Social unrest spatial contagion model.

The model is defined on a square grid of sites, which represents the division of a country into urban clusters (see the section Parameter Estimation for details). Each site of the grid can be in one of three different states: empty (“white”), susceptible to social unrest (“red”), and involved in social unrest (“crowd”). Sites are one grid step apart if they are directly adjacent to each other in either the horizontal or vertical direction, or are connected through weak links (e.g., site E4 is connected to site F2). The weak links are formed by associating with each site, with probability , a single link to a site selected uniformly at random from the grid. The grid is updated synchronously according to the following rules: at each time step, empty sites become susceptible with probability , and susceptible sites become involved in social unrest with probability . Unrest contagion occurs on a short time scale as follows. If a site is involved in social unrest (e.g., sites B2 and E4), the unrest activity spreads with probability to susceptible sites that are one grid step apart (e.g., sites C2, B3, or D5), which in turn can lead (with probability ) to further instabilities of susceptible sites that are two grid steps apart, three grid steps apart, and so on. Each of the sites involved in social unrest during a time step contributes to the size of the unrest contagion.