Model reduction for agent-based social simulation: Coarse-graining a civil violence model

Agent-based modeling (ABM) constitutes a powerful computational tool for the exploration of phenomena involving emergent dynamic behavior in the social sciences. This paper demonstrates a computer-assisted approach that bridges the significant gap between the single-agent microscopic level and the macroscopic (coarse-grained population) level, where fundamental questions must be rationally answered and policies guiding the emergent dynamics devised. Our approach will be illustrated through an agent-based model of civil violence. This spatiotemporally varying ABM incorporates interactions between a heterogeneous population of citizens [active (insurgent), inactive, or jailed] and a population of police officers. Detailed simulations exhibit an equilibrium punctuated by periods of social upheavals. We show how to effectively reduce the agent-based dynamics to a stochastic model with only two coarse-grained degrees of freedom: the number of jailed citizens and the number of active ones. The coarse-grained model captures the ABM dynamics while drastically reducing the computation time (by a factor of approximately 20).

Figure 1

Arrest probability from Eqs. (1) and (2) (see the text).

Figure 2

Flow chart for the model simulation procedure within a one-day period.

Figure 3

(a) Time history of the numbers of jailed and active citizens extracted from the full agent-based simulation and (b) their two-dimensional phase plane projection.

Figure 4

(a) Partition of the $H$-$R$ domain into three regions and (b) allocation of $H,R$ pairs in each region to each citizen type (active, inactive, and jailed).

Figure 5

Simulation domain in the $X_1$-$X_2$ phase plane that contains all trajectories. The central white region stands for the simulation domain. The surrounding green region is not accessible by $X_1$ and $X_2$ pairs.

Figure 6

Snapshots and heterogeneity measures of police officer position patterns between time 0 and 200 in a single realization of simulation. (a) Time history for populations of jailed and active citizens. Green vertical bars stand for the time steps at which the police officer position patterns shown have been obtained. (b)–(k) Snapshots of police officer position patterns in a $10\times 10$ grid of uniform cells. (l) Heterogeneity measure $\sigma$ of the police officer position patterns. Drastic police officer movement can be seen between times 4 and 20. More homogeneous and stationary position patterns subsequently arise between times 20 and 200.

Figure 7

Two snapshots of active citizen position patterns at times 1 and 3 in the simulation of Fig. 6. Red, cyan, and black dots represent active citizens, inactive citizens, and police officers, respectively. The active citizen positions can be approximately enclosed by a circle with the radius $r$ calculated from Eq. (8).

Figure 8

Distribution of residual jail terms for jailed citizens from regions 2 and 3 in Fig. 4a: (a) distribution for region 2 jailed citizens at time 50, (b) distribution for region 2 jailed citizens at time 100, (c) distribution for region 3 jailed citizens at time 50, and (d) distribution for region 3 jailed citizens at time 100.

Figure 9

Stochastic path of jailed and active citizens obtained from the simulation of the effective reduced SDE (3) subject to the boundary conditions (9).

Figure 10

Comparison of the outburst time distributions from (a) the effective reduced SDE and (b) the original ABM.

Figure 11

Comparison of the PDF of the numbers of jailed and active citizens between the true (ABM) and the reduced (SDE) models. (a) Three-dimensional (3D) surface of ${\text{log}}_{10}$ (PDF) simulated using the SDE, (b) 3D surface of ${\text{log}}_{10}$ (PDF) simulated using the ABM, (c) 2D projected image of ${\text{log}}_{10}$ (PDF) simulated using the SDE [the region in the red box is zoomed in and shown in (e)], (d) 2D projected image of ${\text{log}}_{10}$ (PDF) simulated using the ABM [the region in the red box is zoomed in and shown in (f)], (e) zoomed-in image of the PDF simulated using the SDE in the subdomain $[150,300]\times [-5,50]$, and (f) zoomed-in image of the PDF simulated using the ABM in the same subdomain.

Figure 12

(a) Finite-element mesh for solving Eq. (12) and (b) resulting stationary PDF of the numbers of jailed and active citizens.

Figure 13

(a) Finite-element mesh for solving Eq. (17) and (b) mean exit times computed as a solution of Eq. (17).