  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.
  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.
