Majority Bootstrap Percolation on $G(n,p)$

TL;DR

This paper investigates the dynamics of majority bootstrap percolation on Erdős-Rényi random graphs above the connectivity threshold, revealing results comparable to those on hypercubes, and enhances understanding of epidemic spread in random networks.

Contribution

It provides new insights into majority bootstrap percolation on $G(n,p)$, especially for small $p$, extending known results from hypercube studies to Erdős-Rényi graphs.

Findings

Percolation occurs with high probability above the connectivity threshold.

Results for small $p$ are similar to hypercube percolation outcomes.

The study extends understanding of epidemic processes in random graphs.

Abstract

Majority bootstrap percolation on a graph $G$ is an epidemic process defined in the following manner. Firstly, an initially infected set of vertices is selected. Then step by step the vertices that have more infected than non-infected neighbours are infected. We say that percolation occurs if eventually all vertices in $G$ become infected. In this paper we study majority bootstrap percolation on the Erd\H{o}s-R\'enyi random graph $G(n,p)$ above the connectivity threshold. Perhaps surprisingly, the results obtained for small $p$ are comparable to the results for the hypercube obtained by Balogh, Bollob\'as and Morris (2009).