A Fractal Concept of War Maurice Passman Adaptive Risk Technology Ltd.

A Fractal Concept of War

Maurice PassmanAdaptive Risk Technology Ltd

Early Combat Theory

Frederick W Lanchester

Frederick Lanchester during the First World War proposed two systems of equations describing attritional warfare. These systems depended on whether fighting was ‘collective’ or not: •The linear law described ‘ancient’ combat•The square law described ‘modern’, collective combat.•These laws were based upon ‘common sense’ as he saw it.

Linear Law

Square Law

( )

( )

B

R

dRk B t

dtdB

k R tdt

( ) ( )

( ) ( )

BR

RB

dRk B t R t

dtdB

k R t B tdt

The Complexity of Fractal Structure

Transmural Pressure Gradient Fractal Lung Structure

Lanchester Square Law• With firearms engaging each other directly with aimed fire from a distance, they can attack

multiple targets and can receive fire from multiple directions. The rate of attrition now depends only on the number of weapons firing.

• Lanchester determined that the power of such a force is proportional not to the number of units it has, but to the square of the number of units. This is known as Lanchester's Square Law.

• More precisely, the law specifies the casualties a firing force will inflict over a period of time, relative to those inflicted by the opposing force. In its basic form, the law is only useful to predict outcomes and casualties by attrition.

• It does not apply to whole armies, where tactical deployment means not all troops will be engaged all the time. It only works where each man (or ship, unit or whatever) can kill only one equivalent enemy at a time (so it does not apply to machine guns, artillery or, an extreme case, nuclear weapons).

• The law requires an assumption that casualties build up over time: it does not work in situations in which opposing troops kill each other instantly, either by firing simultaneously or by one side getting off the first shot and inflicting multiple casualties.

• Note that Lanchester's Square Law does not apply to technological force, only numerical force; so it takes an N-squared-fold increase in quality to make up for an N-fold increase in quantity.

Lanchester Interpretation

Linear Law

Square Law( )

( )

B

R

dRk B t

dtdB

k R tdt

( ) ( )

( ) ( )

BR

RB

dRk B t R t

dtdB

k R t B tdt

Linear LawPoint fire interpretation: Targets few or difficult to locate so each attacker locates targets at a rate proportional to the number of targets present.Area fire interpretation: Each attacker engages all targets in a certain area per unit time, while targets are dispersed over a region maintaining a constant area occupied so that a reduction in the number of targets reduces target density.

Square LawPoint fire interpretation: Targets are sufficiently numerous or the ability to locate them is sufficiently good that each attacker locates targets at a constant rate.Area fire interpretation: Each attacker engages all targets in a certain area per unit time while targets are dispersed over a region maintaining a constant density so that a reduction in the number of targets reduces the area occupied.

So What Went Wrong?Lanchester equations relate to grinding attritional warfare where large blocks of forces interact.

It’s been difficult to find historical evidence where Lanchester equations consistently and uniformly describe the complicated process of combat.

Open Systems• Workers post-Lanchester have expanded his equations into areas where they were originally not

envisioned.• Current operations – low intensity conflicts, operations other than war etc., emphasise the lack of

applicability of the Lanchester equations.• We therefore have to look for a way to mathematically model combat that not only encompasses

the ‘common sense’ of the Lanchester equations but also encompasses ‘information age’ conflict.• Moreover we need to find simulation tools that can accurately model real combat whether

armoured warfare on a major scale or low intensity conflicts and human effects such as morale and cohesion.

Where Do We Go Now?

• Realize that the mathematics of conflict is that of open systems.• Notice that other workers – outside of military operational

research – have had similar ideas:– E.g. Theodore Modis and his S-shaped growth curves that

can be mathematically described by log-log/power law behaviour (Conquering Uncertainty, 1998)

• Use the Complexity based approach first expounded by Moffat and Passman (Metamodels and Emergent Behavior in Models of Conflict, OR Simulation Study Workshop 2002).– This approach is based upon the evidence that the statistics conflict, e.g.

in Turcotte and Roberts Fractals 6, 4,1998 demonstrates power low behaviour.

– Create a ‘metamodel’ that describes the nature of conflict but also encompasses Lanchester.

Dimensional Analysis

• The whole metamodel methodology is based upon the dimensional analysis ideas expounded by G I Barenblatt (Scaling, Self-similarity and Intermediate Asymptotics, CUP 1996).

• This allows the consideration of three types of metamodel:

– One whose characteristic function tends to a non-zero finite limit and thus that can be determined by dimensional analysis.

– One whose characteristic function tends to power law asymptotics.

– One where power law asymptotic behaviour is not observed and the characteristic function has no finite limit different from zero.

A Deterministic Approach

• A dimensional analysis approach illustrated by Moffat and Passman was to regard the Lanchester equations as exponent type metamodels:

HGFR

EDCB

tRtRkdt

dB

tRtBkdt

dR

)()(

)()(

A Deterministic Approach 2• Extensive work by Hartley (Report K/DSRD-263/R1, Martin Marietta

Centre for Modelling, Simulation & Gaming, 1991) in mapping conflict databases demonstrated that for the linear relationship for exponents (D-G) =(H-E) = α-1 and (C-F) = β held with α=1.35:

)()(

)(

)(

0

022

0

220

RatioForceLnRatioHelmboldLn

B

RLn

tBB

tRRLn

The Fractal Attrition Equation

• Lauren’s combat attrition metamodel was developed from examining the MANA cellular automata combat model (See Fractals 10, 481-490, 2002).

• A number of authors have developed Lauren’s work in expressing expected losses in terms of elements that are associated with cellular automata based on two dimensional surfaces i.e. fractal dimension, average cluster sizes etc. (See Moffat et al, TTCP Report JSA TP3, July 2005).

• This work has been extended, notably by Perry to develop a Fractal Attrition Equation that has Lanchester’s Laws as limiting cases for the Equation (N Perry, DSTO-TR-1822, 2006).

The Time Series Approach

• Whilst the FAE gives excellent insight into the fractal processes of cellular automata models it does not give direct tools in predicting combat casualty results from historical data. Moffat and Passman, therefore proceeded in a slightly different way (See Moffat, Complexity Theory and Network Centric Warfare, Information Age Transformation Series, 2003).

• The ethos adopted was to build a practical research tool for predicting combat casualty rate patterns from real casualty data. Kuhn WWII data (Ground Force Casualty Patterns, Report FP703TR1, 1989) was used as being the most comprehensive and readily available.

• Success was to be determined not by exactly fitting (or over fitting) the real data but rather by observing if the predicted results were of the same form as the real data (i.e. gaining insight into the Physics of the process was the aim).

The Time Series Approach 2

• The first part of the times series (up to day 38) was used to train a number of different time series predictive methods.

• The ‘fractal’ predictive method used was based upon the assumption that circumstances remained sufficiently constant so that the power spectrum of the process was linear when plotted on a log-log scale.

• A commercially available package – the Chaos Data Analyser – produced by the American Institute of Physics was used and a number of predictive methods (included in the package) were also undertaken as a comparison.

Results 2nd Armoured Division• The plots are casualties per

1000 on the y axis and days on the x axis.

• The first plot is the actual data• The second plot is the

maximum entropy prediction• The third plot is the neural net

prediction• The final plot is the fractal

prediction• Results taken from Moffat,

Complexity Theory and Network Centric Warfare, Information Age Transformation Series, 2003.

Results 9th Armoured Division• The plots are casualties per 1000 on

the y axis and days on the x axis.• The first plot is the actual data• The second plot is the maximum

entropy prediction• The third plot is the neural net

prediction• The final plot is the fractal

prediction• Results taken from Moffat,

Complexity Theory and Network Centric Warfare, Information Age Transformation Series, 2003.

• Note apparent loss of data on predicted results is due to graphical representation scale effects.

• Note for both 2nd and 9th armoured divisions the ‘jerkiness’ of predicted results and general pattern are similar to actual results.

Conclusion• Efforts over the past decade or so have attempted to use Complexity

Science to increase our knowledge of the processes that make up combat.

• Experiments with cellular automata models such as MANA, together with mathematical techniques such as dimensional analysis, have linked historical data to fractal fingerprint behaviour.

• What still seems a long way off is the development of tools to examine, model and predict the wide range of combat and associated processes:

– Operations other than War– Low Intensity Warfare– Aggregation and synergistic effects within combat units– Unit effectiveness and organisational structures– Effects of morale and cohesiveness on combat