# Bootstrap percolation on the stochastic block model with k communities

## TL;DR

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

## Contribution

It provides the first rigorous analysis of bootstrap percolation on SBM, identifying phase transition thresholds based on community structure and initial activation.

## Key findings

- Existence of a sharp phase transition in the final active nodes
- Characterization of sub-critical and super-critical regimes
- Dependence of percolation 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\"{o}s--R\'{e}nyi random graph that allows representing the "community structure" observed in many real systems. In the SBM, nodes are partitioned into 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 system parameters, we prove the existence of a sharp phase transition for the final number of active nodes and characterize sub-critical and super-critical regimes in terms of the number of initially active nodes, which are selected uniformly at random in each community.