Threshold Models of Cascades in Large-Scale Networks

The spread of new beliefs, behaviors, conventions, norms, and technologies in social and economic networks are often driven by cascading mechanisms, and so are contagion dynamics in financial networks. Global behaviors generally emerge from the interplay between the structure of the interconnection topology and the local agents' interactions. We focus on the Threshold Model (TM) of cascades first introduced by Granovetter (1978). This can be interpreted as the best response dynamics in a network game whereby agents choose strategically between two actions. Each agent is equipped with an individual threshold representing the number of her neighbors who must have adopted a certain action for that to become the agent's best response. We analyze the TM dynamics on large-scale networks with heterogeneous agents. Through a local mean-field approach, we obtain a nonlinear, one-dimensional, recursive equation that approximates the evolution of the TM dynamics on most of the networks of a given size and distribution of degrees and thresholds. We prove that, on all but a fraction of networks with given degree and threshold statistics that is vanishing as the network size grows large, the actual fraction of adopters of a given action is arbitrarily close to the output of the aforementioned recursion. Numerical simulations on some real network testbeds show good adherence to the theoretical predictions.