ions zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA ISMOR - - PDF document

ISMOR Self Organised Criticality, Manoeuvre Warfare, and Peace Support perat ions zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

Jim Moffat, CDA Maurice Passman, CDA 1

I

Self Organised Criticality zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

• The

Sand pile zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

0 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

In order to illustrate the idea of self organised criticality, consider a pile of sand which is built up by adding grains of sand from the top. Each addition causes a local change, but with no long range global effect until the pile reaches a critical size and slope. At that point, avalanches of all sizes are possible - a global emergent phenomenon. The pile has self organised itself towards this critical state. The question to answer is whether we can predict such emergent avalanche behaviour theoretically, given assumptions about the local interactions of the grains of sand. We then look at the relevance of this to providing an underpinning theory for various emergent processes in conflict, including the irruption process of manoeuvre warfare, and the movement of confrontation lines in peacekeeping operations. Application of these ideas to various physical systems by Per Bak of Brookhaven Labs, together with a number of collaborators, has led to the development of a number of theoretical models of this process, based on mathematics derived from particle physics known as ‘Reggion Field Theory’. All of these models can be shown to be related to each other through a number

• f ‘scaling constants’.

The Bak-Sneppen Evolution Model zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

From our research, it is apparent that two of these models are particularly relevant to providing an underpinning theoretical understanding of conflict. The first of these - the Bak Sneppen Evolution Model, is directly relevant to the 'irruption' process of Manoeuvre warfare (described by David Rowland of CDA from Historical Analysis) and the interaction between attrition and

• manoeuvre. The Sneppen Depinning Model is directly relevant to the

movement of confrontation lines in Peacekeeping scenarios (a subject also modelled by Gass using Fluid Dynamic equations). The basic Bak-Sneppen Evolution model works in the following way. Define a finite lattice in D dimensions. (D=2 in the picture above ). At each lattice point locate a random number f between 0 and 1. This can be thought of as a measure of fitness (or effectiveness) of that lattice point. Sample without replacement to ensure all values are different. Scan the lattice and identify the unique lattice point with minimum value off . Call this f(0) (we could redefine this so it is the point of maximum effectiveness - the maths goes through in the same way). This represents the starting point for an avalanche. At this point and each of its nearest neighbours in the lattice, replace the f values with new random numbers drawn between 0 and 1 .(We could insist that the new numbers are at least as great as the old numbers - again the effect is essentially is the same).

The Gap Equation zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

• 1

Time (iterations) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

Now scan the complete lattice again and choose the minimum point. If one of these new numbers is the new minimum point for the lattice, we have the first step of an avalanche process. Thus zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA adjacent points of the lattice form the steps

• f an avalanche process. Otherwise a new avalanche starts from a new

minimum point. At each step of an avalanche, we generate random numbers which are smaller than the original value f(0). Thus if an avalanche stops, all values across the lattice must be bigger than f(0). The minimum value f(0) then increases to a new minimum value f( 1) which is the new smallest random number for a lattice site across the whole lattice. The move from f(0) to f(

1) is shown by the

white line step change in the figure above. The red line indicates an avalanche process which occurs at each step until all values lower than the critical value f(j) have been eliminated. This process continues until a final critical point f (crit) (between 0 and 1) is

• reached. At this point avalanches of all lengths to infinity are possible. This

corresponds to the point at which the sandpile collapses through a global emergent process.

The zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

Gap Equation zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

• 2

Gap Size G(s)

nimal site value zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

I

f l zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

i

,

Time (s) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Time is measured by the number of iterations of the process (s). At any timestep j, we have a minimum value fo), and the Gap is defined by f(crit) - fG): the difference between the current minimum value fo), and the ultimate critical value f(crit). We want to determine the rate at which the Gap size G(s) tends to zero. 5

The Gap Equation zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

• zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

3 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

The rate of change of the Gap size G(s) with time s is shown here, in the continuous approximation to the discrete steps (s). This is derived by Bak and co-workers using mathematics from particle physics denoted ‘Reggion Field Theory’ . In this process, each avalanche step corresponds to the creation of a ‘particle’. The denominator on the right hand side is the product of L superscript d (the size of the Lattice) and <S> , (the average length of an avalanche which can be generated at time s).

t

Interpretation for Conflict

I zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

Lattice site = firefight at that location

I

minimal value f = l/(effectiveness of firefight)

I

advance to the critical value = self organisation of ‘irruption’ process of manoeuvre warfare zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

I time series of avalanches = casualty creation zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

In relating this to the effect of manoeuvre warfare, we consider the process of ‘irruption’ described by Rowland as a massive breakthrough related to high casualties and large rates of advance, and wish to relate this to the ideas of Ancker,Speight and Rowland, on warfare as a series of interacting ‘firefights’. A lattice site, we hypothesise, corresponds to a firefight. The value f(s) at the site corresponds to the effectiveness of the firefight. The rate of closure of the Gap size G(s) corresponds to the rate of growth of the ‘irruption’ process. A fast rate of growth relates to ideas of ‘shock’ and tempo. We can also show, in the discussion which follows, that real casualty statistics over time have the characteristics corresponding to self organised criticality - they are created by a series of ‘avalanches’.

Rate of closure of the gap is inversely proportional to avalanche size zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

1

Looking at the Gap size equation, we can see that the rate of closure of the Gap (G dot s) is inversely proportional to avalanche size zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA <S>. We can see that as we approach the critical value f(crit) the expected avalanche size tends to infinity, as the Gap size rate of change tends to zero. 8

The Signature of Self Organised Criticality zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

• a scaling relation
• ccurrenc zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

In general, it can be proved that, for such processes, the size of avalanches is related to their frequency through a power law relationship - in fact this is the signature of a Self Organised Critical Process - see picture. On a Log -Log scale, the relationship is a straight line. We shall show empirical evidence that this is the case for casualty production. 9-

I

Example: All recorded occurrences

• f wars

from 1820 to 1945. Number of fatalities versus frequency zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

I

I zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

I 1 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA IO IO IO IO IO 10

I .o

I

3 4 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 5 6

7 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 8

F - Fatdiria

pcr D B m Y

QvMel

Figure 14: Frequency o f

"deadly quarrels" (Richardson. 1960) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

with various numbers of fatalities

'

• ;
• U

This data is taken from reference: Richardson L F. (1 960) 'Statistics of Deadly Quarrels'. Chicago, Quadrangle Books. The graph is taken from Dockery JT and Woodcock AER. (1993) 'The Military Landscape' Woodhead Publishing Ltd, Cambridge, UK. The smallest value recorded is 3 16, and the two world wars account for zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

75%

• f the total fatalities.

It is clear from the Log- Log plot and the resultant straight line that the generation of casualties in conflicts zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

• f

zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

all sizes is a Self Ordered Critical process. Other examples of such Log-Log plots are given in Woodcock and Dockery (ref. as above).

SOC is a Space-Time Fractal and hence a fractal process in time

Time zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

I zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

## ##

Space

b zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

Looking again at the lattice, and how the avalanches are generated over time, for each timestep s we can plot the location of the minimum value f(s) shown by a ## on the picture. It can be shown that the set of all these points forms a space-time fractal. This has implications for how we can aggregate forces by the use of scaling and fractal methods. This is currently being written up by us as a paper for publication in the peer reviewed open literature. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA A cut in the time direction of the space time fractal produces a fractal time series, of exactly the same form as the time series for casualty production. This insight allows us to develop time series methods for predicting casualty production or other fractal properties of the process, over time, and this is discussed next.

U S zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

W 2

data on casualties suffered zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

I

2nd Armoured Division data

20

15

10 5

'

1 5 9 13 17 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

21 25 29 33 37 41 45 49 53 57

I zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

This is the actual empirical data. The y axis is casualties per thousand, and the zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA x

axis is in days. The data are drawn from an

extensive body of historical data amassed by zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA G Khun. The first part of this time series (up to day 38) was used to train a number of different time series prediction methods, and these have been compared with the predictions for the days 39 onward based on the assumption of a self

• rdered critical process.

Detailed assumptions for the various altemative fitting methods are obtainable from Maurice Passman. We are assuming that the circumstances remain sufficiently constant that we can fit a single SOC process to this data (this corresponds to a power spectrum which is linear when plotted on a Log-Log scale).

When this is not the case, we have found that the power spectrum breaks up

into a number of linear segments, each of which is SOC. 12

Prediction based on a neural net

2nd Armoured Division Neural Net Prediction

20 15 10 5 0 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

I

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57

A zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA neural net zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

gives a flat prediction. 13

Prediction based zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

• n

Power Spectrum

2nd Armoured Division zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA I Power Spectrum Prediction

20 15 10

5

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

A power spectrum approach also zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

gives an essentially flat prediction 14

Prediction based on non linear methods

2nd zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Armoured Division Non-Linear

Prediction zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

20 10 - 0 - 1 5 9 13 17 21 25 29 33 37 41 45 49 53 57

B) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

(s

KG51

A zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA prediction based on non linear potential methods gets better 15

Prediction based on SOC

2nd Armoured Division SOC Prediction zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

I zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

L U

15 10 5 1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

The prediction based on Self Organised Criticality predicts two peaks in about the right places. We have repeated this on a number of data sets with the same order of result.

The Sneppen Depinning Model zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

The second class of SOC model developed by Bak and his collaborators which we have selected as having potential for conflict analysis is the Sneppen Depinning Model. This model has been successfully applied to physical systems where there is a boundary changing within the system (such as percolation into a substrate). We wish to apply it to a situation previously studied by Gass using a fluid dynamics approach. Gass took the view that the movement of a boundary (in

• range on the picture) between two ethnic groups or nationalities in a

peacekeeping scenario could be modelled as caused by a difference in pressure, and this pressure could be related to a small number of driving

• parameters. This would cause the boundary to move smoothly since it is

modelled using first order differential equations. Anecdotal evidence suggests that such boundaries tend to move in jerky steps, as would be predicted by a model based on SOC, hence the attempt to apply the Sneppen Depinning Model. By exploiting the fact that all of these models can be related to each other by scaling relationships, we can use the same ideas and maths from the Bak- Sneppen Evolution model, with some slight changes. 17 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

I

Predicted Movement of Confrontation Line zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

This picture shows what the theory would predict zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

• sudden jerky change from

the solid to the dotted boundary. This is caused by an SOC avalanche in the same way as for the evolution model. All along the boundary on each side we define a set of ‘depinning forces’. This is a small local force defined at each lattice point adjacent to the boundary, on each side of the boundary. At each timestep we choose the smallest force and change it randomly. The neighbouring forces are then also changed by amounts which reflect some continuity assumptions about the boundary force. Again using the Reggion Field Theory approach described for the evolution model, we can describe theoretically the movement of the system through a series of minimal values to a critical point at which global boundary changes of all sizes are possible. We emphasise that it can be shown that the mathematics for this case corresponds to a scaled version of the previous model. (i.e. the equations are factored by a number of constants of known value). 18

Gass Prediction of Bosnia Confrontation

Gass Bosnia Conflict Prediction zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA This figure shows the movement of the confrontation line between the Bosnian Serb and Muslim populations predicted by Gass. Time is marked in years on the x axis, and incursion of the confrontation line into Muslim ‘territory’ is shown by movement down the page. Prediction is from time period 15

• nwards. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

As can be seen, his approach predicts smooth transition.

.

Sneppen Depinning Model prediction

Bak-Sneppen Depinning Bosnia P red ictio n zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

Using the same time series for the first 15 time periods, the prediction of the Sneppen Depinning Model shows a jerky transition. Further work is required to show that this is a realistic assessment of what might happen. If successfully validated, this approach could be used to assess the stability of a region in confrontation. 20

.

Conclusions zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

• 1

I

Self Organised criticality zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

is shown to be relevant to

the understanding of both Manoeuvre warfare, and the movement of confrontation lines in peacekeeping

• perations.

between local ‘agent’ interactions (depinning forces

• r local effectiveness) and global emergent behaviour

(the ‘avalanche’ process)

I

These models allow an explicit link to be made zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Conclusion are shown on this and the following slide.

Conclusions zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

• 2 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

I SOC predicts that the system evolution is a fractal in

space-time

I This fractal behaviour in space and time has

implications for force aggregation and disaggregation

I The time slice of this fractal gives insight into the

evolution of casualties as a function of time.

• At any point, the forward time slice is a fractal time series process zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

This completes the briefing of the research completed so far. It is anticipated that this work will lay a firm foundation for further theoretical development of the relation between local effects and global behaviour. This will help to develop a theoretical understanding of the effect of more loosely coupled future Command and Control structures, such as those based on network centric warfare concepts. 22