About the CCRP

The Command and Control Research Program (C

has the mission of improving DoD’s understanding 

national security implications of the Information 

Focusing upon improving both the state of the art a

state of the practice of command and control, the 

helps DoD take full advantage of the opportunities af

by emerging technologies. The CCRP pursues a 

program of research and analysis in inform

superiority, information operations, command and c

theory, and associated operational concepts that e

us to leverage shared awareness to improv

effectiveness and efficiency of assigned mission

important aspect of the CCRP program is its ab

serve as a bridge between the operational, tec

analytical, and educational communities. The 

provides leadership for the command and control res

community by:

• articulating critical research issues;

• working to strengthen command and control res

infrastructure;

• sponsoring a series of workshops and symposi

• serving as a clearing house for command and c

related research funding; and

• disseminating outreach initiatives that include th

CCRP Publication Series.
This is a continuation in the series of public

produced by the Center for Advanced Concept

Technology (ACT), which was created as a “skunk w

with funding provided by the CCRP under the auspi

the Assistant Secretary of Defense (NII). This progra

demonstrated the importance of having a res

program focused on the national security implicati

the Information Age. It develops the theo

foundations to provide DoD with information supe

and highlights the importance of active outreac

dissemination initiatives designed to acquaint 

military personnel and civilians with these em

issues. The CCRP Publication Series is a key elem

this effort.

Check our Web site for the latest CCRP activities and public

www.dodccrp.org
DoD Command and Control Research Pro

Assistant Secretary of Defense (NII)

&

Chief Information Officer

Mr. John P. Stenbit

Principal Deputy Assistant Secretary of Defense (NII)

Dr. Linton Wells, II

Special Assistant to the ASD(NII)

&

Director, Research and Strategic Planning

Dr. David S. Alberts

Opinions, conclusions, and recommendations expres

implied within are solely those of the authors. They 

necessarily represent the views of the Departm

Defense, or any other U.S. Government agency. Clea

public release; distribution unlimited.

Portions of this publication may be quoted or re

without further permission, with credit to the DoD Com

and Control Research Program, Washington, D.C. Co

copies of reviews would be appreciated.

Library of Congress Cataloging-in-Publication Data

Moffat, James, 1948-

 Complexity theory and network centric warfare / James Moffat.

 p. cm. -- (Information age transformation series)

 ISBN 1-893723-11-9 (pbk. : alk. paper)

 1. War--Mathematical models. 2. Information warfare. 3. Compu

complexity. I. Title. II. Series.

 U21.2.M584 2003

 355.4'8--dc21

2003

September 2003
Complexity

Theor

and

Network Centric Warfa

James Moffat
This book is dedicated to my wife Jacqueline and my chi

Louise and Katherine.
NOTES TO THE READER

Although I use the term Complexity Theory as if it was a co

body of scientific theory, this area of research is in fa

both young and evolving. I use it therefore as a sho

term to cover a number of areas, each with its own d

heritage. Broadly, it covers fractal structures, non

dynamical systems, and models of self-organisation an

organised criticality. 

The research on which this book is based could not hav

carried out without the help of a number of other p

Their contributions are, I hope, suitably acknowledged 

text. I would like to thank particularly Walter Perry (R

Susan Witty (Dstl), David Rowland, and Maurice Passm

their contributions. I am also most grateful to Professor H

Jensen for contributing the Foreword.
i

Foreword ........................................................

CHAPTER 1

COMPLEXITY IN NATURAL

AND ECONOMIC SYSTEMS.........................

Open, Dissipative Structures .....................................

The Far-from-Equilibrium State ...............................

Self-Organisation in Nature–An Example ................

Clustering in Space and Time ...................................

Movement of a Boundary..........................................

An Example of Complex Behaviour and 

Fractal Time Series in Economics........................

Summary ...................................................................

CHAPTER 2

CONCEPTS FOR WARFARE FROM

COMPLEXITY THEORY..............................

Forest Fires, Clusters of Trees, and Casualties in Wa

CHAPTER 3

EVIDENCE FOR COMPLEX EMERGENT

BEHAVIOUR IN HISTORICAL DATA ..........

Introduction...............................................................
ii

Further Historical Data on the Processes of 

“Irruption” and Breakthrough .............................

The Fractal Front of Combat ....................................

Power Law Relationships in Combat Data ...............

CHAPTER 4

MATHEMATICAL MODELLING OF

COMPLEXITY, KNOWLEDGE, 

AND CONFLICT ..........................................

Introduction...............................................................

Control and Fractal Dimension.................................

Wargames as Open Systems Sustained by 

Knowledge Flowing Across the Boundary ...........

Wargaming with FASTHEX.....................................

The Decision Problem...............................................

A Simple Example .....................................................

Multiple Sweeps.........................................................

False Target Detections/Identifications ....................

Knowledge Representation .......................................

Combat Cycle Knowledge.........................................

Quantifying the Benefit of Collaboration 

Across an Information Network...........................
iii

AN EXTENDED EXAMPLE OF THE

DYNAMICS OF LOCAL COLLABORATION

AND CLUSTERING, 

AND SOME FINAL THOUGHTS ..................

Clustering and Swarming ..........................................

Final Thoughts...........................................................

APPENDIX

OPTIMAL CONTROL WITH A

UNIQUE CONTROL SOLUTION .................

Pontryagin’s Maximum Principle..............................

Determining the Extremal Controls..........................

Uniqueness of the Extremal Control for 

a Linear System....................................................

ABOUT THE AUTHOR ....................................

CATALOG OF CCRP PUBLICATIONS......... C
v

Figure 1.1: Two Horizontal Plates Containing a 

Layer of Fluid ...............................................

Figure 1.2: Schematic of an Open System......................

Figure 1.3: The Lattice of Species Interactions in a 

Model Ecosystem..........................................

Figure 1.4: Movement of the Ecosystem towards a 

Self-Organised Critical Point........................

Figure 1.5: Local Pinning of a Fluid Boundary...............

Figure 1.6: The Roughness “W” of an Interface ............

Figure 2.1: Information Leading to Emergent Behaviou

Figure 3.1: 2nd Armoured Division Data .......................

Figure 3.2: 2nd Armoured Division–

Power Spectrum Prediction..........................

Figure 3.3: 2nd Armoured Division–

Nonlinear Prediction ....................................

Figure 3.4: 2nd Armoured Division–

Neural Net Prediction...................................

Figure 3.5: 2nd Armoured Division–

SOC Prediction ............................................

Figure 3.6: 9th Armoured Division Data ........................

Figure 3.7: 9th Armoured Division–

Neural Net Prediction...................................
vi

Power Spectrum Prediction..........................

Figure 3.9: 9th Armoured Division–

SOC Prediction ............................................

Figure 3.10: The Statistics of Linear Irruption ...............

Figure 3.11: The Statistics of Radial Irruption ...............

Figure 4.1: Area of Operations........................................

Figure 4.2: Five Configuration Classes............................

Figure 4.3: Plot of y = f(x) and y = x - weak control .......

Figure 4.4: Plot of y = g(x) and y = x - strong control.....

Figure 4.5: Recursive Calculation of the 

Probability of Weak Control.........................

Figure 4.6: FASTHEX Game Cycle Sequence ..............

Figure 4.7: A Wargame as an Open Dynamical Process 

Figure 4.8: An Example Two-Cycle Game.....................

Figure 4.9: BLUE Commander’s Allocation Strategy ....

Figure 4.10: BLUE Commander’s Situation 

Assessment Problem .....................................

Figure 4.11: Developing a Refined Estimate ..................

Figure 4.12: Refined Probability Assessments.................

Figure 4.13: Knowledge and Entropy for Example 1.....

Figure 4.14: Experimental Assessment of 

Campaign Level Knowledge and 

Attrition of Enemy Forces ............................

Figure 4.15: Experimental Assessment of the 

Effect of Campaign Level Knowledge on 

Own Force Casualties...................................

Figure 4.16: The Critical Path ........................................
vii

Figure 4.18: The Logistics S-Shaped Curve....................

Figure 5.1: Screenshot of the Start of a 

Typical ISAAC Simulation Run ..................

Figure 5.2: Nearest and Next Nearest 

Neighbour Clustering ...................................

Figure 5.3: Largest Cluster Size as a Function of 

Simulated Time (First Iteration, Red Agents

Figure 5.4: Largest Cluster Size as a Function of 

Simulated Time (First Iteration, Blue Agent

Figure 5.5: Largest Cluster Size as a Function of Time 

(40th Iteration, Red) .....................................

Figure 5.6: Largest Cluster Size as a Function of Time 

(40th Iteration, Blue).....................................

Figure 5.7: Frequency Distribution of the 

Largest Cluster Size for Red Agents.............

Figure 5.8: Frequency Distribution of the 

Largest Cluster Size for Blue Agents ............

Figure 5.9: Distribution of Cluster Sizes 

(2nd Replication, Red Agents)......................

Figure 5.10: Distribution of Cluster Size for Red Agents 
ix

Table 1.1: Decade-by-Decade Behaviour of 

Daily Returns from the Dow Jones Index......

Table 2.1: Relation Between Complexity and 

Information Age Warfare ..............................

Table 3.1: Geometric Mean Area/Attack Front 

at Breakthrough (Miles)..................................

Table 3.2: Geometric Mean Area Per Day/Attack Fron

at Breakthrough (Miles/Day).........................

Table 3.3: Geometric Mean √Area Per Day 

at Breakthrough (Miles/Day).........................

Table 4.1: Refined Probability Assessments: Example 1 

Table 4.2: Multiple Sweeps Case 1 .................................

Table 4.3: Multiple Sweeps Case 2 .................................

Table 4.4: Total Knowledge: Example 1........................

Table 4.5: Total Knowledge ...........................................
xi

FOREWORD

or the last couple of decades, attempts have

been made to develop some general understanding, and ultimately a theory, of systems

that consist of many interacting components

and many hierarchical layers. It is common to

call these systems complex because it is impossible to reduce the overall behaviour of the

system to a set of properties characterising the

individual components. Interaction is able to

produce properties at the collective level that

are simply not present when the components

are considered individually. As an example, one

may think of mutuality and collaboration in

ecology. The function of any ecosystem depends

crucially on mutual benefits between the different species present. One example is the relation

between legumes, such as peas and beans, and

their associated nitrogen-fixing bacteria: the

bacteria collects nitrogen for the legume, which

in turn produces carbohydrates and other

organic material for the bacteria. Clearly this

F
xii

the legume and neglecting the bacteria; the ecologica

tion emerges first when the different components are b

together and interaction is taken into account.

Another important feature of complex systems is their 

tivity to even small perturbations. The same action is 

to lead to a very broad range of responses, making it e

ingly difficult to perform prediction or to develop any t

experience of a “typical scenario.” This must necessari

to great caution: do not expect what worked last time t

this time. The situation is exacerbated since real s

(ecological or social) undergo adaptation. This impli

the response to a given strategy most likely makes the 

egy redundant. An example is the effect of using the 

type of antibiotic against a given type of bacteria. Evo

soon ensures that the bacteria develop resistance and 

the specific type of antibiotic useless. That complex s

adapt and change their properties fundamentally as a 

of the intrinsic dynamics of the system is clearly ext

important. Nevertheless, for the sake of simplicity adap

is often neglected in model studies. Sometimes assumi

existence of a stationary state might be justified (e.g., if 

interested in “toy” models of the flow of granular m

under a controlled steady input of grains). But if one i

ing with more complex situations such as in ecolog

even more when considering social and political sy

ignoring adaptation is very likely to lead to erro

conclusions.

We know from studies of Self-Organised Critical m

which the present book alludes to (for more see P. Bak

Nature Works, Oxford University Press, 1997 and H.J. J

Self-Organized Criticality, Cambridge University Press, 
xiii

model systems are entirely determined by the as

boundary conditions or the applied drive. The lesson 

learned from this is that complex systems cannot be s

independently of their surroundings. Understandin

behaviour of a complex system necessitates a simulta

understanding of the environment of the system. In 

studies, one assumes often that the surroundings can be 

sented by one or the other type of “noise,” but this is 

trick that allows one to proceed with the analysis w

understanding the full system under consideration. It 

important to appreciate that the “drive” or the “nois

equally crucial to the understanding, as is the analysis 

“system” itself. One should bear in mind that the sepa

into system, drive, noise, surroundings, etc. is rather ar

and is far from representing a complete analysis.

From these considerations, we see that it is vitally imp

to consider warfare as a complex system that is linke

interacts (in a coevolving way) with the surrounding 

economical and political context. From that perspecti

present book is a “work in progress” and a prelimina

step along the road in helping to analyse and structure 

difficult and serious issues. Forgetting that war and w

are an intimate part of a much larger complex syste

lead to incomplete and even dangerously incorrect c

sions. Applying the approach of Complexity The

warfare leads one to the self-consistent realisation tha

fare will have to be analysed in its larger context. F

work will need to examine how coevolution across the 

network of military, socioeconomical, and political in

tions leads firstly to emergent effects at higher levels, 

equal importance how such effects lead to coevolution 
xiv

ness of such networks, and their vulnerability to damag

Henrik Jeldtoft 

Professor of Mathematical P

Department of Mathe

Imperial College, L
1

CHAPTER 1

COMPLEXITY IN

NATURAL AND

ECONOMIC SYSTEMS1

n this chapter we consider some of the key

ideas of Complexity Theory as applied to

natural systems. Having established these key

ideas, in Chapter 2 we begin to see how these

ideas map across, in a broad conceptual sense, to

the dynamics of conflict in an Information Age

environment. In later chapters, we will look in

more detail at the evidence and the type of modelling that emerges from this conceptual

connection. In an organisational context, we

argue that complexity provides an explanatory

framework of interrelationships, both metaphorically and analogously, of how individuals and

military organisations interact, relate, and evolve

1

The contribution of Dr. Maurice Passman to this 

chapter is gratefully acknowledged.

I
gypy py 

ventions may have unanticipated consequences, bu

explains how combat effects follow from these consequ

The intricate interrelationships of elements within a co

system give rise to multiple chains of dependencies. C

happens in the context of this intricate intertwining 

scales. We become aware of change only when a differe

tern becomes discernible. But before change at a macr

can be seen, it is taking place at many micro levels si

neously. Hence, microcomponent interaction and c

leads to macrosystem evolution.

In a previous book,2 we considered some of the issue

addressed at the political/military level as a conseque

such emergent behaviour and the “resultant likeliho

complex and unexpected interactions, arising from 

ously unexpected sources.” The diligent reader is direc

that work for further discussion of these issues. Here, w

ply wish to add weight to the points made by Professor J

in his Foreword to this present work. In all that follow

recursion of the process up to this political/military 

must be kept in mind, and will be one of the key ar

future research.

Our lead is taken from current military doctrinal th

both in the UK and in the United States (particula

U.S. Marine Corps [1]). The Chief Analyst of th

Defence Science and Technology Laboratory (Dstl), 

Forder, makes the following point in his discussion 

future of defence analysis [2]:

2

Chapter 1 of: Moffat J (2002). Command and Control in the Informatio

The Stationery Office. London, UK.
fff f

a degree of complexity into the situation such that the emergen

behaviour of the system as a whole is extremely difficult to pre

dict from the characteristics and relationships of the system 

elements. Detailed simulation, using agent-based approaches, 

always possible but the highly situation-specific results that it 

provides may offer little general understanding for carrying for

ward into robust conclusions of practical significance. Usable 

theories of complexity, which would allow understanding of 

emergent behaviour rather than merely its observation, would 

therefore have a great deal to offer to some of the central problem

facing defence analysis. Indeed they might well be the single m

desirable theoretical development that we should seek over the 

next few years.

A similar thought was aired at a Royal United Service

tute (RUSI) conference on future Intelligence, Survei

Target Acquisition, and Reconnaisance (ISTAR) [3]

Admiral Cebrowski,3

 U.S. Navy, centred his keynote a

on Network Centric Warfare (NCW) as the capstone c

for the U.S. Navy after Next. He described it in terms 

achievement of Information Superiority with characteri

gross asymmetries and a diversity of “players.” These id

explicitly derived in his description from the new phy

nonlinearity, complexity, and chaos as exemplified b

Santa Fe Institute corpus of ideas. The relationship b

complexity and “information-based” warfare is (

described it) less deterministic and more emergent; less f

on the physical, and more behavioural; less focused on 

and more on relationships. Command and Contro

3

Now head of the Office of Force Transformation, The Pentagon, 

DoD.
pp, g, 

ADM Cebrowski defined NCW as the robust network

well-informed, geographically dispersed forces.

Where do these ideas come from?

In looking at where these key ideas of complexity come 

let us make a start by considering systems and their beh

in the natural world. In the classical view, such physical 

logical processes are reducible to a few funda

interactions. This leads to the idea that under well-define

ditions, a system governed by a given set of laws will fo

unique course (like the planets of the solar system). More

slight change in the cause will likewise produce a slight c

in the effects (i.e., the system is linear in nature). Recen

increasing amount of experimental data challenging th

has become available and this imposes a new attitude co

ing the description of nature. In natural systems, 

appropriate conditions, a multitude of self-organisatio

nomena on a macroscopic scale (a scale of an or

magnitude larger than the range of fundamental intera

in the form of spatial or temporal patterns may be gener

A SIMPLE EXAMPLE

To illustrate this, let us consider a simple thermody

thought experiment. Imagine a layer of fluid limited b

horizontal parallel plates whose lateral dimensions are 

longer than the width of the layer, as shown in Figure 1.

Figure 1.1: Two Horizontal Plates Containing a Layer of Flui
q, py 

homogeneous state that is statistically identical. The ho

neity of this system extends to all of its properties, parti

to its temperature, which will be the same at all parts 

fluid and equal to the temperature of the limiting pla

alternatively, to the temperature of the “external” world

these properties are characteristic of a system in a par

state, the state of equilibrium, for which there is neithe

motion nor temperature difference with the outside 

What occurs if, for example, the temperature on a sm

tion on one of the plates is temporarily perturbe

equilibrium, this temperature perturbation has no infl

since the temperature rapidly becomes uniform agai

equal to its initial value. In other words, the perturbatio

out; the system keeps no track of it. Such a state is said 

asymptotically stable.

From the standpoint of a very small observer inside th

tem, not only does the homogeneity of the fluid m

impossible for the development of an intrinsic conc

space, but also the stability of the state of equilibrium e

ally makes all time instances identical. It is the

impossible for this observer to develop an intrinsic c

tion of correlation or coincidence (e.g., things happen

the same time, or in the same place). We can increa

complexity of the system by, for instance, heating th

layer from below. In doing this, we communicate ene

the system in the form of heat. Moreover, as the tempe

of the lower plate is now higher than the upper, the e

rium condition is violated. In other words, by ap

external constraints to the system, we do not permit the 

to reach equilibrium. The presence of an external con

therefore implies energy flux and vice-versa.
pp, 

in temperature, , is small. The system will again a

simple and unique state in which the only active proce

transfer of heat from the lower to the upper plate, from 

heat is lost to the external world. The only difference fro

state of equilibrium is that temperature, and consequent

sity and pressure, are no longer uniform. They vary from 

regions to cold regions in an approximately linear fashion

phenomenon is known as thermal conduction. In this new 

that the system has reached in response to a constraint, s

will prevail again and the behaviour will eventually be as 

as at equilibrium. However, by removing the system from 

librium further and further, through an increase in 

observe that suddenly, at a value of that we will call 

matter begins to perform a bulk movement. Moreov

movement is far from random; the fluid is structured in a 

of small structures known as Benard cells.

Owing to thermal expansion, the fluid closer to the lowe

is characterised by a lower density than that nearer the 

plate. This gives rise to a gradient of density that oppo

force of gravity. This configuration is thus potentially un

Consider a small volume of the fluid near the lower 

Imagine that this volume is displaced upward by a per

tion. This volume, now in a colder and hence denser 

will experience an upward Archimedes force, amplifyi

ascending movement further. If, on the other hand, a 

droplet initially close to the upper plate is displaced 

ward, it will penetrate an environment of low density a

Archimedes force will tend to amplify the initial descen

fluid thus generates the observed currents. The stab

effect of viscosity, which generates an internal friction 

ing movement, counteracts the destabilising effects. Th

∆T

∆

∆T
, gy p

difference between the displaced droplet and its environ

explains why currents do not appear as soon as 

strictly zero.

Benard cells also show the complexity of movement. Th

unfold along the horizontal axis, adopting successively 

handed or left-handed rotation. Our very small observ

now locate his position in space by considering the rota

the cell he occupies and by counting the number of c

passes through. The emergence of this notion of sp

known as symmetry breaking. When is below the 

value, the homogeneity of the fluid in the horizontal di

renders its different parts independent of each other. I

trast, beyond the threshold, it is as if each volume elem

watching the behaviour of its neighbours and is taking th

account in order to play its role adequately and to part

in the overall pattern. This suggests the existence of corr

of statistically reproducible rate relations between distan

of the system. The characteristic space dimension of a B

cell is in the millimetre range, whereas the characteristic 

scale of the intermolecular forces is in the Angstrom 

That large numbers of particles can behave in a coheren

ion at this long range, despite random thermal motion, 

of the principal properties characteristic of such self-orga

and emergent complex behaviour.

This experiment is reproducible; the same convection p

will appear at the same threshold value and the process 

ject to a strict determinism. However, the direction 

rotation of the cells is unpredictable. The form of the p

lar perturbation that prevails at the moment of the expe

will decide whether a given cell is right- or left-handed. 

the constraint is sufficiently strong, several solutions are 

∆T

∆T
Open, Dissipative Structures

p

decide which of these solutions is realised. In this way, t

tem has been perturbed from a state of equilibrium or 

equilibrium to a state of self-organisation, with a number 

sible modes of behaviour.

What happens to the Benard cell system when the th

constraint is increased beyond this first threshold? For 

range of values the Benard cells will be maintained gl

but some of their specific characteristics will be modifie

ther constraint induces the system to move beyond a

critical point and turbulence is witnessed. Note that all o

critical behaviours are different from the phase chan

normally associate with closed thermodynamic system

reason behind this is that a nonequilibrium constraint is 

applied. For example, the dendritic structure associate

snowflakes has nothing to do with the structure of the un

ing ice-crystal lattice. The scale, size, and spacing 

emergent structure is of an order of magnitude larger.

To summarise, nonequilibrium has enabled the syst

transform part of the energy communicated from the 

ronment into an ordered behaviour of a new typ

dissipative structure. This regime is characterised by sy

breaking, multiple modes of behaviour, and correlation. Such a 

is called “open” since it is open to the effect of energy or 

mation flowing into and out of the system. It is also 

“dissipative” because of such energy flows, and the re

dissipation of energy.

OPEN, DISSIPATIVE STRUCTURES

Consider then a system embedded in an environmen

which it communicates through the exchange of certain 
Open, Dissipativ

f

system (in contrast to a closed or isolated system). Figur

a schematic representation of such an open system, com

cating with the environment through the exchange o

properties as mass, energy, or information. The rate of a

transported per unit surface is the flux of the corresp

property across the system. In our simple example abo

amount of heating is the flux of energy into the system.

Figure 1.2: Schematic of an Open System

As a result of these exchanges, the variables describi

instantaneous state of the system, {Xi

}, vary in time and 

values typically different from those characterising the s

the environment {Xi,e}. Whatever the detailed interpr

of Xi

 might be, the evolution of the system under conside

may be described in the following general form:

Rate of change of the system state = function of (system 

and control variables).

Thus we have:

( ) ,... ; ,... , ( 1,.... ) 1 1 F X X i n

dt

dX

i n m

i = λ λ =
Open, Dissipative Structures

i

g g

system, and are a set of parameters present 

problem, which can be modified by the external world. 

quantities are known as control parameters. Under certain 

tions, this relation will have a single solution that min

some measure of negative utility (which we call a loss fu

This solution is then a unique optimal control for the syste

each time t, it defines an optimal value for the settings 

control parameters .

AN EXAMPLE

If we think of a guided missile attempting to man

towards a target, the measure of loss is the miss distanc

tive to the aim point. The control parameters are the s

for the missile fins at a given time t. For simple forms o

guidance (e.g., early forms of laser-guided bombs), this l

what is called bang-bang control, where the missile fins “

from one extreme setting to another in order to keep th

sile on course. The Appendix goes into this in more det

shows that such solutions correspond to maximising or 

mising a Hamiltonian function. This is due to Pontr

maximum principle. Applied to a linear control system

maximum principle leads to the solution of bang-bang c

A characteristic feature of many of the systems encount

nature, however, is that the F’s are complicated nonlinea

tions of the X’s. The equations of evolution of this t

system should then admit, under certain conditions, 

solutions (rather than just the one optimal solution) s

multiplicity of solutions is the most typical feature of a n

ear equation. Our assumption will be that these so

represent the various modes of behaviour of the underlying s

λ λ m ,... 1

λ λ m ,... 1
Open, Dissipativ

MODES OF BEHAVIOUR IN NONLINEAR SYSTEM

How do these different modes of behaviour arise in 

nonlinear system? Thermodynamic equilibrium is cha

ised by detailed balance, i.e.

Probability of a “direct” process = Probability of a “reverse” 

We can understand that in such a state, any attempt at d

fication and self-organisation will be smeared out immed

equilibrium is a state of full homeostasis, characterised b

uniqueness and robust stability properties. Our aim 

extend our ideas of equilibrium to the nonequilibrium d

ics of an open, dissipative system.

The most useful view of equilibrium is as follows. We rep

the evolution of the system in a space spanned by the sta

ables (phase space). An instantaneous state of the system 

represented in phase space by a point. As the system 

over time, a succession of such states is produced, giving 

a curve in phase space, which is called a phase space trajecto

dissipative dynamical system, as time progresses, the 

space trajectory will tend to a limit representative of the 

reached by the system when all transients die out. We c

regime the attractor. The attractor representing an equil

position is unique and describes a time-independent sit

This gives a phase space point towards which all possible 

ries converge monotonically. The state of equilibri

therefore a universal point attractor. The goal of self-organis

thus the search for new attractors that arise when a sy

driven away from its state of equilibrium. For example, co

the type of coupled chemical system studied by Ilya Pri

[4] (for which he received the Nobel Prize in Chemistry):
Open, Dissipative Structures

Here the concentration [x] of product X is taken to be th

state variable, being understood that A and B are contin

supplied from or removed to the outside to maintain fixe

centrations. At equilibrium, detailed balance implies th

rate equation is:

These relations fix the equilibrium value [xe

] of x unique

impose a condition on the concentrations of constituents A 

In a dissipative stationary state far from equilibrium, it 

necessary for each individual reaction to balance in both 

tions. Cancelling the overall effect of the two forward re

by that of the backward reactions is sufficient, and this y

cubic equation:

. 

A B X

k

k

3

2

1

←

→

+

X B

k

k

←

→

4

3

3

2

2

1 k [a][x] = k [x]

[ ] [ ] k3 x = k4 b

4 1

3 2

[] [] [ ] e e

e

kb ka

x

k k = =

2 4

1 3

[ ]

[ ]

k k

k k

a

b

e

e =

3 2

2 1 34 [ ] [ ][ ] [ ] [ ] 0 s ss − + + −= k x ka x k x kb
The Far-from-Equili

qy p (

sible modes of behaviour) for certain values of [a] and 

may therefore be said that nonequilibrium reveals the p

alities hidden in the nonlinearities, which remain “dor

at or near equilibrium.

The monotonic character of the approach to the state o

librium implies that the evolution laws of the system 

obey, in the neighbourhood of equilibrium, very particular 

tions. Introducing the deviations of Xi

 from the equil

values, Xi,e:

the evolution of {xi

} near equilibrium can then be writte

.

 is a thermodynamic potential taking its minimum a

librium and is a symmetric matrix. This symmetry 

traced back to the property of detailed balance or the pr

of the invariance of the equilibrium state to time revers

can see from this that the system response is essentially 

near to equilibrium (i.e., small changes lead to small effe

THE FAR-FROM-EQUILIBRIUM STATE

The search for a generalised thermodynamic potentia

the nonlinear range well away from the equilibrium sta

attracted a great deal of attention, but these efforts h

far, not made much progress. Typically, therefore, beyo

linear domain for such irreversible processes, the abov

trol equations are expected to break down. A first conseq

is that the steady state point attractor of the system (i.

i Xi Xi e x = − ,

∑ 















∂

∂Φ = − Γ

j j

i j

i

dt x

dx ,

Φ

{ } Γi, j
The Far-from-Equilibrium State

pyy

over time), extrapolating the state of equilibrium as t

tance from equilibrium is increased, can now be appro

through damped oscillations. This behaviour heralds 

more interesting possibility in which the oscillation eve

becomes sustained. Topologically, this implies the eme

of a new one-dimensional attractor in phase space. The 

attractor for the state of the system is essentially replace

circle. In the limit, the system moves endlessly around t

cle, which is thus known as a limit cycle.

By allowing the intrinsic nonlinearity to be manifested 

regime of detailed balance, nonequilibrium can also l

the coexistence of multiple attractors in state space. Th

space can then be carved up into a set of basins. Each o

corresponds to the set of states that, if the system were t

from there, would evolve to a particular attractor. The

known as the basins of attraction. The ridges separating 

basins of attraction are called separatrices. The coexiste

multiple attractors constitutes the natural mode of s

capable of showing adapted behaviour and of perfo

regulatory tasks. We would thus expect to see the system 

ing within one basin of attraction (corresponding to res

to change) and then at some point switching between 

ent attractors (corresponding to a change in the lon

mode of behaviour) as we further vary the initial state 

system. The existence of one-dimensional attractors 

and circles) suggests the possibility of higher dimen

attracting objects in phase space. These model multip

and chaotic behaviour, which is observed under appro

experimental conditions.

Nonequilibrium phenomena show a variety of behaviou

therefore correspond to the movements of the system to
The Far-from-Equili

pp

the bifurcation diagram, where a single control paramet

thermal gradient in the case of Benard cells) affects the d

ics of the system.

BIFURCATIONS AND UNIVERSALITY LIMITS

Consider a system described by a set of evolution laws 

form of the equation:

where, as before, Fi

 are the rate laws, and are the c

parameters. In a typical natural phenomenon, the num

variables n is expected to be very high. This will consid

complicate the search for all possible solutions. Howeve

pose that by experiment we know one of the solutions

standard method, known as Linear Stability Analysis, we ca

determine the parameter values for which this so

regarded as the reference state switches from asympto

bility to instability.

ROBUSTNESS AND LINEAR STABILITY ANALYSIS

Stability or “robustness to change” is essentially determi

the response of the system to perturbations. It is therefo

ural to transform the dynamical laws into a form in wh

perturbations appear explicitly:

and

.

( ) ,... ; ,... , ( 1,.... ) F X1 X 1 i n

dt

dX

i n m

i = λ λ =

λi

( ) ( ) , X t X x t i = i s + i

({ }, ) ({ }, ) = i i,s + i λ − i i,s λ i F X x F X

dt

dx
The Far-from-Equilibrium State

qg

hand side vanishes if all xi = 0. Expanding, we may write

where Lij are the coefficients of the linear part and hi a

nonlinear part. It is assumed that the asymptotic stab

the reference state (i.e., X = Xs

 or x = 0) of the system i

tical to that of the linearised part:

.

This is reasonable provided that the perturbation is n

large and the system is “well behaved.”

In general, a multivariate system gives rise to a wide spe

of values for the rate of change of perturbations. For a u

control variable , two cases can be distinguished

the perturbations are nonoscillatory and the bifurcation

alternative modes of behaviour) will correspond to s

state point attractors, or secondly, the perturbations are 

latory and the bifurcations will correspond to time-p

solutions in the form of limit cycles. More intricate so

can also be envisaged leading to secondary, tertiary, or 

order interactions; however, the complete stable unfold

the problem remains an open question. It is left to 

methods, such as explicit simulation using cellular aut

(i.e., agent-based simulation) to attempt to solve the pr

of describing the dynamic behaviour of nonequilibriu

tems. The use of such a simulation approach can be des

as “experimental mathematics.”

L x h x i n

dt

dx

i j

j

ij j

i

= ∑ (λ) + ({ },λ) = 1,2,.....

L x i n

dt

dx

j

ij j

i

= ∑ (λ) = 1,2,.....

λ = λc
Self-Organisation in Nature–A

SELFORGANISATION IN NATUREAN EXAM

Consider an ecosystem consisting of a large number of 

acting species, each evolving in response to the enviro

created by the rest of the ecosystem (i.e., each species is 

ing). Such a system consists of many components that in

through some kind of exchange of forces or informati

addition to the internal interactions, the system may be 

by some external force–natural selection in this case. T

tem will now evolve over time under the influence 

external driving forces and the internal interactions. 

happens when we observe such a system? Is there some 

fying mechanism that produces a typical behaviour sha

large classes of such systems? The mechanism, it turns 

that of clustering.

A simple cellular automaton model of such an ecosys

given by the Bak-Sneppen evolution model. It is an exam

the experimental mathematics we described earlier, as 

of analysing the coevolution of such a complex system 

time. A description of this model of coevolution within a

system was given in the last chapter of [5] as part 

introduction to ideas from complexity mathematics a

development of mathematical “metamodels” of future 

mation Age conflict. For ease of reference, some o

description is repeated here.

THE BAK-SNEPPEN EVOLUTION MODEL

In this automaton model, we have a d-dimensional lattic

ure 1.3) and random numbers f

i

 drawn without replac

from the interval [0,1] occupy the lattice sites. At each 

step, the extremal site (that is, the one with the smallest v

f

i

) is chosen, and then it and its 2d immediate neighbou
Self-Organisation in Nature–An Example

g

values f

i

 correspond to “fitness” values. Changing both t

and neighbouring sites captures the process of local coevo

It follows from [6] that the set of such active sites is a fra

space-time (see particularly Figures 1 and 28 of that refer

Figure 1.3: The Lattice of Species Interactions in a Model Ecosys

As described by [5, Chapter 6], the approach to the 

attractor of the process (at which avalanches/clusters 

sizes are possible) is controlled by the “gap equation”:

where G(s) is the maximum extremal value f

i

(s) at time 

responding to the gap opened up between the existing s

the system and the zero or ground state, L is the linear 

the lattice, and is the average avalanche size a

step s. (An avalanche or cluster consists of a set of extrem

ues f

i

, each of which is a neighbour of the previous ex

value. The size of an avalanche is the number of timest

( )

( ) 1 ( )

G s

d L S

G s

ds

dG s − =

G(s)

S
Self-Organisation in Nature–A

p) y, 

following picture of the process (Figure 1.4):

Figure 1.4: Movement of the Ecosystem towards a Self-Organised Criti

Figure 1.4 shows how the equation (the hatched line) ap

mates the self-organised movement of the system, via a 

of avalanches/clusters towards the critical attractor of t

tem at which the system has optimal flexibility (in the 

that clusters of all sizes can be created). This critical poi

responds to a fitness value f

c. At this point, there are no 

values below this critical value and a flat distribution of 

values in the range from f

c to 1.0. This is in complete c

to the behaviour of a closed system such as an ideal ga

isolated container, where the gas evolves from (for ex

being partitioned in part of the container to the equil

state where it is spread equally throughout.

Such critical systems are of particular scientific interes

tems in critical states do not have any characteristic sca

may therefore exhibit the full range of behavioural chara

tics within the particular system restraints. This mean

systems at the point of criticality are in a position of o

flexibility in some sense, as we have noted. It could t
Self-Organisation in Nature–An Example

g[] qy 

is to so arrange things that the forces collaborate local

thus self-organise into this optimal state.

From the previous equation, the rate of change of the 

inversely proportional to the average avalanche size:

. 

Thus at the critical value and near to c

ity, the average avalanche size satisfies the scaling law:

for some exponent (as has been confirmed by experim

Before the critical point is reached, at some time t, if f0 

smallest random number on the lattice in the evolution 

then random numbers created at the next time-step w

continue the avalanche process if they are smaller th

Thus the value f0 can be viewed as the branching prob

of a random process over time. This will give informat

the avalanche size. If f0 = branching probability, th

larger f0 we have larger avalanches. We thus assume [6] 

ing relation of the form:

to describe the probability distribution of avalanches/c

of size S corresponding to an extremal value f0. Such a r

has also been confirmed by simulation experiments. In 

ular, when f0 equals fc, the critical value, the probabili

cluster of size S is given by . We can think 

function g as a ‘cutoff ’ corresponding to the fact that th

( )

1 ( )

G s S

G s ∝ •

→ ∞ ( ) , G s c f S

S ( ) c i f f −γ −

γ

( , ) ( ( ) ) 1

0 0

τ σ P S f S g S f f = c − −

(, ) PS fc S −τ
Self-Organisation in Nature–A

critical state, this cutoff dies away. We shall call the cl

exponent for this coevolving ecosystem. In the cont

manoeuvre warfare, this describes the statistics of local c

ing/collaboration at a transient point f0 heading towar

critical attractor value fc. The parameters and are 

dependent and g is our “scaling function.” Note again th

average size of the f0 avalanche diverges as , i.e.

.

If we mark each of the minimal sites on the lattice as it i

tified as an extremal value f0, then the set of marks gen

over time forms a fractal in space-time [6] as we have a

noted. Cuts of this fractal in the space direction at a give

identify the site that is “active” (i.e., chosen as the minim

at that time. Cuts in the time direction produce a fract

series. In Chapter 3, we show that the time series of cas

in conflict has the characteristics, in some cases, of such 

tal time series. One interpretation of such effects is that i

dynamics of local clustering (by one side) that is lead

casualties to the other side. This can also be shown to o

cellular automata models of conflict such as the ISAAC 

produced by the U.S. Marine Corps Combat Develo

Centre, as discussed in Chapter 6 of reference [5].

SELF-ORGANISED CRITICALITY

In a paper published in 1987, Bak, Tang and Weisenf

first proposed the hypothesis that a system consisting of 

interacting constituents may exhibit, in certain cases, a s

general emergent behaviour characteristic of the system

described the behaviour of this type of system by the ter

organised criticality (SOC). Self-organisation has for many 

τ

τ σ

c f → f 0

S ( ) c i f f −γ −
Self-Organisation in Nature–An Example

y q

tems to develop structure and patterns. The word critica

a very precise meaning in thermodynamics. It is used i

nection with phase transitions. At all temperatures othe

the transition temperature, perturbations of the syste

only locally influence system components. At the critica

perature, the perturbation affects the whole system

though only the nearest neighbour system components 

act directly. The system becomes critical in the sense tha

the members of the entire system influence each other. F

example ecosystem above, the system self-organises itse

the critical state corresponding to this ability of the ent

tem to be influenced through the propagation of 

coevolution influences and the resultant clusters/avalan

species that coevolution created.

SEPARATION OF INTERNAL AND EXTERNAL

TIMESCALES

Such a self-organised dynamical state requires the sepa

of the external and internal timescales. For example, th

in the earth’s crust, built up over a period of time, is of a 

ent scale to the subsequent earthquake lasting merely m

or seconds. The force applied to an individual tectoni

must overcome a threshold in order to produce an earth

This means that the plate may exist in a multitude of in

diate states–metastable states–on the way to criticality. Am

of the metastable states, some are of particular impo

These states are marginally stable; a slight disturbance ma

to a wide variety of responses. Bak envisaged that thes

ginally stable states are characterised by the lack of any 

time or length. This configuration is of a similar type 

seen in the thermodynamics of phase changes. The lack 
Clustering in Spac

g

with power-laws.4

 Hence, the distribution functions desc

the frequency with which SOC events occur exhibit p

law characteristics. For our earthquake example, if E 

energy released during an earthquake, then the probab

an earthquake of that size is given by the power-law re

ship . As noted in [8], such a simple law 

have an elegant explanation. We shall see later, in Cha

when examining the evidence for such emergent behav

warfare, that the number of casualties in war also has 

power-law distribution. However, as yet no such e

explanation is available, either for earthquakes or wars.

CLUSTERING IN SPACE AND TIME

The formation of clusters that are fractal in both spa

time is common in natural systems, as we have already s

was this type of behaviour that Bak wanted to explain. W

see that these relate to ideas of correlation in space or ti

contrast to coincidence in space or time). Correlation in 

or time is a signal of local clustering and collaboration sp

(e.g., across a battlespace) or in time (e.g., across an in

tion grid–reading e-mail creates a correlation in time b

individuals, taking a phone call creates a coincidence in 

The properties of fractals and their link to chaotic beh

have been examined intensively over the last two dec

4

A power-law has the property that the relative change

is independent of x. Power-laws, in this sense lack characteristic sc

5

See for example the standard text Chaos and Fractals by Heinz-Ott

Peitgen, Hartmut Jürgens and Dietmar Sauape. Springer. 1992.

a f (x) = x

ak

f x

f kx = ( )( )

B P E E− ( ) ~
Clustering in Space and Time

p, y () 

fractals form. Fractal structures are not the lowest energ

figuration that can be selected in, for example, th

dynamic systems, therefore some kind of dynamic selec

configuration must be taking place.

Bak explains the connection in the following way. A sign

be able to evolve through the system as long as it is able 

a connected path of above-threshold regions. When the 

is either driven at random or started from a random 

state, regions that are able to transmit a signal will form 

kind of random network. This network is correlated 

interaction of the internal dynamics with the externa

The complicated interrelation between the two d

dynamics means that a complex, finely-balanced system 

duced. As the system is driven, after this marginally stab

organised state has been reached, we will see flashes of a

as external perturbations interact with internal drivers to 

off avalanches (i.e., clusters) of activity through different 

in the system. Bak’s assertion is that the structure 

dynamic network is fractal. If the activated clusters con

fractals of different sizes, then the duration of the induce

cesses travelling through these fractals will vary g

Different timescales of this type lead to what is termed 1/

1/f noise is a label used to describe a particular form 

correlation in nature. If a time signal fluctuates in a see

erratic way, the question is whether the value of the 

 at time has any correlation to the signal measu

time ( ). The amount of causation is cha

ised by a temporal correlation function:

.

( ) 0 N τ 0 τ

0 τ τ + ( ) 0 N τ + τ

2

0 0 0 0 0

( ) ( ) ( ) ( ) τ τ G τ = N τ N τ + τ − N τ
Clustering in Spac

, y p, 

the power spectrum as follows:

where the power spectrum is defined in terms of the 

amplitude of the Fourier transform of the time signal:

.

1/f noise corresponds to the case where and 

sponds to a fractal clustering of the signal amplitude in t

FRACTAL CLUSTERING IN SPACE

For a system that is not poised in a critical state and th

about to change its mode of behaviour, the reaction of t

tem is described by a characteristic response tim

characteristic length of scale over which the perturbation 

However, for a critical system, the same perturbation a

at different positions or the same position at different tim

lead to a response of any size. The average is not ther

useful measure of response. The amount of the system in

is a cluster in the spatial dimensions of the system.

AN EXAMPLE–A PILE OF DRY SAND ON A BEAC

The nature of this critical state can be illustrated using 

ple example–a pile of sand. Anyone can try this expe

on a sunny day by the seaside. Normally, the grains o

only interact locally and nothing surprising happens. 

∫

∞

= τ τ π τ

0

S( f ) 2 d G( ) cos(2 f )

2

2 ( ) 1 ( ) lim ∫

−

π τ

→∞ = τ τ

T

T

i f

T d N e

T

S f

1 S f ( ) f =
Clustering in Space and Time

, (y p

perturbed by the addition of one grain of sand to a ran

chosen position on top of the pile, the extra grain will 

an avalanche (a cluster) characterised by spatial and tem

characteristics such as the total number of sand gr

involved in the avalanche and the lifetime of the avala

The statistical distributions describing the respon

denoted by P(s) and P(t). In the critical state, we expect 

power-law distributions of the form and 

The particular values of are then character

how the system can create such correlations in tim

space. These distributions will typically be bounded by 

and temporal lower and upper cut-offs. For example, an ava

cannot involve the displacement of less than one grain o

and the duration of an avalanche cannot be shorter th

time it takes one grain to move a distance equal to the si

single grain.

In the thermodynamic theory of phenomena, the spatial 

lation function plays a fundamental role. If the sys

described by a space-time field n(r,t), the spatial corr

function is defined as:

where a thermal average and an average over the posit

is assumed.

Away from the critical temperature, the correlations 

exponentially, , beyond the corr

length . The correlation length diverges, , 

critical temperature is approached. At the critical 

, the correlation function changes functional beh

from exponential to algebraic dependence upon 

Ps s ( )~ − 

β 

P(t) 

α β and 

2

0 Gr n n n () ( )( ) ( ) = +−0 0 0 0 r r r rr r

G(r) ~ exp(−r / ξ)

ξ −ν

ξ ~ T − Tc

T = Tc
Clustering in Spac

gg

lack of a characteristic length scale. Self-organising s

may be characterised and examined in this way.

CELLULAR AUTOMATA

The introduction in 1987 of the Self-Organised Criticali

cept employed the language of avalanches (clusters) o

grains. It proposed a simulation model (a cellular automata 

to be precise) of the most essential features of sand dyn

This cellular automata model is indeed characteris

power-laws and exhibits critical behaviour. As an exam

us look again at the modelling of an ecosystem consistin

number of coevolving species (the Bak-Sneppen evo

model) that we mentioned earlier. This automaton p

considers the points of a grid and has a simple set o

determining how the system changes from one time-step 

next, to represent the changing fitness of the species itse

the coevolutionary impact of that change on the fitn

closely linked species in the ecosystem. Note that this lin

one of local species influence and coevolution. It does n

essarily assume physical closeness. Although these ru

simple, the emergent behaviour of the system is compl

surprising–a characteristic of such nonlinear interaction

AN EXAMPLE: CLUSTERING AND COEVOLUTIO

A GLOBAL INFORMATION GRID

We can use the evolution model just described to gain 

into the effect of a Global Information Grid. Imagine 

grid in two dimensions. At each grid point is position

element of our force. Each such force element has a “f

value corresponding to its ability to evolve and adapt t

() 
Movement of a Boundary

the grid. We assume these fitness values are random a

At each step of the process, we assume that the force e

with the smallest fitness is likely to have to adapt fastes

local environment. In so doing, it will change the fitne

ues of the units closest to it on the information grid (i.e

is local coevolution). Note that these force elements m

separated by large and varying distances in space. With 

assumptions, over time the force elements will form clus

coevolution of the form predicted by the Bak-Sneppen 

In particular, the statistics of emergent cluster size can b

dicted mathematically to converge to a power-law. Th

probability of a cluster of size S, P(S), is of the form 

where is a characteristic cluster exponent for this par

information grid.

Thus far, we have looked at complex effects and corre

across space and time. What happens if we restrict ourse

looking at the boundary between two different regime

as two different nationalities or two opposing armed f

and how this would move over time depending on th

coevolution of the elements involved?

MOVEMENT OF A BOUNDARY

In natural systems, we can consider the movement of a b

ary through a medium (for example, the boundary 

atomic surface, the boundary of a growing cluster of ba

or the front of advance of a fluid “invasion” of a medium 

as a crystalline rock). This has been studied extensively i

tion to the laying down of single atom surfaces using mo

beam epitaxy [9]. The most relevant case from our p

view is the front of advance of fluid “invasion” of a m

As described in [9], we can represent the medium itself a

τ
Movement of 

g , 

represents the fact that that cell can be wetted. The prop

of cells containing a “1” is defined as p. For large con

tions, we can also interpret p as the probability 

particular cell contains a “1.” A “0” represents the fact t

cell cannot be wetted–it thus “pins” the advance of th

through the medium, at least locally. In the standard Di

Percolation Depinning (DPD) model, we start off wi

example) a two-dimensional square lattice of cells, 

labelled with 1s and the rest with 0s to represent the di

tion of these “pinning” forces throughout the m

Initially, we wet one edge of the lattice. In Figure 1.5, w

the wetted right-hand edge. In the standard DPD mod

of the unpinned cells in the next column is chosen at ra

and the fluid invades that cell.

Figure 1.5: Local Pinning of a Fluid Boundary
Movement of a Boundary

, pg p

greater than a critical value pc, the growth of the inter

halted by a spanning path of pinning cells. Such mo

interface or boundary movement exhibit fractal prope

the interface, as discussed in detail in [9]. We shall see 

effects later in our discussion in Chapter 4 of the contro

battlespace using ideas based on preventing the flow of 

ing forces and/or third parties through the space. Rathe

choosing the next cell to invade at random, as in the 

model, we can use a model of the process that is more 

the manoeuverist principle of applying your strength 

the opponent is weak–in other words, the cell next to b

ted is the one where the local pinning force of the med

weakest. Such a model of the boundary movement is th

sion Percolation model. We can create a model of this proce

way that is consistent with our description of the Bak-Sn

evolution model of local coevolution [6]. We start by 

ing, as with the Bak-Sneppen evolution model, ra

numbers f

i

 between 0 and 1 to the points of a d-dimen

lattice. Initially, one side of the lattice is the wetted 

The random numbers at the boundary of the wetted 

are examined. At each update step s, the site with the sm

random number f

i,s on the boundary of the wetted 

located and added to the cluster. In this case, we can in

the values f

i

 as the values of the local pinning force, a

cluster advances at those points where the pinning f

smallest. As noted in [6], an important physical realisa

invasion percolation is the displacement of one flu

another in a porous medium. The boundary of the clust

ated by this process is fractal and has a fractal dimension 

range 1.33-1.89 dependent on the exact definition 

boundary [6, Appendix].
Movement of 

AN ANALYTICAL MODEL OF THIS PROCESS

In natural systems, the boundary of such an interface 

moving through a medium can be characterised by its “

ness.” This is defined as follows.

We assume that the interface is defined across a linear sp

length L (as shown in Figure 1.6). The roughness of the in

at a given time t and for a span of length L, is defined [9

where the interface length is divided into a number o

ral cells of unit length, h(i,t) is the height of the ith colu

time t (as indicated in Figure 1.6) and is the aver

these heights at time t. The roughness W(L,t) is thus th

dard deviation of the height at a given time. For 

natural systems, the roughness first goes through a tra

period before stabilising at an equilibrium value. Once 

stabilised, the expected behaviour is that this “satur

value Wsat scales with the interface width L, i.e. we 

the relationship Wsat .

Figure 1.6: The Roughness “W” of an Interface

2

1

1 ( , ) ( ( , ) ( ))

L

i

W Lt hit ht

L =

= − ∑

h t( )

L

α
Movement of a Boundary

, p

exponent. It is then a characteristic exponent of the in

process under study. A typical value of is about 

shown in [9]). This idea can also be used to chara

Brownian motion, as we discuss later in this chapter 

context of stock price dynamics.

By using symmetry arguments, we can derive [9] an ana

expression of the rate at which such an interface 

through a medium. (In terms of conflict, this correspo

the rate of advance of a combat front.) This is given 

Kardar-Parisi-Zhang (KPZ) equation:

.

The first term in this equation represents linear effects 

interface growth, the second captures nonlinear effec

the third is a noise term. This thus represents the startin

for an analytical expression (i.e., a metamodel) of the ad

of a conflict front through a locally controlled area, 

cussed below.

AN EXAMPLE: FORCE CONTROL ALONG A

BOUNDARY

The concept of pinning a fluid locally is similar to the 

trying to exert local control over a boundary to preve

flow of other forces or third parties across that bounda

will see later (in Chapter 4) that the idea of control as th

vention of such flows through an area has imp

implications for the emergent behaviour of a force (or tw

peting forces) attempting to exert control over a battlespa

α

2 2 (,) ( ) (,)

2

hxt v h h xt

t

λ

η

∂ =∇ + ∇ +

∂
An Example of Complex Behaviour and Fractal Time Series in 

g y, py 

in terms of the invasion percolation model, can be u

analyse the effect of two forces (an attack and a defence 

interacting across a boundary, when the boundary mo

the point where the defending (pinning) force is weakest

defence pinning force is coevolving locally, then the bou

should form a fractal with a fractal dimension in the 

1.33-1.89 [6, Appendix], as we have seen. In Chapter 

show that there is historical evidence in warfare for su

effect, and for values in this range.

AN EXAMPLE OF COMPLEX BEHAVIOUR AN

FRACTAL TIME SERIES IN ECONOMICS

BROWNIAN MOTION, FRACTALS, AND SIMILAR

Mandelbrot [10] has considered the movement of stock 

using fractal ideas and we start by using a simple ex

based on Brownian motion (since this is the basis of mo

rent predictors of stock price volatility). Consider 

Brownian motion in one space variable, thus the mo

particles is restricted to a line. The impacts affect the p

only from the left and right, causing a displacement of l

in either direction. Can any prediction be made abo

total displacement after a number of time-steps n? Fir

total expected displacement is zero, as all displacements 

or –l, both with equal probability 0.5. Consider, instea

square of the displacement. The average of these squa

placements, called the mean square displacement, indicate

much the particles spread in a given number of time-st

the average. Its value is nl2

. In terms of Brownian motio

number of steps corresponds to the number of impact

particle and cannot be directly measured in an exper
An Example of Complex Behaviour and Fractal Time Series in Economics

, , g 

number of n impulses during a timespan t, the particle tr

total length nl. If v is the average speed of the particle, th

have vt = nl. The mean square displacement nl2

 thus equ

Therefore, the mean square displacement is proportio

the timespan. Is there anything else that can be said reg

the distribution of the displacement after time t? Expe

tally, it is found that the statistics for a simulated 

dimensional Brownian motion follow a Gaussian or n

distribution. This is not surprising as a normal distri

arises where independent and identically distributed ra

events are averaged and is an implication of the Centra

Theorem of Statistics.

Brownian motion is also associated with scaling relationsh

fractal behaviour. As a nonfractal object is magnified, no ne

tures are revealed. As a fractal object is magnified, finer 

are revealed. The size of the smallest feature of a non

object is called the characteristic scale. A measurement m

finer resolution will include more of these smaller pieces

the value measured of a property will depend upon the 

tion used to make the measurement. How a me

property depends on the resolution used to make the me

ment is called the scaling relationship. A fractal obje

features over a broad range of sizes. Fractal phenomeno

characteristics are:

1. SELF-SIMILARITY: behavioural characteristics are 

lar” at different resolutions.

2. SCALING: the value measured for a property depe

upon the resolution at which it is measured.

3. DIMENSION: the dimension of an object gives a qu

tative measure of self-similarity and scaling. It tells 
An Example of Complex Behaviour and Fractal Time Series in 

y pj

viewed at higher magnification.

4. NONSTATISTICAL PROPERTIES may be observed. 

Moments may be zero or nonfinite (e.g., the mean 

towards zero and variance tends towards infinity).

There are two types of self-similarity:

1. GEOMETRICAL: pieces of the object are exact sma

copies of the whole object.

2. STATISTICAL: the value of the statistical property 

measured at resolution r is proportional to the val

Q(ar) measured at a resolution ar such that Q(ar) = 

For statistical probability distribution functions (pd

this implies that: pdf[Q(ar)] = pdf[kQ(r)].

Statistical self-similarity is possible in both space (spati

time (temporal).

In summary then, such self-similarity implies a scalin

tionship. The simplest form of the scaling relationship 

the measured value of a property Q(r) depends on the 

tion used to make the measurement, as in the eq

Q(r) = Brb

, i.e. LogQ(r)=LogB+bLog r, where B and b ar

stants. Hence for self-similarity, we observe log-log 

behaviour. Experimentally, this implies that for Bro

motion, there should be a scaling factor, r, that yields 

that are visibly identical, i.e. when scaling Brownian m

in time, by a factor say of t and in amplitude by a fact

we should see no difference. This transformation is c

scaling collapse since it has the effect of collapsing the cu

different scales of measurement on top of each other in

normalised relationship.
An Example of Complex Behaviour and Fractal Time Series in Economics

g p y 

cally for Brownian motion. It follows from the mean sq

analysis that the mean squared displacement, , 

Brownian motion X(t), is described by . Now co

the rescaled random function:

,

i.e. the graph of X is stretched in the time direction by 

tor a and in the amplitude by r. The displacements in 

time differences t are the same as those in X multiplie

for corresponding time differences t/a. Thus, the sq

displacements are proportional to r

2t/a. In order to 

the same constant of proportionality as the original B

ian motion, we require or . For ex

when replacing t by t/2, i.e. stretching the graph by a 

of 2, we have a=2 i.e., .

In this broader context of fractal processes, ordinary Bro

motion is a random process X(t) with Gaussian incremen

 where H (the Hurst 

nent) = 1⁄2. We can consider (as discussed in [9]) the Bro

motion time series as describing an interface (betwe

parts above and below the series) stretching between th

points t1 and t2. In terms of our previous discussion of in

roughness, we now have L=t2-t1. The standard deviatio

the roughness of the interface generated by the Bro

motion time series) over this timespan L has the for

where in this case equals the Hurst exponent H (as w

see from the expression for the variance above in terms 

Thus, standard Brownian motion corresponds to a rou

exponent of 1⁄2. Other values of H are possible, corresp

to rougher or smoother forms of time series.

2 ∆

∆ ∝ t 2





 



 = a

t Y(t) rX

/ 1 2r a = r = a

r = 2

H

X t X t t t

2

2 1 2 1 var( ( ) − ( )) ∝ −

α
An Example of Complex Behaviour and Fractal Time Series in 

STOCK PRICES AND BROWNIAN MOTION

Mandelbrot considered the movement of stock prices u

Brownian random walk process as his starting point [10

behaviour of such a variable z can be understood by co

ing the changes in its value in small intervals of time. Co

such a small interval of time and define as the c

in z during . must have two basic properties

 where is a random draw from a standa

normal distribution (i.e., a normal distribution with mea

and standard deviation of 1.0); and second, the values 

for any two different short intervals of time are in

dent. It follows that has a normal distribution with 

zero, standard deviation of and variance . The 

property implies that z follows a Markov process. Now 

sider the change in the value of z during a relativel

period of time T. This can be denoted by z(T)-z(0)

change can be regarded as the sum of the changes in 

small time intervals of length , where . Thus:

z(T)-z(0) = 

where (i = 1,2,...N) are random drawings from a sta

ised normal distribution. It follows that z(T)-z(0) is no

distributed with mean zero, variance of N = T and 

dard deviation . The process described so far has 

rate of zero and a variance rate of 1. The expected val

at any future time is equal to its current value and the va

of the change in z in a time interval T equals T.

A generalised Wiener process for a variable x generalises th

cept of the drift rate and variance of such a process, an

be defined in terms of dz as where a and 

∆t ∆z

∆t ∆z

∆z = ε ∆t ε

∆t

∆z

∆t ∆t

∆t

t

T N

∆ =

t

N

i

∑ i ∆ =1

ε

i ε

∆t

T

dx = adt + bdz
An Example of Complex Behaviour and Fractal Time Series in Economics

pp

of a per unit time. Without the bdz term, the equation be

, i.e. .

The bdz term may be regarded as adding noise or variab

the path followed by x. For a small time interval, 

change in x, , is given by where 

random draw from a standardised normal distribution. 

a normal distribution with mean of a , standard devia

, and a variance of . Similarly, the mean chan

for any time interval T is normally distributed with 

change in x given by aT, standard deviation of chang

given by , and variance of change in x as b2T.

Even more generally, a stochastic process is defined as 

process if .

A basic simulation of stock price movement would be to 

generalised Wiener process. This is clearly inadequate 

assumes that both a constant drift rate and constant va

rate occur, i.e. the percentage stock return is dependen

stock price. The constant expected drift rate assump

inappropriate and is replaced by the assumption th

expected drift, expressed as a proportion of the stock p

constant. Thus, if S is the stock price, the expected drift 

S is for some constant parameter and for a sma

interval, , the expected change in S is . If the va

rate of the stock price is always zero, then:

 or 

i.e., 

dx = adt a

dt

dx =

∆

∆x ∆x = a∆t + bε ∆t 

∆t

b t ∆ b ∆t 2

b T

dx = a(x,t)dt + b(x,t)dz

μS μ

∆t μS∆t

dS = μSdt dt

S

dS = μ

t S S eμ = 0
An Example of Complex Behaviour and Fractal Time Series in 

p

when the variance rate is zero, the stock price gro

declines) at a continuously compounded rate of p

time. In practice, as stock prices exhibit volatility, a reas

assumption is that the variance of the percentage retur

short period of time, , is the same regardless of stock 

Define as the variance rate of the proportional cha

the stock price. Thus is the variance of the propo

change in stock price, S, during time . The instant

variance rate for S is . This implies that S can be 

sented by an Ito process that has an instantaneous drift r

and instantaneous variance rate . This can be writt

or

and

.

The change in ln S between times t and T is thus no

distributed:

where is the stock price at a future time T, S is th

price at the current time, and denotes a normal 

bution with mean m and standard deviation s.

S0

μ

∆t

2 σ

∆t 2 σ

∆t

2 2 σ S

2 2 σ S

dS = μ Sdt +σ Sdz

= μ dt +σ dz

S

dS

d S dt σdz σμ + 













 = − 2

ln

2





 



 − − 













 S − S ≈ − T t T t T σ

σ φ μ ( ), 2

ln ln

2

T S

φ(m,s)
An Example of Complex Behaviour and Fractal Time Series in Economics

thus ln St

 is normally distributed so that ST has a log-n

distribution.

Under closer examination, stock prices in fact depart fro

normal behaviour. Examination of movements in stock 

presents changes greater than our model predicts. 

returns exhibit leptokurtosis, i.e. the likelihood of return

the mean and of large returns is greater than our Bro

motion model predicts, whilst other returns tend to 

likely [11, 12]. Table 1.1 gives a précis of the statistics 

non-normal behaviour.

Table 1.1: Decade-by-Decade Behaviour of Daily Returns

from the Dow Jones Index

CLUSTERING IN TIME

What is of particular interest is that Turner and Weigel

strongly suggest the occurrence of temporal clustering

Decade Mean

%

Standard 

Deviation

%

Skewness Kurtosis Max

%

1920s 0.0058 2.1453 0.1321 11.9272 11.6396 -1

1930s -0.0216 1.8233 0.2856 4.5422 13.8635 -8

1940s 0.0098 0.7618 -1.0890 12.8516 6.5275 -7

1950s 0.0470 0.6578 -0.9201 7.3343 4.0476 -6

1960s 0.0063 0.6558 0.0437 5.4873 4.5787 -5

1970s 0.0025 0.9262 0.2693 1.7889 4.9517 -3

1980s 0.0489 1.1607 -4.3115 99.3933 9.6661 -2

Overall 0.0142 1.1329 -0.7919 28.8256 13.8635 -2





 



 − − 













 S ≈ S + − T t T t T σ

σ φ μ ( ), 2

ln ln

2
An Example of Complex Behaviour and Fractal Time Series in 

2p, 2p

positive jumps in the S&P 500 occurred within one an

days respectively of another positive jump in equity 

Positive jumps in the Dow Jones were similarly clustere

11.3 percent of the positive jumps taking place within o

and 36.2 percent transpiring within 5 days of each other

The second defect with the Brownian motion model is 

the model holds, then stock returns should be proportio

elapsed time and the standard deviation of returns sho

proportional to the square root of elapsed time. This is 

upon the scaling properties of Brownian motion. Turn

Weigel’s data also demonstrated that monthly and qu

volatilities are higher than annual volatilities and conv

that daily volatilities are lower than annual volatiliti

their research shows that the stock returns do not sca

Brownian motion sense.

The answer to these problems can perhaps be found b

sidering the mathematical form of these distributions [1

have shown that scaling is observed within Brownian m

and that this may be characterised by a Gaussian prob

distribution plot. A log-normal distribution may be c

assumed to be a Gaussian plot with an increased tail. Th

of Pareto distributions also die off much more slowl

Gaussian or log-normal tails. Such fat tail probability di

tions are thus increasingly describing the greater and g

volatility of the system. Current financial models rely 

extension of the basic Brownian motion model, either 

ing a different volatility distribution (as in “jump vo

calculations”) or attempting to empirically analyse and 

the observed volatility. For example, for a Pareto prob

distribution of the random variable y, with y>1 then we o
Summary

(y) ,

i.e. a power-law. For further discussion of these and 

ideas from Complexity Theory and dynamical systems u

financial mathematics, see references 10 and 13.

SUMMARY

In summary, we have looked in some depth at the co

behaviour of natural biological and physical systems. 

our analysis of these open and dissipative systems, it i

that there are a number of key properties of complexi

are important to our consideration of the nature of futur

fare. Such futures, involving the exploitation of 

coupled command systems such as Network Centric W

will have to take account of these key properties. A list o

is given here, and then discussed further in Chapter 2 

context of Network Centric Warfare.

1. NONLINEAR INTERACTION: this can give rise to s

ing and non-intuitive behaviour, on the basis of sim

local coevolution.

2. DECENTRALISED CONTROL: the natural systems 

have considered, such as the coevolution of an eco

tem or the movement of a fluid front through a 

crystalline structure, are not controlled centrally. T

emergent behaviour is generated through local 

coevolution.

3. SELF-ORGANISATION: we have seen how such na

systems can evolve over time to an attractor corre

ing to a special state of the system, without the ne

guidance from outside the system.

yα
Q(p

space and time correlations) inherent in an open, 

pative system far from equilibrium.

5. ADAPTATION: we have seen how such systems are 

stantly adapting–clusters or avalanches of local 

interaction are constantly being created and disso

across the system. These correspond to correlatio

effects in space and time, rather a top-down impo

of large-scale coincidences in space and time.

6. COLLECTIVIST DYNAMICS: the ability of element

locally influence each other, and for these effects t

ple through the system, allows continual feedback 

between the evolving states of the elements of the 

system.

REFERENCES

1 HOFFMAN F G and HORNE G E (1998). Maneuver Warfare Science 

Dept of the Navy, HQ U.S. Marine Corps, Washington DC.

2 FORDER R (2000). “The Future of Defence Analysis.” Journal of D

Science. 5, No. 2. pp. 215-226.

3 CEBROWSKI A (2000). “Network Centric Warfare and Informatio

Superiority” Keynote address from proceedings, Royal United Serv

Institute (RUSI) conference “C4ISTAR; Achieving Information 

Superiority.” July 2000. RUSI, Whitehall, London, UK.

4 PRIGOGINE I (1980). From Being to Becoming; Time and Complexity in 

Physical Sciences. W H Freeman and Co., San Francisco, USA.

5 MOFFAT J (2002). Command and Control in the Information Age; Represen

Impact. The Stationery Office, London, UK.

6 PACZUSKI M, MASLOV S, and BAK P (1996). “Avalanche Dyna

Evolution, Growth and Depinning Models.” Physics Review. E 53 No

pp. 414-443.

7 BAK P, TANG C, and WIESENFELD K (1987). Self-Organised Critic

Explanation for 1/f Noise. Physics. Review Letters, 59. pp. 381-384.
Additional Reference

J ()

Noise.” Nature. Vol 410. pp. 242-250.

9 BARABASI A L and STANLEY H E (1995). Fractal Concepts in Surfac

Cambridge University Press. Cambridge, UK.

10 MANDELBROT B (1997). Fractals and Scaling in Finance. Springer-V

11 TURNER A L and WEIGEL E J K (1990). “An Analysis of Stock M

Volatility.” Technical Report, Frank Russell Co., Tacoma WA. USA

12 TURNER A L and WEIGEL E J K (1992). “Daily Stock Market Vo

1928-1989.” Management Science. 38. pp. 1586-1609.

13 MANTEGNA R and STANLEY H E (2000). An Introduction to Econo

Correlations and Complexity in Finance. Cambridge University Press. Cam

UK.

ADDITIONAL REFERENCE

14 PEITGEN H-O, JURGENS H, and SAUAPE D (1992). Chaos and F

Springer-Verlag.
45

CHAPTER 2

CONCEPTS FOR

WARFARE FROM

COMPLEXITY THEORY

s a starting point for our journey,

Chapter 1 established in some depth and

detail the key ideas and methods from Complexity Theory that we can bring to bear in thinking

about and modelling future warfare. In his RUSI

keynote talk [1], ADM Cebrowski (Head of the

Office of Force Transformation, U.S. DoD)

indicated that Network Centric Warfare is an

emerging theory of war based on the concepts of

nonlinearity, complexity, and chaos. It is less

deterministic and more emergent; it has less

focus on the physical than the behavioural; and

it has less focus on things than on relationships.

It is clear from the discussion and modelling in

Chapter 1 that Complexity Theory is the

essence of these ideas.

A
p[2] p

the effects of command and control in agent-based simu

of Information Age warfare. This is done by representi

process as the interaction of top-down and bottom-up 

These are described as Deliberate Planning and Rapid 

ning. Deliberate Planning is appropriate when ample 

available for the consideration of a number of alte

courses of action by either “side” and a course of action 

chosen that is considered to be, in some sense, optimal. 

Planning is appropriate when time is short and exper

sionmaking under stress leads to a pattern-matching app

To quote from [2]:

Combat is, by its nature, a complex activity. Ashby’s Law of 

Requisite Variety...which emerged from the theoretical consider

ation of general systems as part of Cybernetics, indicates that 

properly control such a system, the variety of the controller (th

number of accessible states which it can occupy) must match t

variety of the combat system itself. The control system itself, in 

other words, has to be complex. Some previous attempts at rep

senting C2 in combat models have taken the view that this mu

inevitably lead to extremely complex models. However, recent 

developments in Complexity Theory...indicate another way fo

ward. The essential idea is that a number of interacting units

behaving under small numbers of simple rules or algorithms, c

generate extremely complex behaviour, corresponding to an 

extremely large number of accessible states, or a high variety c

figuration, in Cybernetic terms. It follows that, if we choose th

simple interactions carefully, the resultant representation of C2 

will be sufficient to control, in an acceptable way, the underlyi

combat model. As part of this careful choice, we need to ensur

that the potentially chaotic behaviour generated by the interacti
f ppy p2 

which remains focused on the overall, high level, campaign 

objectives....

...It follows, from what we have just said, that the representati

of the C2 process must reflect two different mechanisms. The 

first is the lower-level interaction of simple rules or algorithms

which generate the required system variety. The second is the 

need to damp these by a top-down C2 process focused on cam

paign objectives. Each of these has to be capable of being 

represented using the same Generic HQ/Command Agent obj

architecture. We have chosen to do this by following the genera

psychological structure of Rasmussen’s Ladder, as a schema fo

the decisionmaking process. At the lower levels of command 

(below about Corps, and equivalent in other environments), th

will consist of a stimulus/response mechanism. In cybernetic 

terms, this is feedback control. At the higher level, a broader 

(cognitive-based) review of the options available to change the 

current campaign plan (if necessary) will be carried out. In 

cybernetic terms, this is feedforward control since it involves th

use of a ‘model’ (i.e., a model within our model) to predict the 

effects of a particular system change.

In the last chapter of [2] (Chapter 6: “Paths to the Futur

following point is made, which is the foundation for all 

work and ideas presented here:

Modelling and analysis to determine the effect of such phenome

underpin our thinking about such future conflict, the representa

tion of information and command being at their heart. A new 

approach to capturing these effects has been put forward in thi

book, and is having a significant influence on the approach to 
g p, pg pf 

intelligent agents in conflict, set within a widely divergent set 

possible futures, leads to a rich set of possible trajectories of sy

tem evolution for analysis to consider. We thus need to 

complement this effort with other work to categorise and under

stand the classes of behaviours which might emerge from such 

complex situation. This is the domain of Complexity Theory.

The overall aim is thus to develop an “Operational Syn

(as discussed in [3]) of both agent-based modelling appr

(as described in [2]) and higher level mathematical met

els based on Complexity Theory. Reference [2] lays ou

initial ideas on how to develop such an understanding 

on a theoretical approach to the development of a highe

“metamodel” of a cellular automaton model of conflict s

the ISAAC model developed by the U.S. Marine Corps 

bat Development Centre. We will develop these ideas 

in Chapters 4 and 5. In Chapter 5 in particular, we wi

sider again the ISAAC model and demonstrate how th

of Complexity Theory lead to an understanding of the c

forming and reforming in such a model, and how they 

to the emergent behaviour of the model.

For the moment, let us consider the list of key concept

Complexity Theory at the end of Chapter 1. These are 

left-hand side of Table 2.1. On the right-hand side is an 

pretation in terms of military behaviour and doctrine 

we have termed “an Information Age force structure”).

The nature of Network Centric Warfare for such future 

mation Age forces can be outlined as: within a broad 

and constraints available to all the forces, the local forc

self-synchronise under mission command in order to a

the overall intent [4].
Table 2.1: Relation Between Complexity and Information Age W

This process is enabled by the ability of the forces invo

robustly network. We can describe such a system as loos

pled to capture the local freedom available to the u

prosecute their mission within an awareness of the 

intent and constraints imposed by high-level command

also emphasises the looser correlation and nonsynchronou

tionship between inputs to the system (e.g., sensor repor

outputs from the system (e.g., orders). In this process, in

tion is transformed into “shared awareness,” which is av

to all. This leads to units linking up with other units, wh

either local in a physical sense or local through (for ex

an information grid or Intranet (self-synchronisation). T

turn leads to emergent behaviour in the battlespace, as 

in Figure 2.1.

COMPLEXITY CONCEPT INFORMATION AGE FORCE

Nonlinear interaction Combat forces composed of a larg

number of nonlinearly interacting 

Decentralised Control There is no master “oracle” dictati

actions of each and every combata

Self-Organization Local action, which often appears 

“chaotic,” induces long-range ord

Nonequilibrium Order

Military conflicts, by their nature, 

ceed far from equilibrium. Correla

of local effects is key.

Adaptation

Combat forces must continually a

and coevolve in a changing 

environment.

Collectivist Dynamics

There is a continual feedback betw

the behaviour of combatants and 

command structure.
Figure 2.1: Information Leading to Emergent Behaviour

Compare these ideas with the broad conceptual framew

Complexity Theory, as summarised below.

THE CONCEPTUAL FRAMEWORK OF COMPLEX

Prof. Murray Gell-Mann [5] traces the meaning to the 

the word. Plexus means braided or entwined, from w

derived complexus, meaning braided together, and the E

word complex is derived from the Latin. Complexity is th

associated with the intricate intertwining or inter-conne

of elements within a system and between a system and it

ronment. In a human system, connectivity means 

decision or action by any individual (group, organisation

tution, or human system) will affect all other 

individuals and systems. That effect will not have equal 

form impact, and will vary with the state of each 

individual and system at that time. The state of an ind

and system will include its history and its constitution, w

turn will include its organisation and structure. Conne

applies to the interrelatedness of individuals within a syst

well as to the relatedness between human social systems, 

include systems of artifacts such as information system

intellectual systems of ideas.

The term complexity is used to refer to the theories of co

ity as applied to complex adaptive systems (CAS). The

dynamic systems able to adapt and change within, or 

of, a changing environment. It is important to note, ho
y y

ment in the sense that a system always adapts to a ch

environment. The notion to be explored is rather that o

tem closely linked with all other related systems making 

“ecosystem.” Within such a context, change needs to b

in terms of coevolution with all other related systems as w

in Chapter 1, rather than as adaptation to a separate a

tinct environment.

COLLECTIVIST DYNAMICS AND FEEDBACK

The phenomenological definition of a complex system is 

exhibits nonlinear, emergent, adaptive behaviour. Non

behaviour is associated with far-from-equilibrium, ope

tems, in that cause and effect are no longer linearly conn

This is ultimately due to the type of internal-external 

interactions (feedback) affecting our system.

SELF-ORGANISATION AND CLUSTERING

Self-organisation in this context is taken to mean the c

together of a group of individuals to perform a particula

They are not directed by anyone outside the group. Thi

the same as “self-management,” as no manager outsi

group dictates that those individuals should belong t

group, what they should do, or how it should be done. I

group members themselves who choose to come togethe

decide what they will do and how it will be done. A fea

these groups is that they are informal and often temp

Enabling self-organisation can often be a source of inno

Military commanders who understand the nature of 

staktik1 have always understood this: a commander 

regard his superior’s intention as sacrosanct and m
Forest Fires, Clusters of Trees, and Casualties in War

yg ppyg 

is given a task and resources and any constraints, and 

this framework he is left to make his plan.

We already know from Chapter 1 that some complexity 

els of natural ecosystems use extremal dynamics 

behavioural driver. In systems where the dynamical evo

is a struggle against various types of thresholds or barrie

action will predominately occur where the net bar

change is the smallest. The Bak-Sneppen evolution 

described in Chapter 1 is an example of such an ex

model. The species with the lowest fitness coevolves firs

ilarly, in considering the movement of a fluid thro

medium, the boundary moves where the pinning fo

smallest. The avalanches of active sites are the clusters c

in such models. In the model ecosystem, the system self-

ises towards a critical point where it has the g

dynamical freedom–clusters of all sizes can potentially b

ated, and the statistics of the emergent distribution o

cluster sizes can be predicted–the distribution is of pow

form. Such extremal dynamics echo one of the key te

manoeuvre warfare–namely, to focus your strength a

your enemy’s weakness.

FOREST FIRES, CLUSTERS OF TREES, AND

CASUALTIES IN WAR

The “forest fire” model is another example of a syste

evolves to a critical point through a process of local in

tion. In this case, we start with an empty two-dimensiona

At each iteration of the process, with a certain probab

1

Auftragstaktik is directed control, as opposed to befehlstaktik (det

order tactics).
Forest Fires, Clusters of Trees, and Casual

(y ), p 

point. If the grid point is blank, the tree is planted. If t

already a tree there, the new tree is discarded. With pr

ity (1-p) we drop a spark onto a random site instead of a 

the site is bare the spark goes out. If there is a tree on t

the tree and all of its immediate neighbours burn. All 

neighbours of these trees then burn and so on until th

plete cluster of linked trees is burned (this is termed a for

The rate (which is [1-p] if each iteration of the pro

counted as a unit of time) of sparks dropping onto the 

termed the sparking frequency. This sparking frequency sp i

driver of the dynamics of the forest ecosystem. If sp is 

very large clusters of trees are allowed to form, which sp

entire grid. When a spark is then dropped, the forest fire 

out an entire forest stretching from one side of the grid 

other. In Complexity Theory, this is known as s

noise [6]. This name comes from looking at the behav

the system over time–large spikes of tree extinction (fore

are created at isolated points in time. If the sparking freq

sp is very large, then tree clusters do not have the cha

grow. Thus, over time, the system produces a large num

small spikes of activity, which are called popping noise. W

is in the intermediate regime, the system self-organise

critical state where the clusters of burnt trees have a di

tion represented by a power law, and clusters of all sizes 

created. Over time, the spikes produced by this p

(i.e., the time evolution of forest fires of various sizes) 

similar dynamic to that produced by the acoustic dynam

crumpling paper [6], and so this regime is termed cracklin

It is possible [7] to relate such self-organised behaviou

forest fire model to the statistics of the scale and inten

conflicts. This is the beginning of an explanation as t
Forest Fires, Clusters of Trees, and Casualties in War

p

in [7], “...the behaviour of the forest fire model c

explained in terms of a cascade model. If trees are ran

planted on a grid, the distribution of cluster sizes is expo

(Poissonian), not power law (fractal). The distribution of 

sizes in the forest fire model is power law (fractal). T

because clusters of trees continuously grow and comb

form larger clusters. Small fires sample this population o

ters, but the loss of trees in fires is dominated by the 

fires. There is a self-similar cascade of trees from small t

clusters. In terms of the forest fire model, a spark ignites 

and the model fire consumes the entire cluster to whi

tree belongs. This is also the case for real forest fires. Ig

of the forest must take place for a fire to take place, and t

will then spread through the contiguous flammable m

A war must begin in a manner similar to the ignition o

est. One country may invade another country, or a prom

politician may be assassinated. The war will then sprea

the contiguous region of metastable countries....” We w

at this fractal and power law behaviour of casualties in 

detail in Chapter 3.

TUNING AND GOAL SEEKING

This leads us to consider the concept of tuning. Self-orga

systems (such as the Bak-Sneppen evolution model of a

system described in Chapter 1) can, as their name im

develop local organisation within the system in order to 

towards an attractor. Tuning can be seen as a directive w

the macrosystem to attempt to influence the behaviour 

microsystem. A controlling intelligence is deemed to be 

sary in order to guide the system towards a particular go

example of this is the tuning of the sparking frequency 
pp 

ity (and hence power law statistics for the size of clus

burnt trees). Varying the tuning parameter (the sparki

quency) of the forest fire model represents intervention 

outside the system in order to ensure that it heads tow

particular goal. This question of tuning makes us consid

boundaries of the systems we are examining, and the 

energy and/or information across the system boundary.

INFORMATION FLUX ACROSS THE BOUNDARY O

OPEN SYSTEM

In such open dissipative systems, there will always be flu

information and/or energy across the system boundary

example of this, we show in Chapter 4 how it is poss

quantify the flux of information across the boundary 

open system (a wargame) using the concept of Inform

Entropy. We can then use this to relate such a flux of in

tion to emergent properties of the wargame (such 

number of casualties suffered). The idea of using entrop

measure of information (and hence knowledge) is then a

in Chapter 4 to show how the benefits of network-cen

opposed to platform-centric) approaches to specific task 

cution can be quantified (this latter based on work 

RAND Corporation).

REFERENCES

1 CEBROWSKI A (2000). “Network Centric Warfare and Informatio

Superiority.” Keynote address from proceedings, Royal United Serv

Institute (RUSI) conference. “C4ISTAR; Achieving Information 

Superiority.” RUSI. Whitehall, London, UK.

2 MOFFAT J (2002). Command and Control in the Information Age; Represen

Impact. The Stationery Office. London, UK.
References

()

Dept of the Navy, HQ U.S. Marine Corps. Washington, DC.

4 ALBERTS D S, GARSTKA J J and STEIN F P (1999). Network Cen

Warfare; Developing and Leveraging Information Superiority. CCRP, DoD. 

Washington, DC, USA.

5 GELL-MANN M (1994). The Quark and the Jaguar. WH Freeman. Ne

USA.

6 SETHNA J P, DAHMEN K A, and MYERS C R (2001). “Cracklin

Noise.” Nature. 410 8. March 2001. pp. 242-250.

7 ROBERTS D C and TURCOTTE D L (1998). “Fractality and Self

Organised Criticality of Wars.” Fractals. 6 No 4. pp. 351-357.
57

CHAPTER 3

EVIDENCE FOR

COMPLEX EMERGENT

BEHAVIOUR IN

HISTORICAL DATA

INTRODUCTION

he first point to make is that the exploitation of manoeuvre warfare is not new.

Commanders have exploited such an approach

in previous generations. For example, LTG Sir

Francis Tuker [1] indicated that at a threedimensional spatial level, manoeuvre warfare is

determined by three conditions:

1. Flanks shall be tactically open or it shall be 

possible to create a flank by break-in and 

breakthrough.

2. The mobile arm shall be predominant.

T
Time Series Behaviour

p

point where it will decide the battle and gain deci

victory.

Examination of historical data should thus give us some 

into these effects. 

Secondly, the idea that execution of military network-

enterprises within a battlespace is of a self-organising 

has interesting consequences for the future of both warfi

and other operations. A system that comprises of c

grouping and regrouping, the increasing tempos of battl

awareness, operations, and responsiveness etc. require th

command decision process (or better, the rate of cha

decision process) is undertaken with a similar tempo. Th

of “battle speed” to “C2 speed’” is thus a critical issue

discussed in [2]. When this ratio is high, the system rev

being self-organising in nature. In considering such self-

ising systems earlier, we indicated that under 

conditions, the time series of outputs from such a 

should have a certain fractal character. In the case of a 

ecosystem such as the Bak-Sneppen evolution model dis

in Chapter 1, in fact, this fractal process is driven by the 

mal (smallest) fitness values on the lattice of interacting sp

TIME SERIES BEHAVIOUR

Applications of these ideas have been made in order to p

combat casualty rate patterns from WWII data [3]. A 

series model based on this fractal self-organising approa

been contrasted with three other prediction methodolo

neural network; use of nonlinear prediction (a predic

made by searching the data for the nearest N points 

dimensional embedding and estimating the behaviour c
Time Serie

p []); y, 

a maximum entropy method to calculate a power spe

from which a linear prediction may be made [5]. All o

approaches are available in a package of time series a

procedures (the Chaos Data Analyser) produced by the A

can Institute of Physics for the analysis of experimental 

natural systems, and that is what we have used here. A 

of the data points from the original data were deleted (th

the end of the time series) and a prediction made of the

points, which is then compared with the original data 

2nd Armoured Division. The plots (Figure 3.1 to 3.5) 

casualties per 1000 on the y-axis and time in days on the 

Figure 3.1: 2nd Armoured Division Data

Figure 3.2: 2nd Armoured Division–Power Spectrum Prediction
Time Series Behaviour

Figure 3.3: 2nd Armoured Division–Nonlinear Prediction

Figure 3.4: 2nd Armoured Division–Neural Net Prediction

Figure 3.5: 2nd Armoured Division–SOC Prediction

The first part of this time series (up to day 38) was in fa

to train a number of different time series prediction me

and these have been compared with the predictions f

days 39 onward. In fitting a prediction based on a self-o

ing criticality (SOC) fractal series, we have assumed th

circumstances remain sufficiently constant that we ca

single SOC process (this corresponds to a power spectru

is linear when plotted on a log-log scale). Comparing the 

iness” of the SOC prediction and the real data, the g

pattern of the process is very similar. 
Time Serie

pp

Kuhn for the 9th Armoured Division, with similar res

summarised below.

Figure 3.6: 9th Armoured Division Data

Figure 3.7: 9th Armoured Division–Neural Net Prediction

Figure 3.8: 9th Armoured Division–Power Spectrum Prediction

Figure 3.9: 9th Armoured Division–SOC Prediction
Time Series Behaviour

ppg

appears to give casualty behaviour that is of the same f

for the real data, at least for these data points.

Kuhn himself suggested that the casualty rate data p

taken from his collection of combat rates from World 

to the present, displayed a move away from linearity. Fi

number of “hotspots” in combat did not increase with 

force concentrations, and second, combat was charac

by high rates of casualties lasting for short periods o

interspersed with low casualty rates. These results indica

turn, a move away from the type of modelling associate

attrition warfare. Quantitative patterns were also found 

based upon two types of operational forms: that of a c

front (characterised by combat on the Western Front in 

War II) and that of a disrupted front, (characterised by c

on the Eastern Front in World War II). Combat casualti

matically increased as a consequence of breakth

operations in disruptive types of operations. Combat, 

fore, could be seen as a process wherein quiet stat

interspersed with “critical” irruptions.

Lauren [6] has also considered casualty data from 

War II and compared it with outputs from the MANA 

(an agent-based simulation similar in character to the I

model we will discuss in detail in Chapters 4 and 5). H

shown that such data display fractal properties and 

spectra that confirm the analysis of casualty data discu

this chapter.
Further Historical Data on the Processes of “Irruption” and B

FURTHER HISTORICAL DATA ON THE

PROCESSES OF “IRRUPTION” AND

BREAKTHROUGH

In an historical analysis study [7] of the operational l

combat, it was found by Rowland that the occurre

breakthrough, defined as the destruction of cohesiveness 

defence, was an important event in the eventual succes

offence. Following breakthrough, 86% of operations we

cessful, whereas if no breakthrough was achieved onl

were eventually successful. Once breakthrough has 

achieved, it becomes possible for the attack force to con

type of operation more in the nature of exploitation than 

bat. Moreover, variations in the time to breakthrough a

to differences in the nature of campaigns; the timi

“immediate” (less than half a day), “quick” (less than 2 

and “prolonged” (over 2 days) were used for study purpo

In studying the course of operations, one of the aspects 

ined was the nature of the movements of attack force

study of the rate of terrain capture with operational 

showed patterns that could be related to breakthrough 

Several different measures were examined, the fir

derived having the dimensions of advance rate (dis

time). Whilst the measures change through time, the fol

results relate to measurement (such as the length of 

frontage) at the time of breakthrough.

AREA TAKEN AND MEAN ADVANCE AT

BREAKTHROUGH

This measure may be taken as a possible index of how ba

defence has fared and of its chances of recovery. Howe

the simple area measure does not allow for variations in 
Further Historical Data on the Processes of “Irruption” and Breakthrough

p, (g)(

has been used here. This has a dimension of length and 

represent “mean advance” on the original attack frontag

The distributions of the measures in Tables 3.1, 3.2, a

are log-normal within a given breakthrough category, 

will show explicitly later. In each case, the table shows th

metric mean of the corresponding log-normal distrib

The discussion of stock price returns and Brownian mo

Chapter 1 gives us some insight into why log-normal di

tions might arise in such cases.

OPERATIONS LEADING TO:

CAMPAIGN SUCCESS CAMPAIGN FAILUR

Prolonged

Breakthrough 29 8

Quick

Breakthrough 5 -

Immediate

Breakthrough 11 2.2

Table 3.1: Geometric Mean Area/Attack Front at Breakthrough 

OPERATIONS LEADING TO:

CAMPAIGN SUCCESS CAMPAIGN FAILUR

Prolonged

Breakthrough 4.5 1.5

Quick

Breakthrough 4.5 1.5

Immediate

Breakthrough 11 2.2

Table 3.2: Geometric Mean Area Per Day/Attack Front at Breakt

(Miles/Day)
Further Historical Data on the Processes of “Irruption” and B

A first analysis of the results in Table 3.1 shows tha

“mean advances” at breakthrough are greater for those a

ing subsequent success than those failing by factors o

immediate breakthrough and approximately 3.6 for pro

breakthrough. This larger factor for immediate breakth

is indicative of the extra “brittleness” that pertains to the

quick breakthrough cases. They also show that these 

advances” at breakthrough are less for immediate than f

longed breakthrough by mean factors of 0.38 for 

achieving subsequent success and by 0.28 for those failin

Moving to the more complicated measure of irrupt

Table 3.2, we move from mean advance to mean advan

day, giving a first order measure of the mean rate of ad

For this case, there is somewhat less variability in 

There is, in fact, no significant difference between qui

prolonged breakthrough effects in this case. These two 

can therefore be pooled. Combined quick and pro

breakthrough results in terms of subsequent success and 

are thus shown in Table 3.2.

We again find differences on this measure (mean rate of ad

between breakthroughs leading to eventual success and 

leading to failure. These differences are by factors of 5 for 

OPERATIONS LEADING TO:

CAMPAIGN SUCCESS CAMPAIGN FAILUR

Prolonged

Breakthrough 4.5 2.1

Quick

Breakthrough 8.8 -

Immediate

Breakthrough 18 5.5

Table 3.3: Geometric Mean √Area Per Day at Breakthrough (Mile
Further Historical Data on the Processes of “Irruption” and Breakthrough

g, pgq

The mean rate of advance for immediate breakthrough is 

than for prolonged/quick breakthrough.

The caricature of movement to breakthrough implied is

Rather than:

However, the movement post-breakthrough is better repre

by the linear model than the radial one for success after 

through and without breakthrough. The exception to 

success after immediate breakthrough, which discrim

success most significantly on the basis of the radial mode

An alternative measure of the same dimensions is thus √ 

Time, again a representation of advance rates but repre

propagation from a point rather than a linear advance.
Further Historical Data on the Processes of “Irruption” and B

q

Table 3.3. The differences between groups show simil

terns to the previous measures, but are greater

significant at the 1% level (t test) between immedia

prolonged breakthrough.

It can thus be observed that, despite the great variations 

of the operations studied, there are patterns to be deduc

these can offer lessons on the nature of this little-studied 

of manoeuvre warfare.

This process of irruption has been identified as one of t

emergent effects of manoeuvre warfare [8]. We consid

whether such a process has scaling properties of the ty

cussed in our general consideration of complexity

historical data indicates (as we have discussed) that 

given type of breakthrough (immediate, quick, or prolo

I, Q, or P), and subsequent effect on the campaign (

quent Success [SS] or Subsequent Failure[SF]), the 

advance at breakthrough turns out to be a log-normal 

bution. Of even more interest to us is the fact that i

distributions are plotted for each of the breakthrough

paign effect categories, then they have a certain 

character, which we now define.

Consider Figure 3.10 (page 71). Here we have plotted t

normal distribution of mean advance at breakthrough fo

of a number of breakthrough categories. The x-axis is 

of the mean advance at breakthrough (in miles), and the 

is such that a cumulative normal distribution of the x v

will give a straight line. We can see that the various cat

of breakthrough produce cumulative curves that are par

each other, with one exception. However, this case has 

cantly less data than the other cases and we take this to 
Further Historical Data on the Processes of “Irruption” and Breakthrough

p

each other means the following: given two such curves, 

sponding to x variables x(1) and x(2), there is a scaling v

λ (which depends on the two categories of breakthrough 

considered) such that 

Log x(1) = λ Log x(2)

i.e. 

and the distribution of mean advance at breakthrough 

cides for the variables x(1) and x(2)λ. In this sense, we c

that x(2) can be scaled by a power transformation so t

distribution collapses onto that of x(1). 

An alternative radial measure of irruption, as we ha

cussed, is √(area at breakthrough)/(days to breakthr

with dimensions of miles per day. If this is plotted on th

basis as the previous figure, we again have evidence for 

of “scaling collapse” of the type discussed above (Figure 

Moreover, the stability of the two sets of data indica

there are (at least) two categories of emergent behavio

irruption and subsequent campaign outcome–linear ad

and radial propagation from a point.

Each data point in Figures 3.10 and 3.11 is a campaig

come, classified in terms of immediate (I), quick 

prolonged (P) irruption, and Subsequent Success (SS) o

sequent Failure (SF). The key to the data points is given 

λ x(1) = x(2)
Further Historical Data on the Processes of “Irruption” and B

Figure 3.10: The Statistics of Linear Irruption

Figure 3.11: The Statistics of Radial Irruption
The Fractal Front of Combat

THE FRACTAL FRONT OF COMBAT

In [9], Lauren discusses the fractal nature of a comba

between two opponents. The idea is that an essentially s

line frontage between two tactical-level opponents will 

into a fractal shape, whose fractal dimension can be calc

as a function of the force ratio of the forces involved (the n

of attackers to the number of defenders) as derived from H

cal Analysis of infantry battles carried out by the UK Dstl

Lauren [9] uses the Historical Analysis result as a basis 

analysis:

F = (number of attack infantry/number of defence infant

where F is a multiplier for the base number of casualties 

attacking force per defence weapon. As a consequence 

Lauren was able to show that the combat front will 

over time and in the limit will have a fractal dim

D = 1.685.

From Chapter 1, if we assume that this process is akin t

sion percolation of one fluid by another in a porous m

the fractal dimension of the boundary of the resulting in

should lie in the range 1.33-1.89, which is what we fin

historical data.

It is possible in this case to derive the underlying dyn

producing this statistical effect. It turns out that this frac

tor is due fundamentally to detection of targets [10

comes from a model of the engagement process that le

the following relationship:

2

1 1 k

T

k

R = +
The Fractal Fron

, 

and T is the number of targets in view [11].

It reflects the asymmetry of the infantry battle in the fol

sense [12]. The attack force aim is to close on the defenc

tion, and fire is used in a general suppressive mode–

casualties caused to the defence are only a small part 

process at this point. However, from the defence persp

the aim is to deter the attack, and casualties to the attac

are very important. Such casualties to the attack force 

direct reflection of the intervisibility of targets to the d

force as discussed above. 

As with most applications of fractal processes, the p

breaks down at some point due to the granularity of the 

tion. In this case, the process remains valid up to abou

closing distance between the attack and defence. At that 

a different mechanism comes into play, leading to local d

surrender and attack overrun of defence positions [12].

More generally, the figure of 0.685 relates to open terr

urban areas it is about 0.5 [13]. The closing to o

appears to occur differently in urban and wooded ter

compared with open terrain [12]. For example, in open 

tions, the closing part of the battle occurs across the fro

contrast, in urban conditions, the attack force is split into 

subunits who individually close on defence locations lea

local surrender and overrun. 

1 2 
Power Law Relationships in Combat Data

POWER LAW RELATIONSHIPS IN COMBAT D

THE HARTLEY MODEL

In [14], Hartley has analysed eight separate databases 

torical combat data. Four were developed by Helmbo

fifth dataset, “Inchon,” was developed by Busse. The las

datasets were developed by Dupuy. These eight dataset

several centuries in time, include both air and land co

and span the range from small to large interactions. De

the databases are given in [14]. On the basis of this ext

extensive set of data, Hartley was able to develop a stabl

ysis of the relationship between casualties in conflict a

initial force ratio based on earlier ideas of Helmbo

defines the following two dimensionless variables:

HELMRAT = 

FORRAT = 

where in each case, x0 is the starting value of force size, a

the final value (similarly for y). Hartley has established a 

law relationship between these two variables, HELMRA

FORRAT, on the basis of the comprehensive dat

described above. He has shown that (in logarithmic term

Ln (HELMRAT) = α Ln (FORRAT) +β

where the expected value of α is approximately 1.35 a

value of β is approximately normally distributed abo

value -0.22 with standard deviation of 0.7. Hartley show

the value of α has the characteristics of a universal co

being stable over four centuries of time [14, Figure 17

stable when considering conflicts of different sizes, r

2 2

0

2 2

0

y y

x x

−

−

0

0

y

x
Power Law Relationships in Co

,,

Tables 4, 5, and 6]. 

If it is assumed that the mechanism that produces this re

ably stable relationship between casualty effects and forc

is of Lanchester type, then Hartley shows that it must be 

ear-logarithmic form. However, the relationship is ba

the empirical data alone, and other explanations are po

For example, in [15] an analysis based on self-organisat

particular the forest fire model of Chapter 2) is put forw

the basis for the equally remarkable scaling of conflict si

thus persuasive that such complexity-based effects, rathe

a Lanchester process, lie at the base of the scaling relati

established by Hartley. This analysis by Turcotte and R

is next discussed.

In their paper [15], they begin by comparing the pred

of the theoretical self-organising forest fire model with t

tistics of the relative sizes of real forest fires. Four data s

considered: 4,284 forest fires in the USA Fish and W

Service Lands during the period 1986-1995; 120 of the 

fire areas in the western USA from tree ring data, sp

the period 1155-1960 (800 years); 164 fires in the A

boreal forests during 1990-1991, and 298 fires in the A

lian Capital Territory during 1926-1991. The results 

good agreement with a power law statistical distribution 

of fire versus frequency, with a power law expon

between 1.3 and 1.5. The remarkable thing is the stab

the trend across such a long period of time, during whic

nology has changed, as have ways of fighting such fire

authors then show that a similar power law relationshi

with an exponent in the same range) holds for the inten

conflict versus its frequency. This work extends the rese

Richardson [16], who also showed a power law relati
References

y qy

Roberts base their results on two data sets: that of Levy, 

tabulates the intensities of 119 wars from 1495 to 197

that of Small and Singer, who considered 118 wars ov

period 1816-1980.

The similarity of the power law exponent for both for

statistics and war intensity leads Turcotte and Rob

hypothesise that war deaths are caused by a self-orga

mechanism akin to that of the forest fire model. This is 

the beginning of an explanation of why casualties i

should give rise to such a simple power law relationship

over centuries of time.

REFERENCES

1 LTG TUKER F (1948). The Pattern of War. Cassell. UK.

2 MOFFAT J (2002). Command and Control in the Information Age: Represen

Impact. The Stationery Office. London, UK.

3 KUHN G W S (1989). “Ground Force Casualty Patterns: The Emp

Evidence.” Report FP703TR1.

4 WAYLAND R, BROMLEY D, PICKETT D, and PASSAMANTE 

(1993). Physics Review Letters. 70. p. 580.

5 LAERI F (1990). Computational Physics. 4. p. 627.

6 LAUREN M and STEPHEN R T. “Fractals and Combat Modellin

MANA to Explore the Role of Entropy in Complexity Science.” Pap

prepared for Fractals. Defence Technology Agency. Auckland, New 

Zealand.

7 ROWLAND D, KEYS M C, and STEPHENS A B (1994). “Breakt

and Manoeuvre Operations (Historical Analysis of the Conditions f

Success) Annex I, Irruption.” Unpublished DOAC Report, Annex I

8 ROWLAND D, SPEIGHT L R, and KEYS M C (1996). “Manoeuv

Warfare: Some Conditions Associated with Success at the Operatio

Level.” Military Operations Research. 2 No 3. pp. 5-16.
()g g 

of Scaling Systems.” Military Operations Research. 5 No 3. pp. 47-58.

10 ROWLAND D (1983). “The Effectiveness of Infantry Small Arms F

Defence – A Comparison of Trials and Combat Data.” Unpublishe

DOAC Memorandum.

11 THODY J H and DOVE H J (1981). “An Analysis of Small Arms F

Infantry in Defensive Positions.” DOAE Unpublished Note.

12 ROWLAND D personal communication.

13 ROWLAND D (1991). “The Effect of Combat Degradation on the 

Battle.” Journal of the OR Society. 42 No 7. pp. 543-553.

14 HARTLEY D S (1991). Confirming the Lanchesterian Linear-Logarithmic M

Attrition. Martin-Marietta Center for Modelling, Simulation and Ga

Report K/DSRD-263/R1.

15 ROBERTS D C and TURCOTTE D L (1998). “Fractality and Self

Organised Criticality of Wars.” Fractals. 6 No 4. pp. 351-357.

16 RICHARDSON L F (1960). The Statistics of Deadly Quarrels. Boxwoo

Pittsburg, USA.
77

CHAPTER 4

MATHEMATICAL

MODELLING OF

COMPLEXITY,

KNOWLEDGE,

AND CONFLICT

INTRODUCTION

nderstanding the behaviour of agent-based

simulation models of conflict is now

becoming more important, especially as (with

improved representation of Command and Control [1]) the agents gain intelligence and try to

outsmart each other, producing potentially very

complex behaviour. In the modelling of natural

systems (such as fluid dynamics, or heat flow), the

principal variables in these models can often be

separated out from the rest of the model to produce a mathematical metamodel that is aimed at

relating the outputs of the model to these driving

U
Introduction

pppy

achieved, it improves our understanding of the system 

likely emergent behaviour, as well as complementing the 

detailed simulation. Such an approach is consistent w

idea of ‘Operational Synthesis’ as espoused by Dr. 

Brandstein, then Chief Scientist, U.S. Marine Corps [2].

As an example, in developing a metamodel we consid

relationship between a key outcome of the model, a, an

of input variables as follows:

(This is easily generalised to an arbitrary number of bs

arguments have independent dimensions. That 

dimension of any a cannot be expressed as a combina

the dimensions of the other as. In contrast, the dimen

each b variable can be expressed as such a combinatio

arguments can be transformed using a gauge transform

so that:

.

These correspond to a change in the “gauge” (e.g., from 

metres to metres or kilometres) in the measuremen

variable. Physically, if a gauge change makes no differe

observed behaviour for all observers using different g

the variable is said to be self-similar.

1 12 ( ,... , , ) a fa a bb = k

1,... a ak

1 11,..., a Aa a A a k kk ′ ′ = =
I

[] 

property:

These “ ” variables are sometimes called similarity var

This is because two natural systems, with different value

and bs but the same value of , will tend to have 

emergent behaviour. An example is the flow of air past 

craft in the atmosphere or past a model of the aircra

wind tunnel. The variable in this case is the Re

number. If this is the same in both cases, then the mo

the wind tunnel will give results relevant to the full-sca

craft in the atmosphere.

Self-similar solutions correspond to problems where the 

of the variable ( for example) tend to zero or infinity. 

possibilities are available (see Reference [1] for 

discussion):

TYPE 1 METAMODEL. tends to a non-zero finite limit 

of its arguments tends to zero or infinity. This means 

most practical cases, this argument can be eliminated fro

relationship, giving a simplified form in equation (1).

TYPE 2 METAMODEL. has power-law asymptotics 

form:

( )

11 2 

11 2 2

1 2

1 12 1

1 1

1 12

1 2

1 2

1 1

( ,..., , , ) ... , ... ...

... ,

where and ... ...

p r

k k prp 

k 

p r

k

pr p r

k k

b b a fa a bb a a

a aa a

a a

b b

aa aa

 

= =Φ 

 

= ΦΠ Π

Π= Π =

Π

Π

Π

2 b

Φ

Φ

1

2

1

12 2

2

(, ) ( ) α

α

Π Φ Π Π =Π Φ

Π

%
Introduction

gy

physics, these systems are examined from a gauge theory 

of view using a “renormalisation group” approach in 

the parameter is considered (through the repeated a

tion of a renormalisation group) at larger and larg

smaller and smaller) gauges, giving an asymptotic exp

of the form required for a type 1 or 2 metamodel. La

will discuss the renormalisation group in more dept

relate it to concepts of control of the battlespace.

TYPE 3 METAMODEL. Neither 1 nor 2 holds and self-sim

is not observed; has no finite limit different from ze

no power-law asymptotics.

This approach directs us (for evidence of metamodels o

1 and 2) to search for evidence of power-law relations

the form , which, if plotted on a log-log scale, 

straight line whose slope is the power-law exponent

expressions arise naturally in certain types of complex sy

particularly where fractal structures are involved, an

referred to as scaling relationships since they have no pre

gauge. Evidence of such scaling relations is thus evidence in suppo

assumption of relative gauge. Chapter 3 shows that there i

evidence for such an assumption in historical conflict da

that metamodels of types 1 and 2 should be expect

agent-based models of conflict. In addition, we should 

to see evidence for normalised “scaling collapse” as ex

fied by the function in equation (2), and we 

expect to see the effect of renormalisation groups.

If we consider an agent-based “distillation” such as the I

model developed under Project Albert by the U.S. M

Corps Combat Development Centre, we can consid

emergent behaviour of such a model in terms of both th

2

Π2

Φ

α y = x

1 2 ΦΠ Π (, )
I

g g, y 

on the opponent. For such a distillation, a metamodel of 

applies [1] that allows us to relate the attrition rate for o

to the clustering dynamics of the opposing side, as me

by the mean fractal dimension of these clustering agent

simple example, (given in [1]), assume that the comman

cess, say for Red, is represented by the following effects:

1. The number of discrete clusters of Red agents at t

N(t), is specified ahead of the simulation. 

2. N(t) is a decreasing function of t. 

These assumptions are meant to suggest that the num

Red clusters decreases in time, reflecting the desire of 

concentrate force. With these assumptions let us 

assume that the smallest cluster of Red agents, X(t), at tim

taken and added to another randomly chosen cluster 

agents. This process thus represents both the concentra

Red force and the reconstitution of force elements. 

Let us now define (expected number of clusters 

agents size x at time t)/(initial total number of clusters 

agents) and N(t)=(the total number of remaining clus

Red agents at time t)/(initial total number of clusters). 

the assumptions and definitions above, it can then be 

that , the cumulative distribution of cluster sizes a

t, approaches a self-similar distribution as time pro

(i.e., a scaling collapse takes place). Thus the cluster size 

bution evolves over time by a scaling relation. ca

be represented in the self-similar form:

φ(,) x t =

≥

φ(,) x t

φ(,) x t

( / ( )) (,) ( )gx Xt x t

X t φ =
Introduction

g p

g(1)=N(t)X(t). This self-similar form means that we can 

the distribution of relative cluster size in a way that 

invariant (although the actual cluster sizes will change). 

Now assume that the evolution of the distribution of 

smooth (a small change in time t leads to a small cha

; this is equivalent to saying that the renormal

group is smooth [1]). If is the expected clus

x at time and is the same expectation at 

then this assumption means we can find a constant b 

order such that:

.

It then follows that: 

for some constant c and the normalised expected cluster 

time t, , varies as a power-law with increasing tim

scaling constant b. 

If is the change in the number of Blue agents, Lau

has shown that:

is proportional to the product of (Red unit effectivenes

probability of meeting a Red cluster)x(the expected num

Red units per cluster). It is assumed that unit effective

constant. Keeping the cluster size constant for the mo

this indicates [3] that the rate of change of Blue agents i

by an expression of the form:

φ

φ(,) x t

φ δ ( ,(1 ) ) x t −

(1 ) −δ t φ(,) x t

φ δφ δ ( , ) (1 ) ( ,(1 ) ) xt b x t =+ −

log ( , ) log φ xt b t c = +

φ(,) x t

∆B

B

t

∆

∆
I

where D is the average fractal dimension of Red (and th

an indication of how Red clusters/collaborates locall

both r and q are exponents. This equation is a fo

Lanchester law where the rate constant is dependent up

clustering of Red agents. If Red cluster size varies accord

where x is the cluster size and X(t) the smallest cluster at 

then we can write:

where N(t), inversely related to X(t), is the normalised n

of clusters of Red at time t and g(y(t)) is the mean of the 

bution of cluster size, which evolves as a power law (as w

shown in certain cases). 

MORE GENERAL DISTRIBUTION OF CLUSTER S

For self-organising groups of agents that are approac

critical point in the form of a Bak-Sneppen evolution 

but are not necessarily attempting to concentrate force 

reconstitute force in the sense described above, our pr

analysis in Chapter 1 indicated that the distribution of 

sizes at some intermediate point s is given by:

B qD rD () () k t

t

∆ = ∆

∆

( ) / () (,) ( )

g x Xt x t

X t φ =

() () ( ) ( ( )) B qD rD k t Nt g yt

t

∆ = ∆

∆

1

, , (, ) (( ) ) PS f S gS f f is c is

−τ σ = −
Introduction

py p 

noted before, this converges to a power-law distributio

tends to infinity.

In [1], we also relate the cluster fractal dimension predic

the Bak-Sneppen evolution model of local species coevo

to that which emerges from initial experiments wi

ISAAC cellular automaton model. (For a descript

ISAAC see the Web site at reference [4].)

THE RENORMALISATION GROUP

The gauge invariant approach to metamodelling ou

above also leads us to consider the role of the renorm

tion group, which explicitly appears in terms of its eff

the distribution function in our characterisa

metamodels. Let us explore this a little more here. Su

that is a function of the two variables x and t. 

the “static scaling” assumption of renormalisation [

assume that we have a group of renormalisations of th

, so that:

i s, f

1 2 ΦΠ Π (, )

uxt (,)

Rb,φ

,

,, ,

*

* *

1 (,) ( , ) ( )

From the group property, 

It follows that ( ) ( ) ( )

Thus ( ) for some exponent 

If ( , ) is a fixed point of the renormalisation, then

(,) (

b

a b ab

R u x t u b x bt

Z b

RR R

Z a Z b Z ab

Zb b

u xt

u xt b u b

φ

φ

φφ φ

α

α φ

α

−

=

=

=

=

=

* *

, )

1 Choose , then;

(,) ( )

x bt b

b

t

x

u xt t u

t

α

φ

∀

=

= %
I

pppp

thus (in the limit) produce a functional relation of the fo

This explains why the type 2 metamodel has the 

assumed, and also why the (renormalisation invariant) 

size distribution of Carr and Pego discussed earlier h

form derived by them. Note the simplification that ha

achieved in going from an unknown function of two va

to a normalised unknown function of only one variable.

The ability of a force to control an area of operations 

related to the fractal dimension of the force through the 

such a renormalisation process as we now show.

CONTROL OF THE BATTLESPACE

Looking now at the phenomenon of control of the battle

we can consider the problem using the renormal

approach as in a type 2 metamodel. Consider, as shown 

ure 4.1, the Area of Operations (AO) of a m

commander. For simplicity we assume this is a square of 

Figure 4.1: Area of Operations
Introduction

this area. Firstly we have to define what this means. Eac

shown by a dot in Figure 4.1, has an area surrounding it 

can control. The size of this area is defined by the nature 

force and its associated sensors [6], giving rise to a “bub

control around the unit. In two dimensions, let this area 

spond to a square of side l. We assume that l is signif

smaller than the dimensions of the AO. (Note: we 

restricted the battlespace here to two dimensions to si

the discussion. However, the same approach should w

three dimensions, corresponding to the complete battles

Now let D be the fractal dimension of the force under th

mander’s control within the AO. Suppose we partition t

into square cells of width l. Let N be the total number o

cells, so that . Let N(l) equal the number of cells 

AO that are occupied by one of the units making up the 

By definition of the fractal dimension, we have that 

(normalising the constant of proportionality to 1). If p 

probability that a cell chosen at random in the AO is 

control, then:

.

Note that D always lies between 0 and 2, so that p 

defined.

In discussion with senior UK commanders who hav

recent operational experience at a high level, the conc

control of an area as corresponding to the prevention o

through an area (flow in terms of an opposing force, o

haps some third party) has been endorsed as a good an

We thus define the commander as having “weak cont

his area of operations if he can to some extent control 

2 Nl A =

( 

N

2 ( ) D D Nl l l

p NNA

− −

= ==
I

gp

determining whether fluid can seep through a block o

porous rock, as discussed in [7] where a renormalisation 

approach was used. We thus define weak control as 

sponding to a span of controlled areas that stretch eithe

side to side or top to bottom of the AO. Following on 

this, we define the commander to have “strong control” 

AO if there is a span of controlled areas stretching bot

side to side and top to bottom of the AO, resulting in a 

constraint on the flow of people through the area.

The question at issue is then: how do these concepts of c

relate to the ability of the force to collaborate locally (th

tal dimension)? 

Consider first a cell of four elements where each ce

square of side l, which a single unit can control. We no

sider the probability of weak or strong control of this 

cell of side 2l in terms of the probability:

of a unit controlling each of the squares of side l. We co

each of the five different classes of configuration for this 

shown in Figure 4.2.

In Figure 4.2, we show the five classes a to e of configu

and mark beside each case whether this gives weak or 

control, by considering the span of controlled areas.

1 p

2 D l

p A

−

=
Introduction

Figure 4.2: Five Configuration Classes

The probability of each configuration can be derived in 

of p. For example, the probability of any of the cases in 

uration d is p3(1-p). By adding up the configu

corresponding to weak control, and taking into accou

probability of each such configuration, we have the relat

.

We can do the same thing for strong control, leading 

relation:

.

Using the renormalisation group approach [5], we ite

increasing levels of cell size, leading to the relations:

• Weak control: 

• Strong control: 

These give rise to the recursive schemes shown in Figu

and 4.4. The relations for weak and strong control abov

respond to the relationships respectively:

2 23 4

1 p p p p pp (weak) 4 (1 ) 4 (1 ) = − + −+

3 4

1 p p pp (strong) 4 (1 ) = −+

2 2

1 (4 4 ) n n nn p p pp + = −+

3

1 (4 3 ) pp p nn n + = −

2 2

3

( ) (4 4 )

( ) (4 3 )

fx x x x

gx x x

= −+

= −
I

Figure 4.3: Plot of y = f(x) and y = x - weak control

Figure 4.4: Plot of y = g(x) and y = x - strong control

The fixed points in the recursive relation of weak contr

respond to the values x shown in Figure 4.3 such that y 

intersects y=x. Similarly for strong control, the fixed 

correspond to the values x such that y = g(x) intersects 

For both weak and strong control, there are stable fixed 

at x = 0 and x = 1. However, there is also an unstabl
Control and Fractal Dimension

pg 

trol. This was calculated to be 0.382 for weak contro

0.768 for strong control.

CONTROL AND FRACTAL DIMENSION

In either case, starting with a given fractal dimension 

force, and the dimensions of the AO, we can calculate a 

sponding starting probability:

.

For side length L of the AO there will be a corresp

value of iteration order n such that . By using the 

sive scheme above, we can calculate for this value of 

corresponding probability of weak or strong control 

AO. Consideration of Figures 4.3 and 4.4 indicates tha

is a critical value of the probability:

such that values above p0 polarise towards very good c

whereas values below p0 polarise towards very poor con

fact, the point p0 corresponds to a phase change in the 

iour of such a system. 

Examination of Figures 4.3 and 4.4 shows that it is ea

iterate towards good weak control than towards good 

control, as we would expect (since weak control is ea

achieve than strong control). Figure 4.5 shows how thi

tion works for a starting probability of 0.65 an

requirement of weak control.

2

0

D l

p A

−

=

2n

l L ≅

2

0

D l

p A

−

=
Control and Fractal 

Figure 4.5: Recursive Calculation of the Probability of Weak Con

LOCKOUT

From a game theoretic perspective, we can see that each 

trying to drive its own value of control up, and the othe

down. The analysis above indicates that there should be 

lockout, i.e. one side should rapidly gain control and lo

other side out.

PERCOLATION THEORY AND THE

RENORMALISATION GROUP

In relating these ideas to the behaviour of natural system

various configurations of 2x2 controlled areas are iden

the porous and nonporous regions in semipermeabl

structures [7]. The study of such processes is referred to 

colation Theory. A good step-by-step introduction to the 

and some working examples of how such processes work 

dimensions are provided at the Web site reference [8]. 

the probability of an individual rock (or crystal) domain 
Control and Fractal Dimension

p, p g pp

behaviour can be tuned in the sense that we discus

Chapter 2. If p is below the critical value p0, then the c

are not large enough to form a path of percolation fro

side of the structure to the other (popping noise). Wh

probability p is above the critical value p0, then sudde

clusters tend to spread from one side to the other, al

percolation throughout the structure (these are thus call

colation clusters) and corresponding to a phase change 

dynamic of the system (snapping noise). Near to the 

point, it can be shown [8] that the distribution in size of 

lating clusters is a power-law, corresponding to a 

distribution of cluster size (crackling noise). In fact, 

slightly greater than p0, the fraction F of individual do

that form part of a percolating cluster takes the 

. From this, we can see that as p appr

the critical value p0 from above, the fraction of domains 

ing part of the percolating cluster tends to zero (for a ver

initial configuration). Thus such clusters can become ve

uous close to the critical point.

These configurations come originally from attempts to 

lattices of magnetic spins in more than one dimensio

Such 2x2 configurations are then referred to as bloc

since they are composed of four individual spins, e

which may be either up or down (the block spin is defi

the sum of the signs of the individual spins, so it still h

value +1 or –1). By developing the idea of the renorma

group, iterating to larger and larger domains, Wilson (bu

on work by Kadanoff), was able to show that such ar

magnetic spins do indeed exhibit phase change effects 

have shown above for the phase change between being 

control and being in control of a region), and that the p

0 0 F Fp p ( )β = −
Control and Fractal 

py

Wilson was awarded the Nobel Prize for Physics.

IMPLICATIONS FOR SELF-ORGANISING

INFORMATION NETWORKS

If a Self-Organising Information Network is thought o

grid of connections, and we have a probability p of cre

link between one element of the network and anothe

percolation across the network corresponds to being a

send a signal from one end of the network to the other

Percolation Theory, we can thus see that we would ex

phase change in the dynamic of such a network. If p is 

then only local connections can be made. However, a

critical value of p, there will be a phase change such tha

nections across the full network can suddenly be made

to this critical value of p, the clusters formed by thos

nected on the network will form a fractal set, an

distribution of such cluster sizes is described by a pow

relationship between size and frequency of that size

questions that arise are: 

1. What is the benefit (and cost) of being above the c

threshold so that the connections are robust?

2. How can we measure the benefit of using the know

obtained by such networked interconnection?

Recent work by Perry [9] has exploited the idea of inform

entropy to address the second question (with a reduc

entropy across the network corresponding to an incre

knowledge, and this then being equivalent to a reduc

delay in prosecuting an action). To understand where th

originates, we first look at the influence of knowledge i

games from an open systems perspective.
Wargames as Open Systems Sustained by Knowledge Flowing Across the Boundary

WARGAMES AS OPEN SYSTEMS SUSTAINED 

KNOWLEDGE FLOWING ACROSS THE

BOUNDARY

As noted recently by Roske [10], a wargame is an open 

of the type introduced in Chapter 1. As he notes,

“In a classic command post exercise, we inject human decision

making into a structured environment to generate open system 

behaviours. Human decisionmaking represents energy crossing 

the structured system boundary...”

In [11], Perry and Moffat developed the idea of using 

mation Entropy (from Shannon Information Theory) as 

of capturing the knowledge available to a military 

mander’s decisionmaking during a wargame. This was i

applied to looking at the benefit to be gained from adv

airborne standoff radars (such as JSTARS in the U

ASTOR in the UK). 

A series of wargames was carried out in the UK to quan

difference in combat effectiveness of a force without ai

standoff radar (ASTOR) in comparison with a forc

ASTOR, or a force with other weapon systems who

cycle costs approximately equalled those of ASTOR

common thread through all of the cases to be examin

the stream of decisions made by the friendly force 

mander. We describe here how we were able to captu

flow of information to the Blue commander during th

games using the concept of Information Entropy, an

turning that into a measure of “knowledge.” This app

allowed us to measure the quantity of knowledge f

across the boundary of the system in order to influen
Wargames as Open Systems Sustained by Knowledge Flowing Across th

y 

lution of the wargame. 

WARGAME STRUCTURE

There are three major problems with the use of warga

support military studies: (1) too little output data, (2) the 

hood of atypical results, and (3) oversimplification. Th

problem stems from the fact that wargames are generall

cumbersome, and resource intensive. Consequently, mo

lysts who use them to support studies plan only a small n

of games, thus precluding significant statistical result

second problem recognises the possibility that the seque

decisions taken by the players in these games represents 

cal outliers. Players may adopt extreme strategies tha

“outside” of what is considered to be a typical m

response. The third problem reflects the fact that huma

ers can only approximate the results of combat operati

our studies, we addressed these problems in three wa

arguing that our wargames are quasi-memoryless processes f

tical situation assessment; by introducing the epitomising 

principle in wargames; and by embedding computer mo

adjudicate engagements in the manual games. We disc

first two of these concepts next.

THE MEMORYLESS PROPERTY OF WARGAMES

The wargames used to support this study were two

zero-sum games played over several discrete time peri

cycles (Bowen [12]). Consequently, the entire campaig

be viewed as a dynamical process in which the state va

are the force levels on both sides. In each of the war

both the friendly and enemy commanders formulated 
Wargames as Open Systems Sustained by Knowledge Flowing Across the Boundary

ppgj

the play of each game, both commanders assessed the 

situation at the strategic and tactical levels. Assessment 

strategic level consisted of examining the need to alt

campaign plan. At the tactical level, it supported force 

tion decisions consistent with the implementation 

campaign plan.

A dynamic process is said to be memoryless or Markovia

each cycle, the state of the system is influenced only 

state of the system in the previous cycle, and not by the s

history of the system (Stark and Woods [13]). The war

conducted to support this study approximately satisfied 

conditions as follows:

• TACTICAL DECISIONS: In all of the wargames play

the strategic situation was such that neither comm

found it necessary to alter their original plan. Con

quently, the commanders’ decisions centred on the 

allocation of their forces. This forced them to focu

exclusively on the tactical situation in the current g

cycle and their assessments of the likely situation in 

sequent cycles. Transitions in the state variables 

therefore depended upon the status of the forces at 

end of the previous cycle, the decisions taken in th

rent cycle, and the combat attrition experienced in 

current cycle.

• CYCLE INDEPENDENCE: Both commanders were 

assumed to act outside the opponent’s decision cyc

That is, the opponent was assumed to have sufficie

time within each cycle to redeploy his forces so tha

ligence on unit identity, type, and location gained i

previous cycle was no longer valid in the current cy

past history had no effect on the commanders’ cur
Wargames as Open Systems Sustained by Knowledge Flowing Across th

p, p

only partially valid, as will be made more apparent 

in the discussion of the UK FASTHEX gaming m

used in the wargames for this study.

Under these conditions, the campaign can be viewe

sequence of decisions taken in a discrete dynamical s

Each commander attempts to select a set of decisions (st

that will maximise the likelihood that he will achieve h

sion and that is consistent with the overall campaign 

Because the decisions are made under a connecting cam

plan, we assert that the process is only quasi-memoryles

assumption is most relevant when the situation on the g

is in a state of rapid flux–the most interesting case.

THE EPITOMISING STRATEGY PRINCIPLE

In the play of the games, we strove to ensure that the 

manders took actions that epitomised the side’s h

conduct in battle. We attempted to avoid bold, darin

liant manoeuvres as well as bungling incompetence. F

purposes of analysis, more cautious conservative strategi

are consistent with accepted doctrine are preferred. The

real danger that uncontrolled play would have resulted 

ducing only outlier results, when what is wanted are 

results. We achieved this by playing the wargames “o

Red and Blue players were able to discuss strategies an

sions in the presence of a game controller and thus we e

that the actions contemplated by either did not con

extraordinary tactics.

1

A wargame is “open to Blue” if the physical state of Red is fully kn

the Blue player (Bowen [12]).
Wargaming with FASTHEX

pg pp, gp

that each side “knew” only those things that would no

be known through the available surveillance and intell

assets. Players were forced to examine the inform

received through sensors and surveillance assets caref

ensure that: 

1. Sound military judgement was used in considerin

decision options available.

2. Players used what they learned and interpreted fr

sensor reports, and not what they saw on the “gam

board.”

3. The information received was consistent with the 

tions and capabilities of the equipment being used

employment of the surveillance assets, and other e

ronmental conditions.

WARGAMING WITH FASTHEX

In the play of the FASTHEX wargame, the continuity 

battle was modelled as a series of discrete timesteps refe

as “game cycles” (Figure 4.6). The length of a cycle can 

by the players, but is usually chosen to be 2 hours. Withi

cycle, a linear sequence of actions is taken and the 

quences of each is evaluated to simulate events with

cycle. Blue and Red alternately begin the sequence in o

smooth out the advantages or disadvantages of “going fi

The flow chart in Figure 4.6 can be thought of as a se

modules in a fully automated simulation of combat, l

decision module. The modules are rather simple repre

tions requiring extensive player input. For example, the 

selects the type of reconnaissance system to be used, sta
The Decisi

, g

game board to be searched. With this information, the 

applies the appropriate probability of detection or recog

and reports the results. Environmental conditions and 

features are not known by the model and are inserted th

rules constraining the play.

Figure 4.6: FASTHEX Game Cycle Sequence

THE DECISION PROBLEM

With the epitomising principle in mind and the constra

gaming imposed by the FASTHEX model, we exami

decisions open to each commander with particular em

on the Blue commander. Great care has been taken to 

that decisions required in actual combat had their equ

2

FASTHEX uses a hexagonal game board much like IDAHEX. Fo

games, each hexagon is 7.5 km from face to face.
The Decision Problem

g py gp

fact that players can override almost all computer-gen

outcomes in the game. Therefore, realism can be impo

those cases where the model obviously strays.

The commander makes operational and tactical decis

each combat phase in the FASTHEX wargame cycle in 

ing with his overall operational and tactical aims (Figur

The one exception of course is the engagement pha

engagement is a consequence of the decisions taken 

Red and Blue commanders in the other phases.3

 There

each cycle, the decision set in FASTHEX, D(t), consis

components, D(t)={d1(t),d2(t),d3(t),d4(t)}, where t indicat

cycle and di

(t) is the decision taken at the ith phase in th

THEX model. The following is a description of the de

taken at each phase:

1. RECONNAISSANCE, d1(t): The commander must d

which of the reconnaissance assets at his disposal t

cate. In most cases, this means deciding where to 

his sensors and how many to commit to the proce

ing account of higher level assets such as satellite 

surveillance). It should be noted here that reconna

sance assets are used primarily to identify valid ta

and main force concentrations.

2. STRIKE, d2(t): The strike decisions can be thought 

the allocation of deep fire assets. The engagement 

adjudicates the close battle, but the deep fire battl

handled separately. 

3

Engagement adjudication is done using look-up tables based on l

level computer modelling of the various types of engagement.
The Decisi

, 

3(

)

of his units to move and how far they are to move

constrained by terrain, the maximum speed of his 

and the degree to which the units are fit (in terms 

damage inflicted) to accomplish movement. Units 

at the lowest level of resolution: the battalion, batt

group, or squadron.

4. POST-ENGAGEMENT MOVES, d4(t): After each eng

ment, the commander assesses the damage done t

units. If the units are not capable of continuing as 

integral force, they can be withdrawn, consolidate

other forces, or both.

OPTIMAL CONTROL FORMULATION

Each of the decisions is taken from among a discrete set 

possible choices described above and therefore the set 

consisting of all possible decisions at cycle t has card

equal to the product of the cardinality of the 4 phase-d

sets. The collection of decision sets at each cycle in the g

then referred to as the decision stream for that game and 

fore, we denote

the decision stream for an m-cycle game. Clearly, the n

of possible decision streams for even a simple wargam

easily become unmanageably large and thus we are bu

by the “curse of dimensionality.” 

The consequences of the commander’s decisions at eac

can be measured in several ways. As mentioned earl

have selected the status of friendly and enemy forces. C

quently, if we let 

DG = { } D( ) () ( ) 0 , D 1 ,L, D m −1

() () () () () () [ 

b 

X t x t x t x t y t y t y 

1 2 1 2 = , ,L, , , ,L,
The Decision Problem

gy 

nents and the r enemy force components at cycle t, th

decision stream can be viewed as a memoryless multistage 

process as depicted in Figure 4.7. At each cycle, the comm

wishes to select D(t) so that a performance function dep

upon the vector X(t) is optimised in some way. 

Figure 4.7: A Wargame as an Open Dynamical Process

In this formulation, is a transition function s

 The status of both Red and Blue 

in terms of combat strength in cycle t+1 depends up

their combat status in cycle t and the decisions made in 

The initial condition X(0) is the total combat strength 

friendly and enemy forces at the beginning of the cam

and X(m) is the status of both at the termination 

campaign.

The commander’s problem then is to select the decision 

DG that optimises the performance (i.e., utility) function: 

Assuming that we are able to find a reasonable represen

for and finding the optimal decision str

then an optimal control problem of the form discus

Chapter 1 and Appendix 1. In this case, the decision va

D(t) at each cycle are the control variables and the X(t) 

θt

X ( ) () () t 1 [ ] X t , D t . + = θt

[ ] () () , [ ] ( ). 1

0

P f X t D t X m m

t = ∑ t + Φ −=

( )⋅ t f Φ[ ] X ( ) m ,
The Decisi

ppp, 

under what conditions we might expect such a prob

have a unique solution. For further discussion of such 

tions, see also Bryson and Ho [14].

THE TWO-SIDED GAME

The problem with a two-sided game is the developme

second performance function for the opposing side. Thi

lem can easily be solved if we design P such that i

chooses to maximise P, then Red will choose to minimis

this formulation, we assume two decision streams, one 

friendly commander BG = {B(0),B(1),...,B(m–1)} and o

the enemy commander RG = {R(0),R(1),...,R(m–1)}. 

assumption implies that the game is zero-sum, that 

increase in P for the friendly force results in an equal de

for the enemy force and vice versa. Consequently, we wi

 

subject to the transition constraint: 

.

Solutions to problems of this type are fairly complex for 

very trivial examples. For example, see Hillestad [15

Berkovitz and Dresher [16]. However, the objective 

not to solve the wargame using one of these techniqu

rather to use the two-sided memoryless multistage o

control formulation as a convenient construct for a 

statement of the problem.

() () ( ), maximise minimise

P

B t ∈ BG R t ∈ RG

X ( ) () () () t +1 = θt[ ] X t , B t , R t , 0 ≤ t ≤ m -1
The Decision Problem

GAMES WITH EQUIVALENT DECISION STREAM

Given the complexity of the two-sided optimal contro

struct for the wargames, we forego attempts to app

closed form solution and rely instead on the replicat

instances of the game from each of the scenarios and 

several cases to be examined. Even this however c

extremely time consuming and therefore we wish to ex

those alternative cases in which the decision streams are 

tially different. That is, if the introduction of alte

weapon systems in a game does not significantly alter th

sion stream, DG, then the two games are cons

equivalent. In this way, the results of one game can be 

under the conditions of the second. Although the resul

vary with each, the decision stream is taken to be consta

A simple example will serve to illustrate the process. Co

a conflict in which Red and Blue commanders have onl

lery and tanks with which to conduct a two-cycle cam

and we focus on the use of artillery resources. Our state 

then is where: 

the number of BLUE and RED a

pieces respectively, and

the number of BLUE and RED 

respectively.

In both cycles, we assume that the decision on both side

allocation of artillery fires. Consequently, 

the Blue commander’s decision at cycle t (t = 0,1

R(t)=[r1(t),r2(t)]T is the Red commander’s decision, 

b1(t),r1(t) is the fraction of Blue/Red artillery allocated to 

Red/Blue artillery, b2(t) = 1–b1(t) is the fraction of Blu

lery to be allocated against Red tanks, and r2(t) = 1–r1(t

() () () () () [ ]T X t x t x t y t y t 1 2 1 2 = , , ,

x1() () t , y1 t =

x2 () () t , y2 t =

() () [ 

B t b t b 

1 

= ,
The Decisi

y g

simplicity, we restrict the domain to 0, .5, and 1. Th

cycle game thus described is illustrated in Figure 4.8:

Figure 4.8: An Example Two-Cycle Game

The transition function, , is simply the combat adjud

model. If we assume a simple Lanchester differential 

and let i = 1 for Blue and i = 2 for Red,

where:

 is the effect of Blue/Red artillery against Red

artillery; and

 is the effect of Blue/Red artillery against Red

tanks.

The s can be thought of as single shot kill proba

(SSKPs) and bj

(t)x1(t) and rj(t)y1(t) represent the number of 

Red artillery allocated to Red/Blue artillery and tanks. 

fore the transition equations become:

θt

[ ] () () () , , [ 

() () () () () () (

, , , 

21 1 1 22 2 1 11 1 1 12 2 t θ X t B t R t = α r t y t α r t y t α b t x t α b t 

0 ≤αi1 ≤1

0 ≤αi2 ≤1

αij

( ) () () ()

( ) () () ()

( ) () () ()

y ( ) () () () t y t b t x t

y t y t b t x t

x t x t r t y t

x t x t r t y t

2 2 12 2 1

1 1 11 1 1

2 2 22 2 1

1 1 21 1 1

1

1

1

1

α

α

α

α

+ = −

+ = −

+ = −

+ = −
The Decision Problem

. In other words, the number of kills 

exceed the number of target weapon systems available.

What remains to be defined is a utility function that i

measure of how well both sides accomplish their missio

this simple problem, we assume that both sides wish to 

mise the number of tanks available at the end of the 

cycle. Their reasoning is that as the opposing sides com

closer contact, tanks are more effective than artillery. A 

function that does this is:

The Blue commander therefore wishes to maximise P a

Red commander wishes to minimise P. Note that the w

0.9 and 0.1, reflect the relative importance assigned to 

and artillery by the two commanders. Of course, this ob

function is not unique. There are several other possibilit

An interesting problem arises when the opposing comm

do not agree on a common objective. That is, what ha

when the game is not zero-sum? In this case, we would n

evaluate a separate objective function for each side. 

example above, this would correspond to having a 

P(Blue) and a utility P(Red) that might correspond (for 

ple) to different weightings for the value of tanks 

artillery due to different perceptions of the endstate (Re

simply wish to survive with a roughly balanced for

example). We would then require some higher level m

of what this set of outcomes implies. In some cases (s

Operations Other Than War), we might be seeking to 

mise the utility of both sides (i.e., a win-win situation 

than the win-lose assumption we made above). 

() () () 

y 

2 j j 1 j

b () () () t x t y t α1 j j 1 ≤ j

() () ( ) ( ) ( ) 

1

22 2 2 1 

0

0.9 2 2 0.1 2 

t P xt yt x y x = = −+ − + −    

∑    
The Decisi

p, 

yse an equivalent decision streams case. The allocat

artillery to enemy tanks and artillery over the two cy

referred to as the allocation strategy. Figure 4.9 lists 

possible strategies by describing the game for the Blue 

mander in extensive form. Note that the diagram reflec

the allocation of Blue artillery to Red tanks, b2(t). 

Figure 4.9: BLUE Commander’s Allocation Strategy

The Blue commander reasons that during the first 

because the opposing forces are not in direct conta

greatest threat to his forces is the enemy artillery. The

he allocates all of his artillery against Red artillery. In t

ond cycle, as the forces begin to close, he sees Red tanks 

more serious threat and therefore allocates all of his res

against Red tanks. Therefore, his decision stream (allo

strategy) is:

BBB G = = { } ( ) () 0 , 1 1,0 , 0,1 . { } [ ][ ]
The Decision Problem

gy p, 

sponds to the bang-bang solution discussed in Appen

where only extreme settings of the control variable are 

(This solution would result from a Lanchester squa

Blue/Red interaction for example, which would give ri

linear Hamiltonian, as explained in Appendix 1.)

Now, assume that a new game is to be played that differ

the current game only in that the Blue force has bee

mented by more tanks. If the addition of these tanks do

alter the decision stream, then we consider them equ

and the game may be replayed with the same decisio

both sides and the outcomes (value of P) compared.

DECISION UNCERTAINTY

ASTOR’s primary function is to contribute to tactical 

tion assessment by observing the battlefield, detectin

identifying enemy units, and reporting on its findings. C

quently, a metric designed to measure how well sit

assessment is accomplished in all cases tested was seen 

ful to this study of the ASTOR sensor system. Such a 

allows us to measure the degree of confidence the commander 

he possesses an accurate picture of the battlefield in his area of 

We would expect that the greater his knowledge abo

location, size, and composition of the enemy force, the g

his confidence in making decisions concerning the allo

of his weapons and the movement of his forces. We also 

nise that information of this type is not all he would r

Information concerning enemy intent gleaned from 

INT, SIGINT, and known enemy fighting doctrine wou

assist in completing the picture.
The Decisi

p

edge the commander has concerning the enemy forces a

against him in his area of interest. The measure is a func

the size, diversity, and effectiveness of the sensor suite, a

effectiveness of the command and control system used 

cess the reported sensor observations. The detailed de

ment of this metric is covered next.

PROBABILITY DISTRIBUTION

We begin by letting the vector U represent the com

hypotheses that any number of enemy units are a

against the friendly commander at time cycle t s

U = {0,1,2,...,n}. Given the level of resolution for the A

games, a unit was taken to be a battalion. We omit th

index, t, for now focusing instead on analysis within a tim

The term arrayed against is taken to indicate the units loca

the battlefield in some area of interest to the friendly 

mander. This may mean along some avenue of approa

defensive operation or blocking a route of advance in an 

sive operation. Figure 4.10 depicts a notional de

campaign situation.

We assume that the friendly commander knows the num

enemy units that might be brought to bear against him 

the campaign. That is, we assume that he knows n. This i

sonable assumption in that it is highly likely that the Intel

Preparation of the Battlefield (IPB) would yield this in

tion. What is unknown is the tactical deployment of the u

each timestep. Tactical situation assessment then is taken 

the process of estimating the enemy’s tactical deploym

time t and the effectiveness of this estimate is the deg

uncertainty associated with his current state of knowledg
The Decision Problem

Figure 4.10: BLUE Commander’s Situation Assessment Problem

BAYESIAN DECISIONMAKING

We begin by analysing the intelligence gathering pro

each timestep. We first assume that a Bayesian update 

odology for tactical situation assessment is appropriate 

a wargame cycle, but not between wargame cycles, giv

assumptions concerning the Markov properties (i.e., l

memory) of the FASTHEX game with 2-hour time

Consequently, the process described here is repeated p

each decision to commit forces.

1. INPUT DISTRIBUTION: The friendly commander 

or may not have some idea of the likely disposition 

enemy units. If he does, we may describe it using 

empirical distribution. However, for this analysis, 

assume that the friendly commander is completely 

4

We later exploit Bayesian updating by assuming multiple sensor s

within a single decision cycle. 
The Decisi

y 

provides us with a worst case situation, correspond

the assumed lack of memory between timesteps. W

P(U = u) represent the probability that the enemy 

mander will commit u of his n units in a specified a

interest in the area of operations AO (avenue of 

approach in Figure 4.10). Assuming that the enem

commander is equally likely to deploy any numbe

units in the area of interest, we have that

P(U = u) = 1/n+1. The friendly commander hope

refine this distribution using his sensor assets. 

2. THE SENSOR MODEL: We next let V = {0,1,2,...,n

resent the number of units detected by the sensor 

allocated to the area of interest.5

 Therefore, P(V = 

the probability that the sensors will detect v of the e

units arrayed against the friendly forces. However

number is conditioned on the number of units in 

area of interest deployed by the enemy command

Consequently, we focus on the conditional probab

. For simplicity, we assume a single 

sor is cued to search the specified area of interest.

further assume that the sensor is capable of detect

unit in the area of interest with probability q and t

there are no false detections from the sensor or els

where.7

 Consequently, the conditional probability 

distribution, , is the binomial 

distribution:

5

By “detect” we mean that sufficient information is provided to all

unit to be targeted by a weapon.

6

This assumption can be relaxed to allow for the characterisation 

multisensor suite, provided that the sensors are independent.

P( ) V = v |U = u

P( ) V = v |U = u
The Decision Problem

.

3. SENSOR OPERATIONS: Our objective is to clarify 

enemy force deployment picture based on the sen

observations by refining the friendly commander’

tial and subsequent probability distributions on U

is, we wish to calculate , where vd 

number of detections reported in the cycle, and th

assess the impact of the evidence provided by the 

on our estimate of the number of enemy units arr

against the friendly forces in the area of interest. O

tionally, we assume that the sensor sweeps the are

interest once in a cycle. As a detection occurs, it is 

immediately reported so that there are vd+1 repor

from the sensor per cycle. The additional report 

accounts for the fact that a report of 0 detections i

initially. Since it is impossible to control the time w

detections occur within a FASTHEX game cycle, 

assume a uniform distribution of reports. That is, 

report of no detections occurs at time t/(vd+1), a r

of one occurs at 2t/(vd+1), etc. The estimate is refi

every subinterval using Bayes’ formula as follows:

7

We later relax this assumption by allowing for the possibility that 

sensor detections/identifications are false, that the command and 

system used to transmit the sensor information may report a false 

detection/identification as real, and that the intelligence processing 

may interpret a false detection/identification as real.

( ) ( )









 − ≤ 















= = =

−

0 otherwise. 

1 for | q q v u

v

u

P V v U u

v u v

( ) d P U = u |V = v

( ) ( )( 

∑ ( )( 

= = = − = =

= = − = = = = = n

i P U i V v P V v U 

P U u V v P V v U u P U u V v

0 | 1 |

| 1 | |
The Decisi

, p

ability, is the knowledge contribu

the latest report (the probability that one more un

detected), and is the posterior pro

ity on U given the last report. Note that 

; that is, the p

distribution before sensors are deployed is flat, as 

described above. This process is repeated for 

v = 0,1,...,vd. Making the appropriate substitutions 

we get:

where v = 0,1,...,vd is the number of units detected b

sensor and at each iteration. Figure 4.11 depi

process diagrammatically. Note the difference betwe

sensor sweep in progress and a report of no detectio

The former is depicted by a flat probability distribut

U whereas the latter is a refinement to the flat distri

( ) U 

u |V v 

P( ) V = v |U = u

P( ) U = u |V = v

P( )( ) ( ) U = u |V = −1 = P U = u =1/ n +1

( )

( ) ()

( ) ()

( )( )

( )( )

,

| 1 1

| 1 1

 

| 1 1

| 1 1

|

0

∑

∑

=

=

−

− 













 = = −

− 













 = = −

=

− 













 = = −

− 













 = = −

= = =

n

i v

i

u

n

i

v 

v u v

q

v

i

P U i V v

q

v

u

P U u V v

q q

v

i

P U i V v

q q

v

u

P U u V v

P U u V v

u ≥ v
A Simple Example

Figure 4.11: Developing a Refined Estimate

A SIMPLE EXAMPLE

The following illustrates the process. Table 4.1 summari

results of a simple situation in which three units are kn

be available to the enemy commander. The sensor syste

a probability of detection/identification of q = 0.8. The 

in the rows are the refined probabilities from 0, 1, 2, 

detections. The first row is the a priori probability asse

on U assuming initial total ignorance. Figure 4.12 depi

results graphically.
Mult

Table 4.1: Refined Probability Assessments: Example 1

Figure 4.12: Refined Probability Assessments

MULTIPLE SWEEPS

We now refine the analysis to show the value that multip

sor sweeps within the same cycle have on refinin

probability estimates for U. Suppose that we assume th

sensors are capable of k sweeps of the area of interest 

the commander’s decision cycle. That is, the sensor ca

form k sweeps of the area of interest before the 

v

- .2500 .2500 .2500 .25

0 .8013 .1603 .0321 .00

1 0 .9218 .0736 .00

2 0 0 .9634 .03

30 0 0 1

P ( ) U = 0 |V P ( ) U =1|V P ( ) U = 2 |V P ( U = 
Multiple Sweeps

y gy

of these sweeps (i), vdi enemy units are detected. We 

assume that the probability estimates are made seque

and that the sweep time is sufficiently small to allow for 

gle “end of sweep” report. Using Bayes formula we get:

where i = 1,2,...,k, and 

Making the appropriate substitutions, (4) becomes:

 

In general, Bayesian updating has a tendency to co

rather rapidly–especially in cases such as this wher

detections/identifications are not allowed: that is, it is im

ble to overstate the number of units actually presen

effect is that subsequent detections that report fewer uni

the previous are totally ignored. To illustrate, consider 

ple case in which n = 3 units. We assume that three 

were conducted resulting in three sequential detections u

sensor with probability of detection: q = 0.8. Table 4.2 

marises the results of applying equation (5) with k = 

number of units detected each time is listed in the tabl

number of units in the area of interest is actually thr

subsequent observations that two units were detected/

fied are completely ignored.

( ) ( ) ( ) ( )

( ) ∑ ( ) ( 

= −

−

= = = =

= = = = = = = n

j d i di

d i di

di

P U j V v P V v U 

P U u V v P V v U u

P U u V v

0 1

1

| |

| | |

P( )( ) U = u |V = vd 0 = P U = u =1

( )

( ) ( ) ( )

( ) ( ) ( )

.

| 1

| 1

|

1

1

∑ = −

−

− 













 = =

− 













 = =

= = = n

j v

j

di

d i

u

di

d i

di

di

q

v

j P U j V v

q

v

u

P U u V v

P U u V v
False Target Detections/Id

Table 4.2: Multiple Sweeps Case 1

Now consider a second case with a somewhat different 

as depicted in Table 4.3. In this case, four sweeps wer

ducted resulting in the sequential detections depicted 

Table. The detection of one unit persisted for three r

Note the rapid convergence of . However, t

gle detection of two units in Sweep 4 shifts the mode 

distribution to U = 2. Because we exclude false detectio

reports less than the current number detected will be ign

Table 4.3: Multiple Sweeps Case 2

FALSE TARGET DETECTIONS/

IDENTIFICATIONS

Up to this point, we have assumed that false targets we

present. It is possible to relax this assumption and rec

that targets can be misclassified in several ways: (1) the 

i

0 - 0.250 0.250 0.250 0.2

1 2 0 0 0.625 0.3

23 0 0 0 

32 0 0 0 

i

0 - 0.250 0.250 0.250 0.2

1 1 0 0.658 0.263 0.0

2 1 0 0.767 0.122 0.1

3 1 0 0.925 0.059 0.0

4 2 0 0 0.855 0.1

di v P( ) U = 0 |V P( ) U =1|V P( ) U = 2 |V P( 

U = 

P( ) U =1|V

di v P( ) U = 0 |V P( ) U =1|V P( ) U = 2 |V P( 

U = 
Knowledge Representation

yy ; (2) 

used to transmit the detection to a central processing 

may have erroneously introduced a false target; (3) th

cessing centre equipment or personnel may have m

preted the data being received; and (4) the enemy m

actively engaging in deception activities (i.e., Inform

Operations). The way in which this is done is descri

detail in reference [11], using a Poisson process to rep

the “flow” of false targets through the sensing process.

KNOWLEDGE REPRESENTATION

It now remains to ascertain the degree of uncertainty e

in the mind of the friendly commander at the time h

take a decision on the employment of his forces. His c

knowledge consists of two components: (1) the fact that h

sor suite detected a number of enemy units in his area o

est; and (2) the refined probability distribution ov

possible number of enemy units that might be in his a

interest based on his most recent sensor report. The v

the first component depends upon whether false detectio

possible. The second depends upon the number of 

units detected and the reliability of the sensor system. T

is to develop a knowledge metric that incorporates these tw

ponents, thereby quantifying the likelihood that the 

mander has a true picture of the number of enemy 

arrayed against him in his area of interest. 

INFORMATION ENTROPY

We draw on information science to develop a knowledg

ric that is a function of the average information present 

set of all possible uncertain events. This quantity is refe
Knowledge Rep

fpyf 

a probability distribution.

The amount of information available from the known 

rence of the event, U = u, i.e. that u enemy units are 

arrayed against the friendly force, is inversely proportio

the likelihood that the event will occur. An event that 

likely to occur provides little information when it does 

On the other hand, an unlikely event provides consid

information when it occurs. Mathematically, we define 

mation as follows:

9

If we now consider all of the events in the refin

, we reason that each occurs with prob

 Therefore, the information available 

the occurrence of each event is:

,

and the expected information from the occurrence o

event is:

8The term entropy is used because the information entropy function 

same as that used in statistical mechanics for the thermodynamic q

entropy. For a more complete discussion of entropy, see Blahut [17

Zurek, ed. [18].

9

In communication theory, the units of measurement are “bits” if 

logarithms are used and “nits” if natural logarithms are used (See 

Kullback [19] p. 7). For our purposes, we will assume a dimension

quantity.

( ) ( ) ln ( ). 1 ln P U u

P U u

I U u = − = = = =

d U |V = v

( ) | . d P U = u V = v

( )( ) d d I U = u |V = v = −ln P U = u |V = v

( )( ) ( ) ( 

| | | ln | 

d d d 

P U = u V = v I U = u V = v = −P U = u V = v P U = u 
Knowledge Representation

qy, g

ability distribution: can be expressed as:

The entropy quantity is the residual unc

regarding U given that V is instantiated to vd. The a

uncertainty then is the sum of the residual uncert

weighted by the probability distribution on the sensor 

tion/identifications:

PROPERTIES OF INFORMATION ENTROPY

Information entropy has properties that make it ideal as 

ric for measuring the commander’s uncertainty pr

making a decision and for measuring the uncertainty 

entire campaign:

1. MAXIMUM ENTROPY: The entropy function is ma

mised when the uncertainty in the distribution is 

greatest. Maximum uncertainty occurs when the 

friendly commander has no sensor assets to deplo

this case, any number of units might be arrayed a

him with equal probability. Mathematically we ha

that . The entropy in this case is

Thus the maximum uncertainty in P(U) is ln(n+1

that as n grows larger, the entropy increases as we 

expect; the more units available to the enemy 

( ) d P U |V = v

[ ] ( ) | ( ) ( )( 

| | ln[ | 

∑=0 = = = = − = = = 

n

i d d d 

H P U V v H U V v P U i V v P U i V 

( ) d H U |V = v

( ) ( ) ( )( 

| | ln[ | 

∑ =0 ∑=0 = − = = = = 

n

i

n

j

H U V P V j P U i V j P U i V 

P( ) U = u =1/ n +1

( ) ln( ) 1 . 1

1 ln

1

1

0 = +

+ + = −∑= nn n

H U n

i
Combat Cycle 

, py

the absence of sensor outputs. In general, a prob

distribution with a wide variance exhibits high en

2. MINIMUM ENTROPY: The entropy function is min

mised at 0. This occurs when P(U = ui

) = 1.0 and 

P(U = uj

) = 0 for all . This represents total cer

or minimum uncertainty.

3. CAMPAIGN ENTROPY: The total campaign entrop

denoted H(U1,U2,...Um), where m is the total numb

game (campaign) cycles satisfies the relation:

The equality condition holds when the process is m

ryless as approximated in the FASTHEX gam

purposes of tactical situation assessment), when th

ation being considered is rapidly changing acro

timespan of the campaign. 

COMBAT CYCLE KNOWLEDGE

For the wargames we were dealing with, we found it 

tant to develop a metric that was capable of provid

ordinal ranking of the wargame cases across all scena

terms of the knowledge possessed by the commander p

making a decision at each cycle. Although entropy is a 

nient measure of decision uncertainty, making 

comparisons among the cases examined could be misle

We need only recall that maximum entropy is H(U) = l

to realise that the varying number of enemy units in t

makes a direct comparison incorrect. In addition, it is i

plete because it addresses only the second know

component, namely the knowledge gained from the r

j i ≠

( ) () , , . 1 1 2 ∑= ≤ m

i

H U U LUm H Ui
Combat Cycle Knowledge

py p

sive metric incorporating the residual uncertainty in the r

distribution, and the detection information gained by the 

report. Having said this, in later applications of this i

quantifying Information Dominance [6] and the benef

network-centric approach to collaboration and task pr

tion [9], the simpler form of Residual Uncertainty (and 

Residual Knowledge–which is a measure of unce

removed) has been found to be adequate.

For the wargaming application then, we let K(U,V = vd) 

sent the knowledge gained from detecting vd enemy unit

there are U enemy units in the area of interest. Symbo

we have:

where is the knowledge associated with the 

ual uncertainty in the refined probability distribution g

sensor report of vd units, and K(V = vd) is the knowledge 

by detecting/identifying vd enemy units. If we can ensu

both and K(V = vd) are confined to the i

[0,1], then we can think of K(U,V = vd) as a probabilit

such, it represents the likelihood that the commander 

complete picture of the battlefield at the time he makes 

sion. This can be a very powerful statistic when cor

with the Force Loss Exchange Ratio, enemy attritio

friendly survivability as discussed below.

1. RESIDUAL KNOWLEDGE: The maximum uncerta

 is ln(n+1). Therefore, maximum cert

10K(U,V = vd) satisfies the probability axioms (see Stark and Woo

p. 9 for instance) and therefore can be thought of as a subjective 

probability.

( ) ( )( ) d d d K U,V = v = K U |V = v K V = v

( ) d K U |V = v

( ) d K U |V = v

( ) d P U |V = v
Combat Cycle 

this quantity provides us with the following definit

residual knowledge:

. 

Residual knowledge is maximised when residual e

is 0 and it is minimised when residual entropy is ln

In general, residual knowledge reflects the amo

uncertainty in the refined probability distribution.

2. DETECTION KNOWLEDGE: Given that vd enemy u

were detected, we are now concerned with the ad

tional information this provides concerning the 

likelihood that there are actually vd or more enemy 

in the area. This is clearly a function of the reliabi

the sensors and the command and control system 

to process the sensor data it receives. Mathematic

we are interested in the information content for th

event: . That is, the information tha

be provided from the detection reports this cycle, 

prior information content of the event, V = vd. This 

culated to be:

If vd = 0, we get no information because 

However, if vd = n, the information content is 

mised at . This is due to t

that decreases monotonically 

11In general, the change in information resulting from detecting V 

units is .

()( ) d 

|

( ) () ( ) d d ∆I U |V = v = H U − H U |V = v

( ) ()( )

ln( ) 1

ln 1 | |

+

+ − = = = nn H U V v K U V v d

d

d d U ≥ v |V = v

( )( ) [ ] [∑ ( 

= ≥ = = − ≥ = = − = 

n

i v d d d d 

d

I U v |V v ln P U v |V v ln P U i |

P( U ≥ 

ln | PU nV v ( ) d − ==    

( ) d P U ≥ u |V = v
Combat Cycle Knowledge

g 

gests the following definition for K(V = vd):

.

(We use vd–1 to ensure that the denominator never g

zero).

The total knowledge gained is then defined to be the p

of residual and detection knowledge:

We can apply equation 7 to the example depicted in 

4.1. The first 5 columns of Table 4.4 repeat the informa

Table 4.1 for convenience. The last two columns conta

entropy and knowledge figures based on the refined di

tions at each iteration, and the intermediate values 

Figure 4.13 depicts the results graphically.

Table 4.4: Total Knowledge: Example 1

v

- .2500 .2500 .2500 .2500 1.3863 

0 .8013 .1603 .0321 .0064 0.6130 

1 0 .9218 .0736 .0046 0.2918 .

2 0 0 .9634 .0366 0.1570 .

30 0 0 1 0 

( ) ( )

( )

ln | 1

ln | 1

d

n

d i v

d

d

PU iV v

KV v

PU nV v

=

  = =−   = =   = =−  

∑

( ) ()( )

( )

( 

( )

ln | 1 ln 1 | , ln 1 ln | 1

d

n

d d i v

d

d

PU iV v n HUV v

K UV v

n PU nV v

=

 

= =− +− =  

== ⋅ +   = =−  

∑

P ( ) U = 0 |V P ( ) U =1|V P ( ) U = 2 |V P ( ) U = 3|V H( ) U |V K
Combat Cycle 

Figure 4.13: Knowledge and Entropy for Example 1

CAMPAIGN KNOWLEDGE

A similar formulation may now be used to calculate ca

knowledge, given that the FASTHEX games are taken 

memoryless processes for tactical situation assessment

sider a campaign consisting of m cycles. At each cycl

enemy units are detected by the sensor. At each cyc

number of possible enemy units arrayed against the f

forces, n, is likely to be reduced as a result of combat dur

cycle so that nt

 is the total number of enemy forces that 

be arrayed against the friendly forces in the area of in

 then represents the residual uncertainty a

cycle and the total campaign uncertainty is expressed as

( ) t dt H U |V = v

( )( 

( )( ) | ln | .

| , | , , | |

1 0

1 1 1 2 2

∑ ∑

∑

= =

=

= − = = = =

= = = = 

m

t t dt

n

i t dt

m

t d d m dm t 

P U i V v P U i V v

H U V v U V v U V v H U V 

t

L
Combat Cycle Knowledge

y g, gp

be defined as:

Detection knowledge can be calculated in a similar w

this case, we are interested in the total information av

from having detected/identified vdt enemy units at each 

m cycles, t. To do this, we rely on the fact that the tota

mation available from the occurrence of m independent 

is the sum of the information available from the occurre

each of them. Therefore we get that:

Detection knowledge for the entire campaign can th

expressed as:

and total campaign knowledge is:

( ) [ 

( )( 

( ) 

ln 1

ln 1 | | , | , , |

1

1

1 1 2 2

∑

∑

=

=

+

+ − 

= = = = m

t t

m

t t t 

d d m dm

n

n H U V 

K U V v U V v L U V v

( 

( )

11 1 2 2 2

1

| 1, | 1, , | 

| 1.

d d d d m dm 

m

t dt dt t

IU v V v U v V v U v V 

IU v V v =

≥ =− ≥ =− ≥ = 

= ≥ =− ∑

L

( 

( )

( )

11 1 2 2 2

1

1

| 1, | 1, , | 

ln | 1

ln | 1

t

dt

d d d d m dm 

m n

t dt t iv

m

t t dt t

IU v V v U v V v U v V 

PU iV v

PU n V v

= =

=

≥ =− ≥ =− ≥ = 

  = =−   =   = =−  

∑ ∑

∑

L

( )

( )( )

( )

( 

( 

1 12 2

1 1

1 1

, , , ,, ,

ln 1 | ln | 

ln 1 ln | 

t

dt

d d m dm

m n m

t 

t t dt t iv t

m m

t t t dt t t

KUV v U V v U V v

n HU V v PU iV v

n PU n V v

= = =

= =

== =

  +− =  

= =

  

= ⋅ + = =

 

 

∑ ∑ ∑

∑ ∑

L
Combat Cycle 

AN EXAMPLE

Consider the example summarised in Table 4.5. The cam

consists of 5 cycles. At each cycle t, the detection probab

the number of units detected vdt and the maximum size n

enemy force is given. The last three columns depict the r

uncertainty in the refined probability distribution, the in

tion available from the detection of vdt enemy units, a

“probability” that the commander has an accurate pic

the number of enemy units in his area of interest. The la

reflects his total campaign knowledge.

Table 4.5: Total Knowledge

The detection probability qt is assumed to change as a 

tion of time t to reflect the changing sensor mix

maximum size of the enemy force nt

 reduces over t

reflect attrition. The fluctuation in numbers detec

(including a complete lack of detections during one time 

leads to a reduced value of overall campaign knowledg

is scaled to vary between 0 and 1, with 0 representing 

plete ignorance, and 1 representing complete knowle

the number of enemy units in the commander’s area o

est at every stage of the campaign. 

t qt vdt nt

1 .6 2 5 0.6122 0.2263 0.13

2 .4 3 5 0.3950 0.3868 0.15

3 .9 3 4 1.0000 0.5693 0.56

4 .9 0 3 0.5572 0 0

5 .3 2 3 0.5166 0.5000 0.25

Total Campaign Knowledge 0.20

( ) t dt K U |V = v ( ) t dt K U = v ( 

t 

K U ,V 
Combat Cycle Knowledge

THE EFFECTS OF KNOWLEDGE

We stated at the outset that it was desirable to assess the 

of increased knowledge on the outcome of the campaign

way to do this is to compare K(U,V = vd) to the Forc

Exchange Ratio (FLER) and the friendly and enemy c

attrition. Comparisons with the FLER measure how 

edge influences the relative losses in combat. Comp

with friendly and enemy attrition measure the degree to 

knowledge enhances the survivability of friendly forces a

destruction of the enemy. Statistically, we have shown 

tive linear relationship when the FLER or enemy attr

compared to knowledge, and a negative linear relati

when friendly casualties are compared to knowledge [11

results also showed that the approach and structure (the 

the epitomising approach, and open gaming) adopted 

wargames was able to produce a set of coherent and qua

alternatives, which formed the basis of the balance of 

ment analysis. As a consequence of this analysis, inform

could be weighed in the same scale as weapon effects, a

benefit clearly demonstrated. 

Figure 4.14 shows the relation between campaign level 

edge and attrition of enemy forces, as assessed usin

FASTHEX wargaming experiments, giving a corr

value of 0.8. 

Figure 4.15 shows the effect of an increase in campaig

knowledge on own force casualties, again as a result 

FASTHEX wargaming experiments, giving a corr

value of –0.4.
Combat Cycle 

Figure 4.14: Experimental Assessment of Campaign Level Knowledg

Attrition of Enemy Forces

Figure 4.15: Experimental Assessment of the Effect of Campaign L

Knowledge on Own Force Casualties
Quantifying the Benefit of Collaboration Across an Information Network

[] p, 

appears to be a point where the knowledge available 

commander exceeds his capacity to act on it, either to 

ate more enemy losses or to prevent further friendly cas

and we need to represent this effect. 

QUANTIFYING THE BENEFIT OF

COLLABORATION ACROSS AN INFORMATIO

NETWORK

In further exploitation of these ideas, Perry [9] has show

this approach can be used as a basis for quantifying the 

of collaborating across an information network. An ex

of the approach is described below, as used in recent w

Dstl in the UK in the context of modelling a time-

operation. Full details of the general approach and othe

of application are in [9].

We assume we have a network of command and control 

that are involved in coordinating a time-critical oper

Each of these nodes has a number of informatio

cessing tasks to perform. If:

is the mean time for node i to complete all of its tas

assume that this completion time is distributed expone

(an exponential distribution is used to model the time b

events or how long it takes to complete a task), so that i

the probability of completing all tasks at node i by time 

.

In general, there will be a number of parallel and seq

nodes in the network sustaining the operation. Let th

1

λi

( ) it

i i ft e λ λ − =
Quantifying the Benefit of Collaboration Across an Informati

p, p

sisting of nodes where is a subset of , as sho

Figure 4.16.

Figure 4.16: The Critical Path

We define the total latency of the path as the sum 

delays (latencies) at each of the nodes, plus the time, d

as tm, required to move a terminal attack system (such 

aircraft) to the terminal attack area. In this sequentia

we thus have that the total expected latency T is the 

the expected latencies at each node on the critical pat

the time tm:

.

If there are sequential and parallel nodes on the critica

these can be dealt with in the way shown by the ex

below:

Figure 4.17: Parallel Nodes on the Critical Path

ρ ρ τ

1

1 m

i i

T t

ρ

= λ

= + ∑
Quantifying the Benefit of Collaboration Across an Information Network

p

.

Returning now to the case of a serial set of nodes that 

tute the critical path, for each such node i on the critic

define the indegree di

 to be the number of command and c

(C2) network edges having i as a terminal link.

For each node j in the C2 network, we assume (based 

earlier discussion of information entropy and knowledg

the amount of knowledge available at node j concern

ability to process the information and provide quality c

ration is a function of the uncertainty in the distribu

information processing time f

j

(t) at node j. Thus the m

know about node j processes, the better the quality of c

ration with node j.

Let Hj

(t) be the Shannon entropy of the function fj(t). 

Hj

(t) is a measure of this (residual) uncertainty defined in 

of a lack of knowledge. By definition of the Shannon e

we have:

If λj

min corresponds to a minimum rate of task complet

node j, then:

12 3 4

11 1 1 max( , ) T tm λλ λ λ

= +++

0

( ) ln( )

Since the differential of ( ) is 

it follows that ( ) ln( )

j j t t

j jj

xx x

j

j

H t e e dt

xe e xe

e H t

λ λ λ λ

λ

∞

− − = −

−

=

∫

1

min λ j
Quantifying the Benefit of Collaboration Across an Informati

ppp

at node j. In order to provide a normalised value of the 

edge Kj

(t) available at node j in terms of the Shannon e

we define this as:

Suppose now that node i is on the critical path, and no

another network node connected to node i. Let cij rep

the quality of collaboration obtained by including nod

this is high, reference [9] assumes Kj

(t) will be close to 

effective latency at node i is thus assumed to be reduced 

factor due to the effect of this high quality 

laboration. The factor is assumed to be 1 if j is one 

nodes directly involved in the time-critical operation (b

on the critical path). It is assumed to be 0.5 if node j is 

the other network nodes, to reflect a lower level of coll

tion with these nodes.

It is important to note that the actual latency may 

reduced by this collaboration, but the ability to use th

more wisely through collaboration (to fill in missing p

the operational picture that are available from 

nodes, etc.) has an impact that can be expressed equival

terms of latency reduction. The use of such time more 

implies a good knowledge of expected time to complet

that can provide such information. This is similar to the 

entropy and knowledge in the FASTHEX wargames 

j

j

j

( ) ln ln min

ln if min min

min

0 if min

1 if e min

j

j j

j

j j

j

j

j

e e K t

e

λ λ

λ

λ λλ

λ

λ λ

λ λ

   = −   

  

 

= ≤≤  

 

= <

= >

(1 ( )) j K t j

ω −

ω j
Quantifying the Benefit of Collaboration Across an Information Network

gy

the enemy force, and hence to wiser use of the comma

own forces. Such wiser use, due to information supe

could be quantified in that case by an equivalent 

improvement in the number of enemy units destroyed

reduction in own force casualties. 

The balance to be struck is that between such enhanc

laboration and the effects of information overload 

increasing network complexity (which we assess separate

function of the number of elements of the network invo

the task).

The total (equivalent) reduction in latency at node i due 

laboration with the network nodes connected to node i 

given by:

Thus the total effective latency along the critical 

accounting for the positive effects of collaboration, is giv

We noted in the experimental data from the ASTOR 

discussed earlier that we need to represent the effect of 

mation saturation. Reference [9] also includes a complexity 

to account for the fact that taking account of addition

1

1

(1 ( ))

i

i

j

d

i ij

j

d

j

j

c c

K t ω

=

=

=

= −

∏

∏

1

1 1

1 { (1 ( )) } i

j

i

c m

i i

d

j m

i i j

c T t

Kt t

ρ

ρ ω

λ

λ

=

= =

= +

= −+

∑

∑ ∏
Quantifying the Benefit of Collaboration Across an Informati

y 

This is the negative effect of collaboration. It leads 

increase in effective latency on the critical path. Followi

we define C to be the total number of network conn

accessed by nodes on the critical path. For each nod

the critical path, this is the indegree di

. Thus:

.

The value of C is then a measure of the complexity of th

work. We assume that the complexity effect associated 

particular value of C follows a nonlinear S-shaped cu

shown below.

Figure 4.18: The Logistics S-Shaped Curve

The equation used to describe this effect is a Lo

equation:

.

The penalty for information overload is then defined as:

.

1

i

i

C d

ρ

=

= ∑

( ) 1

a bC

a bC

e g C

e

+

+ = +

1

1 () − g C
Quantifying the Benefit of Collaboration Across an Information Network

y, g p

and negative effects of C2 network collaboration, is then

.

NETWORK-CENTRIC BENEFIT

This network-enabled approach thus allows us to comp

distribution of the response time of the system as a func

the network assumptions. As we increase the collabo

throughout the network in going from platform-centric 

work-centric to futuristic network-centric (to use the R

categories [9]), so the positive effects of enhanced coll

tion have to balance off against the downside effe

information overload and increasing network comp

Going back to the discussion in Chapter 2 on the Conc

Framework of Complexity, we can call this overall a

performance of the network the plecticity12 of the ne

since it characterises the combined positive and n

effects of network complexity and collaboration.

STOCHASTIC NETWORKS AND NETWORK

VULNERABILITY

So far, we have shown how it is possible to calculate th

tive and negative effects of network complexity 

collaboration based on the use of Shannon entropy as 

sure of (lack of) knowledge. We can extend this 

potentially in a number of ways. The length of the 

path, for example, if the network is adapting over the co

a series of tasks, will be a stochastic variable. We would e

12A term proposed by Perry (RAND Corp.) - personal communic

, 1 ()

c m

cC m

T t T t

g C

− = + −
y 

network (i.e., the number of nodes on the critical path) 

be sampled from a power-law distribution of network siz

exponent of this power-law is then a characteristic mea

the ability of the nodes in the network to form and 

dynamically over time. Similarly, we can consider the in

of a node on the network to be a stochastic variable. 

indegree of a node is sampled from a power-law distribu

the number of links, then the network is said to be “scal

[20]. This corresponds to a network with a small num

nodes with very rich connections, and many nodes with 

connections. (The Internet is an example of a scale-fr

work.) Conversely, if the indegree of a node is sampled 

normal distribution of the number of links, then the netw

of “random” type. Characterising networks in this way 

us to investigate the vulnerability of such networks to att

discussed in [20].

REFERENCES

1 MOFFAT J (2002). Command and Control in the Information Age: Represen

Impact. The Stationery Office. London, UK.

2 HORNE G E and LEONARDI M (2001). Manoeuver Warfare Science 

Marine Corps Combat Development Command. Quantico, VA, U

3 LAUREN M (2002). “Firepower Concentration in Cellular Automa

Combat Models – An Alternative to Lanchester.” J Opl Res. Soc 53 I

pp. 672-679.

4 www.cna.org/isaac (Aug 1, 2003)

5 GOLDENFELD N (1992). Lectures on Phase Transitions and the Renorm

Group. Addison-Wesley. MA, USA.

6 DARILEK R, PERRY W et al (2001). Measures of Effectiveness for the 

Information Age Army. RAND. Santa Monica, CA, USA.

7 TURCOTTE D L (1997). Fractals and Chaos in Geology and Geophysics. 

Edn. Cambridge University Press. Cambridge, UK.
References

ppypyjpp

(February 1, 2003)

9 PERRY W, BUTTON R W et al (2002). Measures of Effectiveness for th

Information-Age Navy: The Effects of Network-Centric Operations on Combat O

RAND. Santa Monica, CA, USA.

10 ROSKE V (2002). “Opening Up Military Analysis: Exploring Beyo

Boundaries.” Phalanx. USA Military Operations Research Society. 3

11 PERRY W and MOFFAT J (1997). “Measuring the Effects of Know

Military Campaigns.” J Opl Res. Soc 48. pp. 965-972.

12 BOWEN K C (1978). Research Games – An Approach to the Study of Decis

Processes. Taylor and Francis, UK.

13 STARK H and WOODS J W (1986). Probability, Random Processes and 

Estimation Theory for Engineers. Prentice Hall, USA.

14 BRYSON A E and HO Y C (1975). Applied Optimal Control. Hemisph

Publishing, USA.

15 HILLESTAD R (1986). SAGE: An Algorithm for the Allocation of Resourc

Conflict Model. RAND working draft.

16 BERKOVITZ D and DRESHER M (1959). A Game-theory Analysis of 

Air War. Operations Research, 17. pp. 599-620.

17 BLAHUT R E (1987). Principles and Practice of Information Theory. Add

Wesley. MA, USA.

18 ZUREK W H ed. (1990). Complexity, Entropy and the Physics of Informat

III, Santa Fe Institute Studies in the Sciences of Complexity Series. 

Wesley. USA.

19 KULLBACK S (1968). Information Theory and Statistics. Dover. New Y

USA.

20 COHEN D (2002). “All the World’s a Net.” New Scientist. 13 April 20

pp. 24-29.
139

CHAPTER 5

AN EXTENDED

EXAMPLE OF THE

DYNAMICS OF LOCAL

COLLABORATION AND

CLUSTERING, AND

SOME FINAL

THOUGHTS1

owards the end of Chapter 4, we discussed

the way in which a particular network could

be analysed using what we called the plecticity of a

network, which includes both the positive effects

of collaboration and the downside effects of information overload. This assumes a network of a

particular size and configuration, and thus raises

1

The contribution of Dr. Susan Witty, Dstl, to this 

chapter is gratefully acknowledged.

T
qg

of collaboration are likely to emerge. In some models of n

systems, we have already seen that the networks of inte

that form (i.e., the dynamics of cluster formation) can b

dicted ahead of time to have a particular form. For exam

Bak-Sneppen model of the coevolution of the species wit

evolving ecosystem (described in Chapter 1) gives rise to c

of coevolution that tend to a power law distribution of 

size. Clusters of burning trees (forest fires) also show such 

law effects, which turn out to be similar to the distribu

casualties in war (Chapters 2 and 3). Clustering of force u

the ISAAC “distillation” model of manoeuvre warfare pr

fractal clustering (Chapter 4). 

We wish to finish with an extended example analysis 

ISAAC distillation; recall that this is a simple agent-based 

of land warfare, incorporating small rule sets that govern 

decisionmaking, movement, and engagement. More d

available in [1]. Figure 5.1 is a screenshot of the start of a 

ISAAC simulation run.

Figure 5.1: Screenshot of the Start of a Typical ISAAC Simulation 
Clustering and 

, p

rules to describe interactions between units. Different s

ios can be set up by changing the parameters of these 

The initial laydown of forces is carried out stochastica

the model within user-defined limits, and so different r

tions of the same basic scenario are possible.

The particular ISAAC scenario we use in this chapter w

plied to us by Dr. Gary Horne of the U.S. Marine 

Warfighting Lab and has three phases. In the first pha

Red forces move to meet the Blue; the second phase 

engagement between Red and Blue forces. It is in th

phase that either the Red forces take control of the Blue 

the top right corner) or the Blue forces retain control o

flag. This scenario is interesting as in all stochastic repli

but one (replication 40), the Red forces are successful in a

ing their goal of taking control of the Blue flag. The a

that follows explores the clustering and “swarming” 

agents and the similarities and differences in the replicati

CLUSTERING AND SWARMING

In order to analyse the swarming dynamics of cluster 

tion and dissolution in ISAAC, we need to consider 

what this means. There are two ways to define a clu

agents. The first, and most usual, is to define neighb

agents only by those that are north, south, east, or wes

cent to the agent in question, known as nearest ne

clustering. The second (and although most intuitive, les

definition is to include all eight neighbours of the centra

as part of a cluster, as shown in Figure 5.2, which is kn

next nearest neighbour clustering.
Clustering and Swarming

Figure 5.2: Nearest and Next Nearest Neighbour Clustering

For a single timestep in any run of a cellular automata

model, the number and size of clusters of agents can be 

mined using a simple algorithm–the Hoshen-Kop

algorithm–for the nearest neighbour case. This algorith

be modified for use with eight neighbours. For details 

algorithm, see references [2,3]. 

CLUSTER DISTRIBUTION

Once the cluster numbers and sizes can be determined

are a number of ways to analyse the data. The first t

look at is the size of the largest cluster. This gives an ind

of the ability of the agents to cluster or the amount of di

of the agents. For example, if the largest cluster size is n

the total number of agents, we know that that is the onl

ter. However, if the largest cluster is small, then we kno

the agents are dispersed in many small clusters.

The following plots are of the largest cluster size agai

timestep for several different replications with differen

dom seeds for the same basic run of the ISAAC mode

clustering algorithm used is that of nearest neighbours a

ferent plots are graphed for Red and Blue agents. The 

can be ordered by state–alive, injured, or dead. The plo

follow are for only those agents that are alive. Fo

timestep, we plot the largest cluster size for Blue and th

est cluster size for Red. For the first iteration of our ex
Clustering and 

, g

cluster size for Red as a function of simulated time. Figu

shows the same thing for Blue.

Figure 5.3: Largest Cluster Size as a Function of Simulated Tim

(First Iteration, Red Agents)

Figure 5.4: Largest Cluster Size as a Function of Simulated Tim

(First Iteration, Blue Agents)

In these two plots (Figures 5.3 and 5.4), it is possible to 

smoothly changing pattern in the largest cluster size 

Red forces, but not for the Blue. Looking at the cluste
Clustering and Swarming

, yp

distinct areas in time by the rate of change of cluster si

slope of the plot. The distinct areas in time of the chan

the slope of the plot of largest cluster size of Red agents 

spond to the three phases of the ISAAC run.

In further replications of the ISAAC run, a similar p

emerges–each time Red succeeding in his objective. Ho

in the 40th replication, Red fails to secure the Blu

Dr. Horne estimates that the Red force is successful in 

of 100 replications, with only this one failure. Figures 5

5.6 show the evolution of the largest cluster size fo

replication.

We can see that there is now no clear evidence of a third 

of operation in the plot of the largest Red cluster size. I

the plot of the largest Blue cluster size is now more stru

and shows evidence of a third phase of the force’s ope

We suggest that Red’s failure is due to the increased clu

ability of the Blue forces, thereby reaching a greater 

cluster size than in other replications. Such behaviour 

sistent with the mathematical metamodel of ISAAC dis

in Chapter 4. This metamodel indicates that agile clu

and reclustering should lead to better local force rati

hence improved ability to cause attrition to the enemy (l

and to thus move freely.

To gain some additional insight, let us now consider th

est cluster size at each timestep of the simulation, an

this as a frequency distribution of cluster size. We hav

this for a number of replications. Figure 5.7 shows t

quency of largest cluster size for Red agents for typic

exceptional replications.
Clustering and 

Figure 5.5: Largest Cluster Size as a Function of Time (40th Iteratio

Figure 5.6: Largest Cluster Size as a Function of Time (40th Iteration

Figure 5.7: Frequency Distribution of the Largest Cluster Size for Red 
Clustering and Swarming

g

spectrum of cluster sizes. Figure 5.8 shows the same p

the Blue agents with the number of each replication 

on the plot.

Figure 5.8: Frequency Distribution of the Largest Cluster Size for Blue 

We can see from Figure 5.8 that the spread of clust

Blue is much smaller in general. However, for the 40th 

cation, Blue is able to generate a wider spread of cluste

thus succeed.
Clustering and 

THE DISTRIBUTION OF CLUSTER SIZE

Let us move on from largest cluster size now and look 

the clusters formed over time in the simulation. Fro

mathematical metamodel of ISAAC discussed in Cha

and from the general emergent behaviour of natural s

we have discussed in this book, we anticipate that the di

tion of cluster size should approximate to a powe

distribution. Thus on a log-log scale, the distribution of 

size should be a straight line, with end effects whe

assumptions break down.

Firstly let us just look at one replication of the simulatio

ure 5.9 shows the distribution of cluster size for Red age

the 2nd replication of ISAAC, plotted on a log-log scale

Figure 5.9: Distribution of Cluster Sizes (2nd Replication, Red Ag

We can see that in the intermediate regime, the plot fo

straight line, confirming the theoretical expectation. 

5.10 shows a similar plot with the other replic

superimposed.
Final Thoughts

Figure 5.10: Distribution of Cluster Size for Red Agents

Finally, from theory and from the analysis in Chapter 

expect that the time series of casualties produced by a 

such as ISAAC should show evidence of fractal cluste

time. In precise terms, this implies that the power spect

the time series of casualties should be related to the c

size as a power law relationship. Using a related dist

model called MANA [4], evidence of this effect ha

found by Lauren [5], as we have already discussed in C

3. This is an area that we intend to investigate further 

context of building metamodelling equations.

FINAL THOUGHTS

We started by considering what we can learn from n

systems: an ecosystem in which species coevolve loc

fluid forming an interface when it is pinned; the effect o

fires. All of these show regularities and emergent beha

of the whole system that can be captured and deduced 

mathematical models. We have also shown how the 

ideas of local coevolution within such “open” systems a

relevant to thinking about the consequences of a networ

tric form of warfare, where units coevolve (self-synch

across an information grid. By exploiting this linkage, it 
qp 

the likely emergent behaviour of such coevolving netw

force interaction. 

This is work in progress that we hope will contribute 

new science of understanding, analysing, and modelli

effects of Information Age warfare. In doing so, we a

remarked in the preface to a previous contribution,2 to “

deeper understanding not only of conflict, but also 

avoidance of conflict, which is the ultimate aim of the 

cal/military art.”

REFERENCES

1 ILLACHINSKI A (2000). “Irreducible Semi-Autonomous Adaptive 

Combat (ISAAC): An Artificial Life Approach to Land Warfare.” M

Operations Research. Vol 5 No 3. pp. 29-46.

2 HOSHEN J and KOPELMAN R (1976). “Percolation and cluster 

distribution. I. Cluster multiple labeling technique and critical conce

algorithm.” Phys. Rev. B14, p. 3438.

3 http://www.splorg.org/people/tobin/projects/hoshenkopelman/

hoshenkopelman.html (Aug 1, 2003)

4 LAUREN M K and STEPHEN R T (2002). “Map-Aware Non-Un

Automata (MANA)–A New Zealand Approach to Scenario Modelli

Journal of Battlefield Technology. Vol 5 No 1. pp. 27-31.

5 LAUREN M K and STEPHEN R T (2002). “Fractals and Combat 

Modelling: Using MANA to Explore the Role of Entropy in Compl

Science.” Paper prepared for Fractals. Defence Technology Agency. 

Auckland, New Zealand.

2

Moffat J. Command and Control in the Information Age; Representing its I

The Stationery Office. London, UK, 2002.
151

APPENDIX

OPTIMAL CONTROL

WITH A UNIQUE

CONTROL SOLUTION

n this Appendix, we investigate the case of a

unique optimal control solution to the problem of system control, and show that this unique

solution is of the form of bang-bang control when

the system is linear in nature.

As in Chapter 1, we assume that our system can

be described by the functional relationship:

where Fi

 are the rate laws, and are the control variables. In matrix/vector notation, we

write this as:

with initial conditions .

( ) 1 1 ( ),... ( ); ( ),... ( ) , ( 1,.... ) i

in m

dX FXt Xt t t i n

dt = = λ λ

( ) i λ t

Xt FXt t & ( ) ( ( ), ( )) = λ

0

0 ( ) Xt X i i =

I
y y 

and sufficient to make the equations “well behaved.”

We consider here analysis of this relationship as time 

over a fixed time interval [t0,t1]. At time t1, the state va

will have values Xi

(t1), and the objective is to maximise o

imise a linear combination of these endstate values

problem can thus be written as:

Optimise 

(where the ci are constant coefficients or “weights”) sub

the constraints:

.

We now define a process [1] that yields a necessary con

for a vector of control variables to optimise the ob

function. In other words, any control vector that gives 

an optimal value of the objective function must satisfy th

dition. Although this does not guarantee that a solutio

satisfying this condition is optimal, other information (s

the uniqueness of such a solution) can be used in par

cases to prove that is indeed an optimal control vec

The first step in this procedure is to introduce a set of 

variables:

that are defined by the relationships:

λ

1

1

( )

n

i i

i

cX t

=

∑

Xt FXt t & ( ) ( ( ), ( )) = λ

0

0 ( ) Xt X i i i = ∀

λ( )t

λ( )t

1( ), ........ , ( ) n ψ t t ψ

1

( ,) () ()

n j

i j

j i

F X

t t

X

λ

ψ ψ

=

∂ = − ∂ & ∑
where the values ci

 are the same as the coefficients app

in the objective function.

A Hamiltonian function H is now defined by:

where < > denotes the inner product of the two v

. 

Thus:

From the definition of H, we have the dual relationship:

The corresponding “boundary conditions” are:

1 ( ) i i ψ t ci =− ∀

HX X (, ,) , ψλψ =< > &

ψ λ and

1

1

( , , ) () ()

() ( , )

n

i i

i

n

i i

i

H X tX t

tF X

ψλ ψ

ψ λ

=

=

=

=

∑

∑

&

i

i

i

i

H X i

H i

X

ψ

ψ

∂ = ∀

∂

∂ =− ∀

∂

&

&

0

0

1

( )

( )

i i

i i

Xt X i

ψ t ci

= ∀

=− ∀
Pontryagin’s Maximum Principle

PONTRYAGINS MAXIMUM PRINCIPLE

Using the notation we have now developed, Pontr

Maximum Principle [2] states that if is a control 

that maximises (resp. minimises) the objective function:

then the Hamiltonian achieves a minimum 

maximum) at the point for any value of X or .

It is worth noting here that if the set U of admissible c

vectors is a topologically compact set, then the cont

function H will realise its minimum or maximum value 

The control vectors that minimise or maximise the H

tonian H are called extremal controls. If we denote this su

U by U*

, then we know from Pontryagin’s Maximum Pr

that if is an optimal control vector, then . T

we know that:

1. An optimal control vector exists; and

2. There is only one extremal control (i.e., U*

 is a sin

point)

then the single element of U*

 must be the optimal contr

tor. Whether (1) and (2) apply depends on the par

problem under study.

DETERMINING THE EXTREMAL CONTROL

The process for determining these extremal controls is 

lows (for definiteness, we assume that we are maximisi

objective function, and hence minimising the Hamiltoni

* λ ( )t

1

1

( )

n

i i

i

cX t

=

∑

H X (, ,) ψ λ

* λ ( )t ψ

λ

* λ * * λ ∈U
Determining the Extrem

p

.

2. Compute the values of which minimise 

for .

These are the extremal controls. They are expre

functions of X and 

.

3. Substitute this extremal control into the relatio

in order to solve for the extremal system traject

and extremal dual function .

4. Substitute these values into the expression for the 

mal control:

to give an explicit formulation of this extremal c

vector.

Knowledge of the extremal trajectory X*

 allow

objective function:

to be evaluated.

H X (, ,) ψ λ

λ H X (, ψ 

λ ∈U

ψ

λλ ψ = (,) X

λ

i

i

i

i

H X

H

X

ψ

ψ

∂ = ∂

∂ = − ∂

&

&

* ψ

* * λλ ψ = (,) X

*

1

1

( )

n

i i

i

cX t

=

∑
Determining the Extremal Controls

LINEAR MODELS

A linear system model is defined as having a relationship 

form:

in a matrix and vector representation, with initial con

.

When the system behaviour is of this form, it is poss

characterise the nature of the optimal controls under 

general conditions. This is particularly the case if the ob

function itself is linear, i.e. it is of the form:

where, as before, < > denotes the inner product of tw

tors, and S and W are time-dependent vectors of known 

Assume then that the objective function is of this form

that the system model is linear in the way that we 

described. Without loss of generality, we can set g(t)=

write the system behaviour model in the form:

where X is the vector of state variables (e.g., force levels) 

is the vector of control variables.

Make the transformation:

.

X t At X t Bt t gt & () () () () () () = ++ λ

0

0 ( ) Xt X i i i = ∀

1

0

, ,

t

t

< >+< > SX W λ ∫

X AX B & = + λ

0

1() , ,

t

n

t

X t SX W + = < >+< > λ ∫
Determining the Extrem

j

Optimise 

and the relationship 

added to the set of equations describing the system beh

(This can be done since the above equation is linear and 

the same form as the others.)

Consider now the form of the Hamiltonian for such a 

system. We have:

Since we are interested in the extremal controls that 

mise or minimise H, only the second term is of intere

first not being a function of .

Let us look at this second term in more detail. We have:

Let: .

Then: .

Let us assume that the objective function is to be maxi

By Pontryagin’s Maximum Principle [2], we thus wish t

sider control vectors that minimise the Hamiltonian H

is then equivalent, as we have seen, to minimising the e

sion .

1 1 ( ) X t n+

1( ) ( ), ( ) ( ), ( 

X t St Xt Wt 

&

n+ =< > + < 

λ

(, ,) ,

,

, ,

HX X

AX B

AX B

ψλψ

ψ λ

ψ ψλ

=< >

=< + >

=< > + < >

&

λ

λ

, () i ij j i ij j

i j ji

< >= = ψλ ψ λ ψ λ BB B ∑ ∑ ∑∑

j i ij

i

φ ψ = ∑ B

, , j j

j

< >= =< > ψ λ φλ φλ B ∑

λ

() () j j

j

∑φ λt t
Uniqueness of the Extremal Control for a Linear System

(provided that the set U of all possible control vectors i

logically compact):

These are well defined since a continuous function will 

its max or min on a compact set.

For any t in the interval [t0, t1] we then have:

.

For any control vector in the admissible set U of c

vectors.

UNIQUENESS OF THE EXTREMAL CONTRO

FOR A LINEAR SYSTEM

If V is any other extremal control vector, then since it 

mises the Hamiltonian H, it must satisfy:

for any in the admissible set U of control vectors.

However, if , then it is clear that:

. 

Otherwise it would be possible to define a vector gi

smaller value of the Hamiltonian, contradicting the ex

nature of V. It follows that every extremal vector mus

the form . 

*

*

If ( ) 0 let ( ) max{ ( ), }

and if ( ) 0 let ( ) min{ ( ), }

jj j

jjj

t t tU

t t tU

φ λ λλ

φ λ λλ

≤= ∈

>= ∈

* () () () () jj jj

j j

∑ ∑ φλ φλ tt tt ≤

λ

() () () () jj jj

j j

∑ ∑ φ φλ tV t t t ≤

λ

φ j() 0 t >

( ) min{ ( ), } Vt t U j j = ∈ λ λ

* λ
Uniqueness of the Extremal Control for a Li

pgg, g

an extreme minimum value to an extreme maximu

vice versa. Such a form of control is known as ban

since the value “bangs” from one extreme possible va

another, and never assumes any intermediate values. 

we have shown is that for a linear system with a linear 

tive function, every extremal control (including therefo

optimal control) must be in bang-bang form. In this sen

optimal control vector always lies on the boundary 

admissible set U. 

If it can be shown that the are unique, then the abov

struction yields a unique control vector that must then 

optimal control.1

 Now, we have that:

.

Thus the uniqueness of depends on the uniquen

.

We have that:

with final conditions:

. 

1

It can be shown that for this type of system, an optimal control must exi

( ) j φ ( ) j 

φ j

j i ij

i

φ ψ = ∑ B

φ j

(1 ) i ψ ≤ ≤i n

( )

( )

i jj

j i

j jl l jm m

j lm i

j ji

j

X

X

AX B

X

A

ψ ψ

ψ λ

ψ

∂ = − ∂

∂

=− +

∂

= −

∑

∑∑∑

∑

& &

11 1 ( ) ( , ...... , ) (0,.....,0,1) n ψ tc c =− =− +
References

, qq

[1] and thus the vector is unique. Hence, the optimal 

for such a linear system that optimises the objective func

precisely defined by the bang-bang control function . 

REFERENCES

1 CONNORS M M and TEICHROEW D (1967). Optimal Control of D

Operations Research Models. International Textbook Co. Pennsylvania, 

2 ROZONOER L T (1959). L.S. Pontryagin’s Maximum Function Principl

Application to the Theory of Optimum Systems–I, II, III. Avtomatika i 

Telemikhanika 20. p. 1320 et seq. Translated in the journal Automa

Remote Control (1959). 20. p. 1288 et seq.

ψ

* λ
rofessor James Moffat is a Senior Fellow of the D

Science and Technology Laboratory (Dstl), UK, a 

of Operational Research, a Fellow of the Institute of M

matics and its Applications, and a visiting Profes

Cranfield University, UK. He was awarded the Pres

Medal of the Operational Research Society in the year 

He holds a first class honours degree and a Ph.D. in M

matics, and was awarded the Napier Medal in Mathema

Edinburgh University. He has worked for the past 20 y

so on defence-related operational analysis problems and 

space technology research. His current research intere

building analysis tools and models that capture the key 

of human decisionmaking and the other aspects of C4IS

P