Behavioral Intervention and Non-Uniform Bootstrap Percolation

Bootstrap percolation is an often used model to study the spread of diseases, rumors, and information on sparse random graphs. The percolation process demonstrates a critical value such that the graph is either almost completely affected or almost completely unaffected based on the initial seed being larger or smaller than the critical value. In this paper, we consider behavioral interventions, that is, once the percolation has affected a substantial fraction of the nodes, an external advisory suggests simple policies to modify behavior (for example, asking vertices to reduce contact by randomly deleting edges) in order to stop the spread of false information or disease. We analyze some natural interventions and show that the interventions themselves satisfy a similar critical transition. To analyze intervention strategies we provide the first analytic determination of the critical value for basic bootstrap percolation in random graphs when the vertex thresholds are nonuniform and provide an efficient algorithm. This result also helps solve the problem of “Percolation with Coinflips” when the infection process is not deterministic – which has been a criticism about the model. We also extend the results to “clustered” random graphs thereby extending the classes of graphs considered. In these graphs the vertices are grouped in a small number of clusters, the clusters model a fixed communication network and the edge probability is dependent if the vertices are in “close” or “far” clusters. We present simulations for both basic percolation and interventions that support our theoretical results.