Scaling theory of armed-conflict avalanches

Armed conflict data display features consistent with scaling and universal dynamics in both social and physical properties like fatalities and geographic extent. We propose a randomly branching armed conflict model to relate the multiple properties to one another. The model incorporates a fractal lattice on which conflict spreads, uniform dynamics driving conflict growth, and regional virulence that modulates local conflict intensity. The quantitative constraints on scaling and universal dynamics we use to develop our minimal model serve more generally as a set of constraints for other models for armed conflict dynamics. We show how this approach akin to thermodynamics imparts mechanistic intuition and unifies multiple conflict properties, giving insight into causation, prediction, and intervention timing.

Figure 1

Cartoon of RBAC model. A growing conflict avalanche spreads out in space to new conflict sites in a fractal manner, generating new events at an ever slower rate. As a result, conflict sites near the core tend to have more cumulative events (thick lines) than peripheral sites (thin lines). The rate at which events are generated at the core is higher than that at the periphery, implied by a site growth exponent exceeding peripheral suppression exponent, ${\gamma }_r>{\theta }_r$.

Figure 2

Number of conflict reports averaged over all conflict avalanches per Voronoi region of Africa. Radii of circles are proportional to number of conflict events. Centers of regions are separated on average by separation length $b=70\mathrm{km}$.

Figure 3

Random variation of “Nice Trees” grown for $T=8000$ time steps with branching number $Q=3$ and varying branch length ratios $B$ [26]. For battles, $B=6.6$. There is one conflict site per unit length.

Figure 4

Dynamical scaling collapse for diameter, extent, fatality, and report trajectories under rescaling of separation time $a$. Inset: Scaling collapse for avalanches of different duration. The particularly poor exception is diameter growth $\overline{l\left(t\right)/L}$. Data points are few for the longest avalanches, but we find long avalanches saturate the maximum diameter suddenly. Inspecting these avalanches in detail, we find they tend to hit hard boundaries like coastlines and national borders they cannot surpass. In the case of the Tunisian and Libyan revolutions, the aggregation of which is included in the shown average for the longest conflicts, the population is largely confined to the coastline. This suggests for conflict avalanches commensurate with geographic or political boundaries, it is essential to account for such boundaries delimiting their maximum extent.

Figure 5

Scaling form predicted in Eq. (12) aligns qualitatively with the data given measured ${\gamma }_r=0.74$. We show bounds on ${\theta }_r$ corresponding to 90% bootstrapped confidence intervals as ${\theta }_r^{-}$ and ${\theta }_r^{+}$ and RBAC simulation (orange). Since for each solution of ${\theta }_r$ there are corresponding fit parameters from Eq. (B1), the bounding lines for ${\theta }_r^{+}$ and ${\theta }_r^{-}$ indicate variability in the shape of the curve but not vertical displacement.

Figure 6

Distributions of average virulence per conflict avalanche $V_r$ and $V_f$ display power-law tails whose measured exponents satisfy self-consistent equations derived from the scaling hypothesis ($p>0.8$ compared to standard significance threshold $p=0.1$) [2].

Figure 7

Dynamical scaling and distributions of conflict avalanche scaling variables generated from RBAC compared with data. Left column: Model simulations (orange) closely mimic calculated exponent relations in Table 1 (dashed black lines) and are similar to scaling in data (blue). Measured dynamical scaling functions are shown after having removed the nonzero intercept at $t=0$ averaged over conflict avalanches with duration $T\ge 4$ days. For $n\left(t\right)$, we also require $N>1$ and for $f\left(t\right)$ that $F>2$ fatalities. Right column: Distributions of scaling variables whose exponents listed in Table 2 align closely with data. Distributions for both data and RBAC are shown above lower cutoffs determined by a standard fitting procedure (see Ref. [2]) and their scales matched such that the lower cutoffs coincide. Shaded regions are spanned by 90% confidence intervals from bootstrapped sampling. See Appendix of Ref. [8] for further details about fitting.

Figure 8

Overview of RBAC model combining a dynamical scaling model with a scale-free distribution of conflict report virulence to generate conflict simulations. Geographic spread of conflict sites involves duration $t$, diameter $l$, and extent $n$, all related by a single exponent. At each conflict site, reports grow in a uniform way, depending only on growth exponent ${\gamma }_r$, peripheral suppression exponent ${\theta }_r$, and report virulence $V_r$. To get total report growth $r$, we sum over the geographic extent of the conflict avalanche. Thus, each aforementioned component contributes an additional exponent to $r$ as indicated by the incoming arrows. In contrast to the other scaling variables (black letters), virulence $V_r$ (red) is quenched, or fixed during the conflict avalanche. The variable $t_0$ (gray) indicates when a site first became infected during the course of a conflict avalanche. A similar descriptions holds for fatality growth $f\left(t\right)$. To obtain the distributions of conflict scaling variables, we further assume a power-law form for report virulence distribution $P\left(V_r\right)$ that modulates conflict density per site over the lifetime of the entire conflict avalanche (gray box).

Figure 9

Cumulative distribution function (CDF) of exponents ${\gamma }_r$ and ${\gamma }_f$ estimated from regression to conflict site growth curves. Given this wide distribution, we take our best estimate of the exponent to be the median with confidence intervals given by the 5th and 95th percentiles as given in Table 2.

Figure 10

Scaling form predicted in Eq. (12) given ${\gamma }_f=0.56$. We show bounds on ${\theta }_f$ corresponding to 90% bootstrapped confidence intervals as ${\theta }_f^{-}$ and ${\theta }_f^{+}$. We compare with simulation (orange).