Thresholds for contagious sets in random graphs

TL;DR

This paper determines the threshold probability for the emergence of contagious sets in random graphs under bootstrap percolation, improving previous bounds and linking the problem to branching process survival probabilities.

Contribution

It identifies a sharp threshold for contagious sets in Erdős-Rényi graphs and provides asymptotic formulas for related branching process survival probabilities.

Findings

Established a threshold for contagious sets of size r in G(n,p).

Improved bounds on the percolation threshold compared to previous work.

Derived asymptotic formulas for survival probabilities of related branching processes.

Abstract

For fixed $r\geq 2$, we consider bootstrap percolation with threshold $r$ on the Erd\H{o}s-R\'enyi graph ${\cal G}_{n,p}$. We identify a threshold for $p$ above which there is with high probability a set of size $r$ which can infect the entire graph. This improves a result of Feige, Krivelevich and Reichman, which gives bounds for this threshold, up to multiplicative constants. As an application of our results, we also obtain an upper bound for the threshold for $K_4$-bootstrap percolation on ${\cal G}_{n,p}$, as studied by Balogh, Bollob\'as and Morris. We conjecture that our bound is asymptotically sharp. These thresholds are closely related to the survival probabilities of certain time-varying branching processes, and we derive asymptotic formulae for these survival probabilities which are of interest in their own right.