Bootstrap percolation on the stochastic block model

TL;DR

This paper studies how bootstrap percolation behaves on the stochastic block model, revealing a sharp phase transition in the spread of activation depending on initial conditions and community structure.

Contribution

It extends bootstrap percolation analysis to the stochastic block model, characterizing phase transitions with community-aware parameters.

Findings

Existence of a sharp phase transition in the SBM.

Characterization of sub-critical and super-critical regimes.

Dependence of activation spread on initial active nodes and community structure.

Abstract

We analyze the bootstrap percolation process on the stochastic block model (SBM), a natural extension of the Erd\H{o}s--R\'{e}nyi random graph that incorporates the community structure observed in many real systems. In the SBM, nodes are partitioned into two subsets, which represent different communities, and pairs of nodes are independently connected with a probability that depends on the communities they belong to. Under mild assumptions on the system parameters, we prove the existence of a sharp phase transition for the final number of active nodes and characterize the sub-critical and the super-critical regimes in terms of the number of initially active nodes, which are selected uniformly at random in each community.