Modeling Human Conflict and Terrorism Across Geographic Scales

Abstract

We discuss the nature and origin of patterns emerging in the timing and severity of violent events within human conflicts and global terrorism. The underlying data are drawn from across geographical scales from municipalities up to entire continents, with great diversity in terms of terrain, underlying cause, socioeconomic and political setting, cultural and technological background. The data sources are equally diverse, being drawn from all available sources including non-government organizations, academia, and official government records. Despite these implicit heterogeneities and the seemingly chaotic nature of human violence, the patterns that we report are remarkably robust. We argue that this ubiquity of a particular pattern reflects a common way in which groups of humans fight each other, particularly in the asymmetric setting in which one weaker but ostensibly more adaptable opponent confronts a stronger but potentially more sluggish opponent. We propose a minimal generative model which reproduces these common statistical patterns while offering a physical explanation as to their cause. We also explain why our mechanistic approach, which is inspired by non-equilibrium statistical physics, fits naturally within the framework of recent ideas within the social science literature concerning analytical sociology, as well as setting our results in the wider context of real-world and cyber-based collective violence and illicit activity.

Appendix

Here we consider the basic version of our model, stripped down to a simple form with no decision-making, and only one population—the Red insurgency. Instead of having cells fragment when interacting with Blue, or when sensing imminent danger, we simply assign a probability for them to fragment. The resulting model yields an exponentially cutoff 2.5-exponent power-law for the distribution of cell sizes. We note that generalizations of this model have appeared in the literature—in particular, [35] contains a number of relevant generalizations, including a variable number of agents in time N(t). A later paper [61] reached similar conclusions to our earlier publication [35] concerning the remarkable robustness of the 2.5 exponent to variations in the model mechanisms. Analysis of a simple version of this model was completed earlier by d’Hulst and Rodgers [8], and real-world applications have focused on financial markets—however the derivation below features general values ν _frag and ν _coal.

At each timestep, the internal coherence of a Red population of N entities (which we refer to as an ‘agents’ to acknowledge application to human and/or cyber systems) comprises a heterogenous soup of cells. Within each cell, the component entities have a strong intra-cell coherence. Between cells, the inter-cell coherence is weak. An agent i is then picked at random—or equivalently, a cell is randomly selected with probability proportional to size. Let s _i be the size of the cell to which this agent belongs. With probability ν _frag, the coherence of a given cell fragments completely into s _i cells of size one. If it doesn’t fragment, a second cell is randomly selected with probability again proportional to size—or equivalently, another agent j is picked at random. With probability ν _coal, the two cells then coalesce (or develop a common ‘coherence’ in terms of their thinking or activities). As discussed in the main text, Kenney provides a wealth of case-study support for thinking of an insurgency as a loose soup of fragile cells [6], as do Gambetta [4] and Robb [5].

The Master Equations are as follows: The equation for the number of cells (i.e., clusters) of strength (i.e., size) s for s ≥ 2 and s = 1 are, respectively:

$$\displaystyle\begin{array}{rcl} \frac{\partial n_{s}} {\partial t} & =& \frac{\nu _{\mathrm{coal}}} {N^{2}} \sum _{k=1}^{s-1}kn_{ k}(s - k)n_{s-k} -\frac{\nu _{\mathrm{frag}}sn_{s}} {N} -\frac{2\nu _{\mathrm{coal}}sn_{s}} {N^{2}} \sum _{k=1}^{\infty }kn_{ k}\,{}\end{array}$$

(11.1)

$$\displaystyle\begin{array}{rcl} \frac{\partial n_{1}} {\partial t} & =& \frac{\nu _{\mathrm{frag}}} {N} \sum _{k=2}^{\infty }k^{2}n_{ k} -\frac{2\nu _{\mathrm{coal}}n_{1}} {N^{2}} \sum _{k=1}^{\infty }kn_{ k}.\ {}\end{array}$$

(11.2)

Here ν _coal and ν _frag are the probabilities per timestep (i.e., rates) of coalescence of two cells, or fragmentation of a cell, respectively. To simplify the limits of the sums, we extend the upper limit to infinity, which is a good approximation for large N. Terms on the right-hand side of Eq. (11.1) represent all the ways in which n _s can change. In the steady state:

$$\displaystyle\begin{array}{rcl} sn_{s} =& \dfrac{\nu _{\mathrm{coal}}} {(\nu _{\mathrm{frag}} + 2\nu _{\mathrm{coal}})N}\sum _{k=1}^{s-1}kn_{ k}(s - k)n_{s-k}\,\quad s \geq 2\,&{}\end{array}$$

(11.3)

$$\displaystyle\begin{array}{rcl} n_{1}& =& \frac{\nu _{\mathrm{frag}}} {2\nu _{\mathrm{coal}}}\sum _{k=2}^{\infty }k^{2}n_{ k}\ .{}\end{array}$$

(11.4)

Consider

$$\displaystyle{ G[y] =\sum _{ k=0}^{\infty }kn_{ k}y^{k} = n_{ 1}y +\sum _{ k=2}^{\infty }kn_{ k}y^{k} \equiv n_{ 1}y + g[y]\, }$$

(11.5)

where y is a parameter and g[y] governs the cell size distribution n _k for k ≥ 2. Multiplying Eq. (11.3) by y ^s and then summing over s from 2 to $\infty$, yields:

$$\displaystyle{ g[y] = \frac{\nu _{\mathrm{coal}}} {(\nu _{\mathrm{frag}} + 2\nu _{\mathrm{coal}})N}G[y]^{2}\, }$$

(11.6)

i.e.

$$\displaystyle{ g[y]^{2} -\left (\frac{\nu _{\mathrm{frag}} - 2\nu _{\mathrm{coal}}} {\nu _{\mathrm{coal}}} N - 2n_{1}y\right )g[y] + n_{1}^{2}y^{2} = 0. }$$

(11.7)

From Eq. (11.5), $g[1] = G[1] - n_{1}$. Substituting this into Eq. (11.7) and setting y = 1, we solve for g[1]

$$\displaystyle{ g[1] = \frac{\nu _{\mathrm{coal}}} {\nu _{\mathrm{frag}} + 2\nu _{\mathrm{coal}}}N. }$$

(11.8)

Hence

$$\displaystyle{ n_{1} = N - g[1] = \frac{\nu _{\mathrm{frag}} +\nu _{\mathrm{coal}}} {\nu _{\mathrm{frag}} + 2\nu _{\mathrm{coal}}}N. }$$

(11.9)

Substituting this into Eq. (11.7) yields

$$\displaystyle{ g[y]^{2} -\left (\frac{\nu _{\mathrm{frag}} + 2\nu _{\mathrm{coal}}} {\nu _{\mathrm{coal}}} N -\frac{2N(\nu _{\mathrm{frag}} +\nu _{\mathrm{coal}})} {\nu _{\mathrm{frag}} + 2\nu _{\mathrm{coal}}} y\right )g[y] + \frac{(N(\nu _{\mathrm{frag}} +\nu _{\mathrm{coal}}))^{2}} {(\nu _{\mathrm{frag}} + 2\nu _{\mathrm{coal}})^{2}} y^{2} = 0. }$$

(11.10)

We then solve this quadratic for g[y]

$$\displaystyle\begin{array}{rcl} g[y] = \frac{(\nu _{\mathrm{frag}} + 2\nu _{\mathrm{coal}})N} {4\nu _{\mathrm{coal}}} \left (2 -\frac{4(\nu _{\mathrm{frag}} +\nu _{\mathrm{coal}})\nu _{\mathrm{coal}}} {(\nu _{\mathrm{frag}} + 2\nu _{\mathrm{coal}})^{2}} y - 2\sqrt{1 - \frac{4(\nu _{\mathrm{frag } } +\nu _{\mathrm{coal } } )\nu _{\mathrm{coal } } } {(\nu _{\mathrm{frag}} + 2\nu _{\mathrm{frag}})^{2}} y}\right ),& & \\ & &{}\end{array}$$

(11.11)

which can be easily expanded

$$\displaystyle{ g[y] = \frac{(\nu _{\mathrm{frag}} + 2\nu _{\mathrm{coal}})N} {2\nu _{\mathrm{coal}}} \sum _{k=2}^{\infty }\frac{(2k - 3)!!} {(2k)!!} \left (\frac{4(\nu _{\mathrm{frag}} +\nu _{\mathrm{coal}})\nu _{\mathrm{coal}}} {(\nu _{\mathrm{frag}} + 2\nu _{\mathrm{coal}})^{2}} y\right )^{k}. }$$

(11.12)

Comparing with the definition of g[y] in Eq. (11.5) shows that

$$\displaystyle{ n_{s} = \frac{\nu _{\mathrm{frag}} + 2\nu _{\mathrm{coal}}} {2\nu _{\mathrm{coal}}} \frac{(2s - 3)!!} {s(2s)!!} \left (\frac{4(\nu _{\mathrm{frag}} +\nu _{\mathrm{coal}})\nu _{\mathrm{coal}}} {(\nu _{\mathrm{frag}} + 2\nu _{\mathrm{coal}})^{2}} \right )^{s}N. }$$

(11.13)

We now employ Stirling’s series

$$\displaystyle{ ln[s!] = \frac{1} {2}ln[2\pi ] + \left (s + \frac{1} {2}\right )ln[s] - s + \frac{1} {12s} -\ldots . }$$

(11.14)

Hence for s ≥ 2:

$$\displaystyle{ n_{s} \approx \left (\frac{(\nu _{\mathrm{frag}} + 2\nu _{\mathrm{coal}})e^{2}} {2^{3/2}\sqrt{2\pi }\nu _{\mathrm{coal}}} \right )\left (\frac{4(\nu _{\mathrm{frag}} +\nu _{\mathrm{coal}})\nu _{\mathrm{coal}}} {(\nu _{\mathrm{frag}} + 2\nu _{\mathrm{coal}})^{2}} \right )^{s}\frac{(s - 1)^{2s-3/2}} {s^{2s+1}} N\, }$$

(11.15)

which implies that

$$\displaystyle{ n_{s} \sim \left (\frac{\nu _{\mathrm{coal}}^{s-1}(\nu _{\mathrm{frag}} +\nu _{\mathrm{coal}})^{s}} {(\nu _{\mathrm{frag}} + 2\nu _{\mathrm{coal}})^{2s-1}} \right )s^{-5/2}\ . }$$

(11.16)

In the limit s ≫ 1, this is formally equivalent to saying that

$$\displaystyle{ n_{s} \sim \mathrm{exp}(-s/s_{0})s^{-5/2} }$$

(11.17)

where

$$\displaystyle{ s_{0} = -\left [\mathrm{ln}\left (\frac{4(\nu _{\mathrm{frag}} +\nu _{\mathrm{coal}})\nu _{\mathrm{coal}}} {(\nu _{\mathrm{frag}} + 2\nu _{\mathrm{coal}})^{2}} \right )\right ]^{-1}\ \ }$$

(11.18)

characterizes the exponential cut-off which appears at very high s. For large cell sizes (i.e., large s such that $s \sim O(N)$) the power law behavior is masked by the exponential function. The equilibrium state for the distribution of cell sizes can therefore be considered a power-law with exponent $\alpha \sim 5/2 = 2.5$, together with an exponential cut-off. In the human context, the fact that the interactions are effectively distance-independent as far as Eq. (11.1) is concerned, captures the fact that we wish to model systems where messages can be transmitted over arbitrary distances (e.g., modern human communications). A justification for choosing a cell with a probability which is proportional to its size, is as follows: a cell with more members has more chances of initiating an event. It will also be more likely to find members of another cell more frequently, and hence be able to synchronize with them—thereby synchronizing the two cells. It is well documented that cells of living objects (e.g., animals, people) may suddenly scatter in all directions (i.e., complete fragmentation) when its members sense danger, simply out of fear or in order to confuse a predator [62]. This model also offers an explanation for Richardson’s finding [17] that the distribution of approximately 10³ gangs in Chicago, and in Manchoukuo in 1935, separately followed a truncated power-law with α ≈ 2. 3.