A simple model of global cascades on random networks

Abstract

The origin of large but rare cascades that are triggered by small initial shocks is a phenomenon that manifests itself as diversely as cultural fads, collective action, the diffusion of norms and innovations, and cascading failures in infrastructure and organizational networks. This paper presents a possible explanation of this phenomenon in terms of a sparse, random network of interacting agents whose decisions are determined by the actions of their neighbors according to a simple threshold rule. Two regimes are identified in which the network is susceptible to very large cascades-herein called global cascades-that occur very rarely. When cascade propagation is limited by the connectivity of the network, a power law distribution of cascade sizes is observed, analogous to the cluster size distribution in standard percolation theory and avalanches in self-organized criticality. But when the network is highly connected, cascade propagation is limited instead by the local stability of the nodes themselves, and the size distribution of cascades is bimodal, implying a more extreme kind of instability that is correspondingly harder to anticipate. In the first regime, where the distribution of network neighbors is highly skewed, it is found that the most connected nodes are far more likely than average nodes to trigger cascades, but not in the second regime. Finally, it is shown that heterogeneity plays an ambiguous role in determining a system's stability: increasingly heterogeneous thresholds make the system more vulnerable to global cascades; but an increasingly heterogeneous degree distribution makes it less vulnerable.

Figure 1

Cascade windows for the threshold model. The dashed line encloses the region of the (φ*, z) plane in which the cascade condition (Eq. 5) is satisfied for a uniform random graph with a homogenous threshold distribution f(φ) = δ(φ − φ*). The solid circles outline the region in which global cascades occur for the same parameter settings in the full dynamical model for n = 10,000 (averaged over 100 random single-node perturbations).

Figure 2

Cross section of the cascade window from Fig. 1, at φ* = 0.18. (a) The average time required for a cascade to terminate diverges at both the lower and upper boundaries of the cascade window, indicating two phase transitions. (b) Comparison between connected components of the network and the properties of global cascades. The frequency of global cascades in the numerical model (open circles) is well approximated by the fractional size of the extended vulnerable cluster (short dashes). For comparison, the size of the vulnerable cluster is also shown, both the exact solution derived in the text (long dashes) and the average over 1,000 realizations of a random graph (crosses). The exact and numerical solutions agree everywhere except at the upper phase transition, where the finite size of the network (n = 10,000) affects the numerical results. Finally, the average size of global cascades is shown (solid circles) and compared with the exact solution for the largest connected component (solid line).

Figure 3

Cumulative distributions of cascade sizes at the lower and upper critical points, for n = 1,000 and z = 1.05 (open squares) and z = 6.14 (solid circles), respectively. The straight line on the double logarithmic scale indicates that cascades at the lower critical point are power-law distributed, with slope 3/2 (the cumulative distribution has slope 1/2). By contrast, the distribution at the upper critical point is bimodal, with an exponential tail at small cascade size, and a second peak at the size of the entire system corresponding to a single global cascade. Above the upper boundary, the global cascade disappears and large cascades are always exponentially unlikely.

Figure 4

Analytically derived cascade windows for heterogeneous networks. The solid lines are the same as Fig. 1. (a) The dashed lines represent cascade windows for uniform random graphs, but where the threshold distributions (φ) are normally distributed with mean φ and SD σ = 0.05 and σ = 0.1. (b) The dashed line represents the cascade window for a random graph with a degree distribution that is a power law with exponent τ and exponential cut-off κ₀, where τ has been fixed at τ = 2.5 and κ₀ has been adjusted to generate graphs with variable z.