Modeling Social and Geopolitical Disasters as Extreme Events: A Case Study Considering the Complex Dynamics of International Armed Conflicts

Abstract

Just as various sorts of extreme climatic events are identified in Earth’s atmosphere, so are some types of extreme events in our sociosphere. A geopolitical conflict that can result in a social disaster is an example. In this chapter, the turbulent-like dynamics of international armed conflicts are treated within the scope of complex multi-agent systems explicitly considering the properties of multiplicative non-homogeneous cascade where endogeny and exogeny are key points in the mathematical model of the phenomenon. As a main result, this study introduces a cellular automata prototype that allows characterizing regimes of extreme armed conflicts such as the 9∕11 terrorist attacks and the great world wars.

Appendices

Appendix 1: The UCDP Data Base

The Uppsala Conflict Data Program (UCDP) database [57] provides one of the most accurate and extensive information on armed conflicts including attributes like conflict intensity based on total number of battle-related deaths; number of conflicts (see Fig. 8); conflict type; details of warring party including geopolitical information; period with specific start and end date, etc. This database is updated annually and considered well-used data-sources on global armed conflicts. Its definition of armed conflict² is becoming a standard in how conflicts are systematically defined and studied. Conflict with minimum of D battle-related deaths per period τ and in which one of the warring party is government of a state is recorded as an Interstate Armed Conflict (IAC) [17]. UCDP DataBase (UDB) categorizes IAC in different intensity levels based on the total battle-related causalities:

• Not active: D < 25 per year.

• Minor: D ≥ 25 per year but fewer than in the extreme event period.

• Intermediate: 25 ≪ D < 1000 total accumulated of at least 1000 deaths, but fewer than 1000 in any given year.

• War: D ≥ 1000 per year.

Appendix 2: Power Laws for Endogenous and Exogenous Time Series

The systemic difference between endogeny and exogeny in abrupt events has been interpreted as a scaling process by Sornette–Deschtres–Gilbert–Ageon (SDGA) [55], where internal perturbations give rise to endogenous extreme events (XE _endo) which is characterized, as shown in Fig. 9a, by smoother average continuous fluctuations that increases slowly and after reaching its highest peak and gradually reduces by itself. Differently, an exogenous extreme event (XE _exo) results from a preponderant external perturbation and can be characterized by a sudden peak followed by unexpected rapid drop in the fluctuations (Fig. 9b).

Fig. 9

Typical time series for (a) XE _endo and (b) XE _exo, according to the SDGA scaling approach

The SDGA model is based on the book sales rank. While the book’s selling rate, which has a XE _endo pattern, only relies on the advertising provided by the common sales system (basically, the publisher’s advertising and, especially, the cascade of information between the readers and likely readers), the sales rate of the book with XE _exo pattern counted on an unusual systemic outsider high cost advertisement via a famous newspaper or TV broadcast interview.

According to the SDGA a time series can be modeled based on social epidemic process where in the beginning, first (mother) agent notices the book in advertisement or news or by chance and initiates buy at time t _i. Subsequent (daughter) generations of agents are build at different time t resulting in an epidemic that can be modeled by a memory kernel ϕ(t − t _i). The net sale is the sum of 1∕f noise processes following a power law distribution that accounts for XE _endo, and impulsive distribution associated with XE _exo. The time series can be described by a conditional Poisson branching process given by

$$\displaystyle \begin{aligned} \lambda(t) = R(t) + \sum_{i/_1 \leq t} \mu_i \phi(t-t_i) {} \end{aligned} $$

(6)

where μ _i is number of potential agents influenced by the agent i who bought earlier at time t _i. R(t) is the rate of sales initiated spontaneously without influence from other previous agents.

For our generic complex systems scenario, the key idea in the SDGA approach is the invariance of the epidemic model but as a non-homogeneous network of potential daughter generations which can be considered through different values of branching ratio. The ensemble average yields a branching ratio, n, that signifies the average number of conflicts triggered by any mother Agent within her contact network and rely upon the network topology and impact of the systemic dissipative behavior. Authors considered the sub-critical regime n < 1 in order to ensure stationarity which accounts for efficient coarse-grained nature of the complex nonlinear dynamics. The exogenous response function is obtained from Laplace transform of the Green function K(t) of the ensemble average.

According to Sornette [55] a bare propagator as ϕ(t − t _i) ∼ 1∕t ^(1+θ) with 0 < θ < 1 corresponds to long-range memory process which provides information on the conflicts propagation:

$$\displaystyle \begin{aligned} C_{exo}(t) \equiv K(t) \sim 1/(t-t_c)^{1-\theta}. {} \end{aligned} $$

(7)

It provides information about the average number of agents influenced by one agent through any possible direct descent or ancestry. And thus average number of conflicts triggered by one agent can be given as:

$$\displaystyle \begin{aligned} \int\limits_{0}^{\infty} K(t)dt = n/(1-n). {} \end{aligned} $$

(8)

Continuous stochastic time series with spontaneous peaks indicates the lack of exogenous shock. Such series can be interpreted as an interaction between external factors over small-scale and enlarged effect of widespread cascade of social influences. This mechanism can explain peak in endogenous time series. Considering results for stochastic processes with finite variance and covariance for average growth of processes prior and later to the peak and applying to λ(t) defined in Eq. (5) one get:

$$\displaystyle \begin{aligned} C_{endo}(t) \sim 1/|t-t_c|{}^{1-2\theta}. {} \end{aligned} $$

(9)

Equations (7) and (9) agree with the prediction that XE _exo should occur faster with exponent 1 − θ compared to XE _endo with exponent 1 − 2θ. Therefore, after characterizing the power laws 1∕(t − t _c)^β with highest correlation coefficient [55], the scaling interpretation presents two different universality classes characterizing XE _endo with β as 1 − 2θ ≈ 0.4 and XE _exo with β as 1 − θ ≈ 0.7. These are compatible values with Eqs. (7) and (9) with the choice of θ = 0.3 ± 0.1.