respond by secreting insulin into the bloodstream. The secreted insulin metabolizes the 

excess glucagon, returning C(t) levels back towards the “normal” concentration range 

Cstable. In this example, the sensor, controller, and actuator functions required for 

homeostasis are all embedded within the pancreatic alpha and beta cells (responding to 

hypoglycemic and hyperglycemic blood concentrations, respectively).

In both examples, a single system state variable is homeostatically controlled at a time. Within a 

typical SoS, however, homeostasis might often involve stable control of a state vector X(t) of 

critical SoS variables (where X(t) = X1(t), X2(t), X3(t),..., XN(t)).

In the most general sense, homeostasis is typically manifested within open systems. To 

differentiate between the concepts of open and closed systems, we turn to the pioneer systems 

theorist von Bertalanffy.

A system is closed if no material enters or leaves it; it is open if there is import and

export and, therefore, change of the components . . . . A closed system must, according 

to the second law of thermodynamics, eventually attain a time-independent equilibrium 

state, with maximum entropy and minimum free energy . . . . An open system may attain 

(certain conditions presupposed) a time-independent state where the system remains 

constant as a whole . . . though there is a constant flow of the component materials. This 

is called a steady state (29).

Walker et al., provide further insight regarding the comparative limitations of closed vice open 

systems.

(When exposed to environmental changes,) a closed system is, or becomes, locked or 

frozen in a particular state and requires no further import or export of information to 

maintain that state. A closed system (or industrial age organization taken to its extreme) 

is therefore unresponsive to environmental change, matched to an optimum means to an 

end within a defined context and slow to change or adapt. To use a computer science 

metaphor, a closed system is an entity that is „programmed‟ while an open system is 

something that „learns‟ (or programs itself) (30).

Although homeostasis can (by design) be present in both closed and open systems, the latter are 

better coupled to a surrounding environment, and are thus more adaptively responsive to 

incoming disruptive phenomena. Within a closed system, a homeostatic process will tend to 

oppose change using every available means; if the system does not succeed in reestablishing a 

stable equilibrium, it can disorganize towards a maximal entropy condition given its exposure to 

persistent functional disturbances. An open system, on the other hand, will tend to adapt itself to 

modifications of the environment and evolve (31).
18

An important attribute of complex spatially-distributed SoS is their dynamic self-organizing/selfsynchronizing nature. Because the homeostasis usually present in this context is dynamically 

sustained, this type of control process is more properly referred to as homeodynamics (32, 33). 

Instead of simply attempting to hold a SoS state variable (e.g., ambient temperature within a 

simulated ecosystem) constant, homeodynamics instead aims to actively reset reference values of 

control variables as a dynamic function of incoming information from the environment while 

simultaneously maintaining SoS organizational structure. Allowing for variation in reference 

stability levels brings adaptability into the homeodynamic SoS, as it can constructively respond 

to novel environmental conditions (34). Thus, homeodynamics characterizes an open (vice 

closed) SoS that can concurrently demonstrate both homeostasis and adaptive behavior (often at 

different space/time scales).

The generalized homeostatic process within a complex adaptive system depicted in figure 4 can 

now be “upgraded” to the similarly generalized homeodynamic process as shown in figure 5. As 

was the case with the system in figure 4, the homeodynamics in figure 5 is again constrained to 

the single system state variable X(t) for purposes of simplicity and clarity. In this open complex 

adaptive system, a multimodal sensor array dynamically accepts information from the 

surrounding environment (e.g., in human physiology, sensory modalities are typically 

categorized as chemoreception, photoreception, mechanoreception, and thermoception). Then, 

the sensor array routes acquired environmental data into a fusion center (e.g., the brain that exists 

in multi-cellular organisms), where a composite informational representation of the sensed 

situation is created. Next, this situational representation is in turn routed to a DMP with the 

ability to adaptively reset the homeostatic reference value Xstable(t) (the latter of which is now a 

dynamic function of a possibly changing situational representation as generated by the data 

fusion center). Once the adaptive DMP has decided to reset Xstable(t) (or not), the updated value 

of Xstable(t) (which may remained unchanged from the old value) is routed into the homeostatic 

process previously depicted in figure 4. Finally, the modulated value of X(t) exits the 

homeostatic actuator (to guide complex system interaction with the environment), and the 

homeostatic cycle is repeated using the current value of Xstable(t) until the latter is again reset by 

the adaptive DMP. It should be noted that, in the case of homeodynamics, the complex system is 

better suited to synchronously manage incoming disturbances from the environment that perturb 

actuator function.

Although a relatively new term in the scientific literature, the concept of homeodynamics is 

demonstrated by a multiplicity of well-known complex adaptive systems. Perhaps one of the 

best-known examples of a well-documented homeodynamic process is the alteration of the 

human body temperature stable reference level Thuman/stable(t) associated with a human 

demonstrating a fever. The increase in body temperature Thuman(t) during illness does not 

represent a control failure in a homeostatic thermoregulatory process. Rather, elevated levels of 

Thuman(t) are homeodynamically regulated by the brain (the source of the adaptive DMP in this 

example), so that conditions in the body where Thuman(t) < Thuman/stable(t) are actively met with 
19

compensatory thermogenic processes (e.g., shivering, metabolic thermogenesis). In fact, the 

elevation of the control parameter Thuman/stable(t) associated with a fever is of benefit in mobilizing 

energy resources critically required when the human body combats infections (35). This 

example illustrates the close correlation that exists between homeodynamics and goal-directed 

behavior (i.e., destroying the invasive infectious agents within the afflicted human so to return 

the body state to a healthy normal condition). Finally, the homeodynamic control of a single 

system state variable in this example will very likely require extension to the control of a state 

vector X(t) when applying this concept to the adaptive management of a typical SoS.

Figure 5. Homeodynamic process within a complex adaptive system.

With respect to a military SoS, there remains the question of how collective “self awareness” 

within and across the population of agents (both human and artificial) making up the SoS effects 

the organizational adaptive efficiency as a function of environmental disruptions. In the context 

of homeodynamic processes within a human, Miller (36) has proposed a qualitative measure he 

calls the range of homeodynamic efficiency (RHE): an abstract characterization of the 

relationship between environmental stressors (including physical, cognitive, and emotional 

types), the state of intra-human mind/body communication pathways, and homeodynamic 

effectiveness, the interactions of which determine potential adaptive outcomes of intentional 
20

situational management by a human under stress. Normally, there can be a fairly wide range of 

disruptions to mind/body communication pathway connectivity that humans can effectively 

tolerate while the overall human system remains relatively effective, but there are also danger 

points that put a person in jeopardy. RHE is also therefore a gauge of how effectively mind/body 

connectivity is maintained in the face of stress, which, in turn, determines how physically and 

mentally healthy a human is. Using this perspective, Miller (36) defines three general levels of 

RHE.

• Ultra-stability: Not the daily state of affairs, the highest level of mental and physical 

integration.

• Average Functioning: Makes allowance for stressors and some communication pathway 

disconnections that still allow for homeodynamic adaptation given minor illnesses.

• Dehomeosis: The most stress combined with the least connectivity among the elements 

constituting a mind-body system, usually resulting in a condition of moderately serious to 

severe illness.

If the environment itself is destructive or toxic to its locale then the adjustments demanded of a 

person will be harsher, and if homeodynamic adjustments are not possible the human system 

itself may follow a dehomeotic course.

Figure 6 illustrates a notional interpretation of the RHE measure as applied to a general 

organizational SoS (i.e., a heterogeneous military force). In this interpretation, RHE is a 

function of both

• the collective “self-awareness” that agents within the organizational SoS maintain 

concerning the operational status of other fellow agents, and

• the level of environmental stress (e.g., problems within the organization, disruptive events 

impinging upon the organization from hostile external agents).

Note that numerical values of RHE presented in the graph (along the vertical axis) are 

normalized to a relative level of maximal homeodynamic efficiency, and should be interpreted as 

qualitative vice quantitative measures of adaptive potential. Here, the organization may be 

considered “ultra-stable” given a very high level of shared situational awareness (SA) distributed 

amongst the SoS population, and a quiescent operational environment. At the opposite end of 

the scale, the organization is collectively in a state of “dehomeosis” when shared SA is at a 

minimum while the SoS attempts to operate within a very hostile environment. Assuming the 

validity of this collective interpretation of RHE, then a SoS is most effectively adaptive when SA 

is optimized across the organization (within practical constraints), particularly that information 

concerning the dynamic operational readiness levels of organizational peers.
21

Figure 6. Range of homeodynamic efficiency (RHE) for a general organizational SoS.

3.3 Sociotechnical Systems Theory

One existential property of a military SoS is that it is a sociotechnical system. Sociotechnical 

system properties are present in any military SoS (37, 38). In a military domain, decision 

making and its consequences are central to the function of the SoS (13). As a rule, in a military 

SoS, people (leaders) make decisions and warfighters use technology to realize their purpose and 

accomplish the task. We therefore hypothesize that, if decision making process and decisions 

made are the bounds that will determine how one can decompose a military SoS, then any 

perspective from which to view the military SoS needs to recognize the sociotechnical properties 

present in a military SoS.

The purpose of this section is to highlight the sociotechnical aspect present in both military and 

non-military SoS. Sociotechnical systems refer to the interaction between the technical system 

and the human. Given a base of knowledge it is the socio part of the system that applies this 

knowledge to satisfy local variations, using the technology in the system. The technological part 

of the system may have a small range of adaptation, but with people in the system a wide range 

of adaptation is possible and in a military system this is the norm. In most definitions of SoS the 

notion of dynamicism induced by human action is only present by implication. The 

interoperability between the people and technology in the system makes for adversarial SoS in a 

competitive environment such as the military.
22

Sociotechnical systems theory is about the behavioral aspects of people and the technical aspects 

of technology. When people and technology interact the outcome is both predictable and 

unpredictable. The predictable interactions are pre-engineered “cause and effect” relationships. 

The socio (people) does not behave like the technical (machines) in the system but the 

interdependence of the socio and technical behavior in the system lends to the undersigned nonlinear relationship in the system that affects the system‟s performance. Consequently, neither the 

socio nor the technical aspect of the sociotechnical system can be optimized individually. The 

aim of the sociotechnical systems approach is to maximize both the outcome of the technical 

system and the people working in the system (joint optimization).

A network enabled military system consists of decision making people using networks 

(technology) to make informed decisions in a competitive dynamic environment. There exists a 

parallel between the military command and control and the design challenges that sociotechnical 

systems theory was developed to answer. In order to survive, any social system must 

demonstrate the following four functions: (1) attainment of organizational goals, (2) adaptation 

to the environment, (3) integration of personnel activities, and (4) maintain continued existence 

of critical organizational roles via recruitment and socialization. This is true in the military 

domain also.

To achieve these system functions, Cherns (37, 38) advocates the following ten design principles 

for sociotechnical systems.

1. Compatibility: The design of the system must be compatible with the system‟s purpose. 

However, including the objectives of everyone in the system can lead to conflict, so 

consensus decision making by the designers of the system is essential.

2. Minimal Critical Specification: To keep system flexibility, specify only those parts within 

an organization that are essential to support purpose and do not specify non-essential 

elements. When non-essential elements of a system are specified, the system‟s flexibility 

is constrained and the system cannot adapt to changing environmental conditions.

3. Variance Control: Variance in sociotechnical terms is an unexpected unaccounted event 

that affects system outcomes. Variance Control principle states that variance needs to be 

minimized and unplanned events must be controlled as close to their point of origin as 

possible. 

4. Boundary Location: Boundary Location relates to boundaries between departments. The 

principle requires that the departmental boundaries within an organization should not

impede the sharing of information, knowledge, and learning. 

5. Information Flow: This principle deals with the dispersion of information in the system. 

Information should be available as needed. This principle assumes that organizational 

information can be used for control, records and action.
23

6. Power and Authority: People who require resources to execute their purpose should have 

access to them and the authority to command them. Another aspect of this principle is that 

people who have access and authority over the resources should be responsible for them 

and their use.

7. Multifunctional Principle: This principle describes the adaptability of the system. 

Elements of organizations must adapt to their environments, of which the most important 

are usually other organizational elements. Adaptation can be via adding new roles or by 

modifying current roles.

8. Support Congruence: This principle refers to the desired similarity between reward 

systems and management philosophy. The systems of social support should be designed so 

as to reinforce the behaviors which the organizational structure is designed to elicit.

9. Transitional Organization: This principle addresses organizations undergoing redesign 

experience and is in a state of change. It is assumed that the transitional organization is 

both different and more complex than either old or new.

10. Incompletion: Design is a reiterative process and the closure of previously-considered 

options opens new one. The main idea purported by this principle is that stability is just a 

moment between transitions and all systems should be able to evaluate and redesign to a 

changing environment.

The sociotechnical system is one method for designing organizations and work systems and this 

method is characterized by the balance of human behavior and technological elements in the 

organization. The design principles aim to jointly optimize people and technology that forms the 

initial conditions for effective network enabled military system. The sociotechnical systems 

theory helps define the notion of shared awareness, peer to peer interaction and adaptability in 

the network enabled military domain.

The classic command and control (C2) model assumes deterministic behavior in the system 

where the whole is equal to the sum of the parts, the outputs are a known function of the inputs 

and the results are repeatable from one application of the model to the next. A recent model of 

C2 developed by the North Atlantic Treaty Organization (NATO) working group SAS-050 (19)

provides three major axes (and a three dimensional space) within which various instantiations of 

C2 can be plotted. The purpose of defining the problem space in terms of these three dimensions 

is to explore alternative paradigms for C2, ones that are becoming increasingly tractable with the 

growth in networked technologies. The formally rational instance of classic C2 is positioned in 

the NATO SAS model as shown in figure 7. This type of organization might be characterized by 

unitary decision rights (in which optimum means to ends are specified at the top of, or at higher 

levels of a vertical hierarchy); tightly constrained patterns of interaction (due to rules, standard 

operating procedures and other means by which an organization specifies optimum means to 
24

ends) and tight control (in which performance can be quantified and controlled through 

intermediate echelons of management).

Figure 7. The NATO C2 conceptual model.

A future network enabled force empowers individual units to interpret the broad command intent 

and evolve a flexible execution strategy with their peers (39). The sociotechnical principles 

provide a basis to study the human aspect of such a SoS. A military SoS is not a sociotechnical 

system in its entirety. The guiding principles of a sociotechnical system emphasize the need to 

maintain a steady state between people and product. In the military domain the primary focus is 

to produce a fighting force that can adapt using information attained from the environment and 

previous knowledge bases while maintaining self synchronization and peer-to-peer interactions.

3.3.1 Multiscale Analysis of Complex Systems

SoS can have complex systems as subsystems. Consequently, a study of methods to analyze 

complex systems can provide insight in the analysis of SoS. A complex system is composed of 

interconnected components that are assembled together to achieve a system behavior that cannot 

be attained from the individual parts that make the complex system. Defense systems consist of 

systems that have nested purposes that the individual systems must accomplish to satisfy a higher 

purpose.

Scale and complexity are inherent characteristics of a complex system. Each system is ideal for 

the context/environment/expression for which it is designed. Its effectiveness is limited when 

the context is changed/modified as it cannot perform all the actions it is designed to take. The 

components of the system interact to complete a task. The interaction and coordination between 
25

the components vary under different conditions. The behavior of the system is the coordination 

of the components that comprise that system coupled with the human action employing those 

components. The number of components that must act together to complete a task is the scale of 

the task. A system is expected to take distinct actions to complete a task. Variety is the measure 

of the distinct actions that the system can take to complete a task. Multiscale analysis of complex 

systems builds on the twin recognitions that scale and variety/complexity are necessary for 

effective performance of complex systems, and is the focus of the remainder of this section (40).

Components in a complex system either work in a coordinated way or independently. When the 

components of the system coordinate with each other, the possible number of distinct actions that 

the components can take is constrained. Coordination reduces variety. A system exhibits high 

behavioral variety when the components in the system work independently of each other. 

Additionally when mesoscale/macroscale collective behavior is required from a system, the 

components in the system have to act coherently. Increasing the variety for one desired scale 

(coordination of components) is possible only when variety is constrained for another scale 

(grouping of components). This implies that there are various degrees of tradeoff that are 

possible to achieve the desired level of behavioral variety from a system.

The key to multiscale analysis of the behavioral variety of any system is that the components of 

the system have a limit or bound on its variety. If the components of a system coordinate other 

components of the system then there has to be limits on what the organizational structure can do. 

It is possible that the variety associated with the coordinated system exceeds the variety of the 

components. A microscale system with independent decision making components has a high 

behavioral variety. As the system size increases, the coordinated system behavior of the 

individual components reduces the behavioral variety of the encompassing system. Complex 

systems have an increasing behavioral variety as the system scale decreases due to coordination 

at various scales.

For a system to be successful its components must be able to work independently and with each 

other. Certain tasks can be accomplished only when a specific number of the system 

components work together. The key issue is in determining the number and components that 

need to work together to accomplish the task. Coordination between components in a system is 

possible only when there is information flow between the components that make up the system. 

At the same time communication and coordination also reduces the behavioral variety of the 

whole system. An understanding of this relationship between scale and variety helps us develop 

a method to analyze SoS. However, it is not the solution because, in information driven military 

SoS, the human components in the system adapt themselves and technology based on the 

information feedback to every action taken. In terms of multiscale analysis this change in 

component systems is equivalent to a change in the whole system.
26

4. A Concept of SoS Analysis

Multiscale analysis of complex systems shows us that each of the components has a limit on its 

variety. When components making up a complex system are acting in a coordinated way, they 

cannot act independently. In a SoS, units often modify the assets (component systems) provided 

to them to survive in a hostile environment. This change of component systems makes 

traditional multiscale analysis technique inadequate for defense SoS.

Purposeful system analysis technique usually involves explaining the component system 

behavior, as in analytical thinking, but this method fails for military SoS. The reason for failure 

is that most military SoS are scenario dependent and the systems in the SoS are driven by nested 

concepts and inter-related purposes in conjunction with a commanders intent and concept of 

operation. Such a SoS loses its essence when it is taken apart or decomposed for analysis. There 

is both a time component and information uncertainty when decisions are made in a military 

SoS. Analysis techniques that include adaptive decision making by complex systems is one 

possible approach to analyze a system in the SoS. However, if any original system(s) in the SoS 

is modified/altered the SoS is altered and the analysis technique is inadequate.

Weapons and equipment are often physically and operationally modified/adapted by the 

warfighter to better suit his objective in a hostile environment. In a military SoS, systems are 

adapted by warfighters to improve their survivability and to achieve their commander's intent 

and at another level of abstraction the warfighters are lost to the overall SoS by death and other 

types of losses. For such a system, we hypothesize that analysis of a SoS is two-fold where the 

analysis takes into consideration the variety and scale of the SoS in a scenario dependant model 

and then uses nested concepts and inter-related purposes to integrate to form composite and 

holistic analysis of the SoS. Furthermore, we contend that if this hypothesis is accepted then the 

output of a SoS model can only be probabilities of possibilities.

5. Conclusions

SoS is a term that has been used frequently in literature to describe systems in many domains. A 

number of people have grappled with the issue of what is and is not a SoS. Our interim report 

proposes that one can view any, and especially military, SoS as possessing fundamentally three 

salient features, they: (1) are social technical systems, (2) are purposeful systems, and (3) 

possess a defined and purposeful relationship among all elements. It is ultimately the interplay 

of these salient features that determines the shape of any analysis of these SoS.
27

First, they are sociotechnical systems that possess some observable structure, for example, a 

brigade combat team, or a business entity of some size. As the term implies, the human element 

provides many of the emergent capabilities often associated with a SoS, and the technical 

elements augment, facilitate, or otherwise promote these emergent capabilities.

Second, military SoS are at their very core, purposeful systems; that is, they are not created as 

static structures, but dynamic entities that must drive to a desired end state that emerges from 

their collected purposeful activity. While they may be “cobbled together” from erstwhile 

previously independent pieces, or designed from scratch with specific properties, they all have a 

purpose, an aim, a goal they desire to achieve. All the social and technical elements that 

comprise a given SoS must derive their activity from their knowledge of the SoS purpose, and 

ultimately, their role in achieving that purpose. Whether they are a private driving a tank, or the 

general commanding the brigade combat team, it is this collected purposeful activity taken in 

light of their perception of local realities that ultimately determine their end state.

Third, while each SoS is a purposeful system, and one can view each entity within a SoS as a 

purposeful system, it is the explicit or implicit relationships between the elements within a SoS 

that ultimately determine its emergent properties. One can view these explicit or implicit 

relationships as an expression of “nested concepts and inter-related purposes.”

Existing analysis techniques apply largely to non-changing systems. Human behavior is 

unpredictable when threatened. Analysis of such SoS is an interplay between a system level 

intent, a purpose, and the activities of the myriad elements that comprise that SoS. The notion of 

“nested concepts and inter-related purposes” allows both the military commander and the analyst 

to form progressively smaller concepts that collectively describe an intermediate end state in 

service to the larger goal, and collected relationships among the entities who must realize that 

concept. By availing themselves of this paradigm, the analyst can both decompose the SoS into 

more easily understood components and then synthesize their results into an overall assessment.
28

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31

List of Symbols, Abbreviations, and Acronyms 

C2 command and control

COA course of action

DMP decision-making process

FCS Future Combat System

FDR flight data recorder

NATO North Atlantic Treaty Organization

RHE range of homeodynamic efficiency

s second

SA situational awareness

SoS system of systems

SoSA SoS analysis
32

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