ions zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA ISMOR - PDF document

ions zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA ISMOR Self Organised Criticality, Manoeuvre Warfare, and Peace Support 0 perat Jim Moffat, CDA Maurice Passman, CDA I 1

Self Organised Criticality zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA pile zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 0 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA - The Sand 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 of ‘scaling constants’.

The Bak-Sneppen Evolution Model zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA ~~ 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 step of an avalanche process. Thus 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 adjacent points of the lattice form the steps of 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.

f l zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Gap Equation zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA The zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA nimal site value zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA - 2 Gap Size G(s) I Time (s) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA i , 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

- zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 3 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA The Gap Equation 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).

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I zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Interpretation for Conflict ‘irruption’ process of manoeuvre warfare zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA I time series of avalanches = casualty creation zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Lattice site = firefight at that location minimal value f = l/(effectiveness of firefight) I advance to the critical value = self organisation of I 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’.

to avalanche size zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Rate of closure of the gap is inversely proportional 1 Gap (G dot s) is inversely proportional to avalanche size zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Looking at the Gap size equation, we can see that the rate of closure of the <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

Criticality zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA The Signature of Self Organised occurrenc zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA - a scaling relation 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. I 9-

from 1820 to 1945. Number of fatalities versus frequency zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Example: All recorded occurrences of wars I zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 4 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 7 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA "deadly quarrels" (Richardson. 1960) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA I I I I .o 3 F - 5 6 8 IO IO IO IO IO 1 0 10 QvMel Fatdiria pcr D B m Y Figure 14: Frequency o f with various numbers of fatalities ' - ; - - U C ' L i U 16, and the two world wars account for zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA This data is taken from reference: Richardson L F. (1 960) 'Statistics of Deadly Quarrels'. Chicago, Quadrangle Books. The graph is taken from generation of casualties in conflicts zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Dockery JT and Woodcock AER. (1993) 'The Military Landscape' Woodhead Publishing Ltd, Cambridge, UK. The smallest value recorded is 3 75% of the total fatalities. It is clear from the Log- Log plot and the resultant straight line that the o f all sizes is a Self Ordered Critical process. Other examples of such Log-Log plots are given in Woodcock and Dockery (ref. as above).