Coupled catastrophes: sudden shifts cascade and hop among interdependent systems

An important challenge in several disciplines is to understand how sudden changes can propagate among coupled systems. Examples include the synchronization of business cycles, population collapse in patchy ecosystems, markets shifting to a new technology platform, collapses in prices and in confidence in financial markets, and protests erupting in multiple countries. A number of mathematical models of these phenomena have multiple equilibria separated by saddle-node bifurcations. We study this behaviour in its normal form as fast–slow ordinary differential equations. In our model, a system consists of multiple subsystems, such as countries in the global economy or patches of an ecosystem. Each subsystem is described by a scalar quantity, such as economic output or population, that undergoes sudden changes via saddle-node bifurcations. The subsystems are coupled via their scalar quantity (e.g. trade couples economic output; diffusion couples populations); that coupling moves the locations of their bifurcations. The model demonstrates two ways in which sudden changes can propagate: they can cascade (one causing the next), or they can hop over subsystems. The latter is absent from classic models of cascades. For an application, we study the Arab Spring protests. After connecting the model to sociological theories that have bistability, we use socioeconomic data to estimate relative proximities to tipping points and Facebook data to estimate couplings among countries. We find that although protests tend to spread locally, they also seem to ‘hop' over countries, like in the stylized model; this result highlights a new class of temporal motifs in longitudinal network datasets.

If the coupling strength σ is increased, then, as Fig. SM-1 illustrates, the S-shaped curves in Fig. 2(b) are stretched vertically, which increases the length of the synchronizing window. In fact, one can calculate the synchronizing window S for System (2); the result is which is illustrated in Fig. SM-1(c). The implication of this expression is that strengthening the coupling makes synchronized regime shifts more likely, in the sense that more paths in parameter space lead to synchronized regime shifts.

SM-2 Other simple couplings
The results for coupling functions C Y (x, y) = σx and C X (y, x) = 0 with σ < 0 can be obtained by reflecting the S-shaped curves b break (σx * ) and b sustain (σx * ) about the lines b = b break (0) and b = b sustain (0), respectively. In this case, the master subsystem facilitates (respectively, impedes) the slave subsystem's sudden shift to its upper branch of equilibria when x * is on its lower (respectively, upper) branch of equilibria. The effects of coupling functions C Y (x, y) = σ|x| can be obtained similarly. They are illustrated in Figure SM

SM-3 Facebook subgraph induced by all countries with protests
In Fig. 4   December 2012, respectively). We also excluded Mauritania because its unemployment was so large (31.1%) that including it in Fig. 4 would obscure the rest of the data.

SM-4 Properties of countries in the hop motifs
Figures SM-4 and SM-5 show properties of the countries in these different roles X, Y, Z in the hop motifs. The "upstream" countries X appear to be relatively close to their tipping points because of their relatively high unemployment, high economic inequality, low GDP per capita, and large youth bulges. Recall that intermediate countries Y may have spread influence to protest from these upstream countries X to "downstream" countries Z. Country Z began to protest before Y did, perhaps because Z was closer to its tipping point [e.g., downstream countries Z had significantly higher unemployment and greater economic inequality (Gini coefficient)] and because intermediate countries tend to have relatively strong economies and significant revenue from oil, indicating a large distance from a tipping point. Intermediate and downstream countries had significantly higher Internet and mobile phone penetration, indicating their greater susceptibility to influence from ongoing protests.
Political rights and civil liberties [1] are too coarse-grained to distinguish among these countries. However, some more specific measures, such as personal autonomy and individual rights, show greater variance among countries involved in the Arab Spring; the fifth row of Fig. SM-5 shows that intermediate countries Y (Kuwait, Saudi Arabia, Egypt; data for U.A.E. is missing) tend to have greater personal autonomy and individual rights, which may play a role in delaying their protests. However, even when these populations were not yet protesting, these citizens may nevertheless have been communicating inspiration to protest to Effect of other couplings on the slave subsystem's bifurcation diagram. As in Fig. 2(b), if the master subsystem's equilibrium x * lies on its lower (respectively, upper) stable branch depicted in Fig. 1(b), then the saddle-node bifurcations of the slave subsystem, b break (σx * ) and b sustain (σx * ), are the blue (respectively, red) curves; the dashed curves correspond to the master subsystem being on its middle, unstable branch of equilibria. If there were no coupling, i.e., if σ were equal to 0, then the saddle-node bifurcations b break (σx * ) and b sustain (σx * ) would be given by the intersections of the blackdashed curves and the a = 0 axis; comparing the blue and red curves with these intersections determines whether the master subsystem facilitates or inhibits a regime shift in the slave subsystem. Panel  Fig. 4 (namely, Djibouti, Israel, Palestinian Territory, Iraq, and Mauritania). The 2010 unemployment rate for Djibouti (30%) was estimated from a linear regression between the available World Bank data [3] and the "fuzzy" data in Table 1 (Table 2). As in Fig. SM-4, the left-hand column shows weighted averages for the countries in the three roles X, Y, Z in the hop motifs (as defined in Definition 1), while the other columns show the countries (and their properties) in those weighted averages. All the data are from 2010 except for press freedom (2013).