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

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 at least half of its neighbours infected become infected. We say that percolation occurs if eventually all vertices in $G$ become infected.

In this paper we  provide sharp bounds for the critical size of the initially infected set in majority bootstrap percolation on the Erdős-Rényi random graph $G(n,p)$. This answers an open question by Janson, Luczak, Turova and Vallier (2012). Our results obtained for $p=c\log(n)/n$ are close to the results obtained by Balogh, Bollobás and Morris (2009) for majority bootstrap percolation on the hypercube. We conjecture that similar results will be true for all regular-like graphs with the same density and sufficiently strong expansion properties.