Bootstrap percolation on G(n,p) revisited

TL;DR

This paper revisits bootstrap percolation on the Erdős–Rényi random graph G(n,p), improving probability bounds for the number of infected vertices at the process's conclusion.

Contribution

It strengthens existing probability bounds for the final infected set size in bootstrap percolation on G(n,p).

Findings

Improved bounds on the number of infected vertices.

Enhanced probabilistic estimates for the infection process.

Refined analysis of bootstrap percolation dynamics.

Abstract

Bootstrap percolation on a graph with infection threshold $r\in \mathbb{N}$ is an infection process, which starts from a set of initially infected vertices and in each step every vertex with at least $r$ infected neighbours becomes infected. We consider bootstrap percolation on the binomial random graph $G(n,p)$, which was investigated among others by Janson, \L uczak, Turova and Valier (2012). We improve their results by strengthening the probability bounds for the number of infected vertices at the end of the process.