Threshold Models of Cascades in Large-Scale Networks

The spread of new beliefs, behaviors, and technologies in social and economic networks are often driven by cascading mechanisms. Global behaviors emerge from the interplay between the interconnections structure and the local agents interactions. We focus on the Threshold Model (TM) of cascades that can be interpreted as the best response dynamics in a network game. 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 TM's evolution on most of the networks of a given size and distribution of degrees and thresholds. Specifically, 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 in the TM dynamics is arbitrarily close to the output of the aforementioned recursion. Finally, we analyze the dynamic behavior and bifurcations of this recursion and complement the predictions with numerical simulations on real network testbeds.