Contagious Sets in Random Graphs

TL;DR

This paper analyzes the minimal size of contagious sets in random graphs under a threshold activation process, establishing asymptotic bounds and threshold probabilities for their existence.

Contribution

It provides the first asymptotic characterization of the minimal contagious set size in Erdős–Rényi graphs for constant activation thresholds.

Findings

Minimal contagious set size is Θ(n/d^{r/(r-1)} log d) with high probability.

Threshold probability for contagious set size r is Θ(1/(n log^{r-1} n)^{1/r}).

Results hold for specified ranges of d and fixed r.

Abstract

We consider the following activation process in undirected graphs: a vertex is active either if it belongs to a set of initially activated vertices or if at some point it has at least $r$ active neighbors. A \emph{contagious set} is a set whose activation results with the entire graph being active. Given a graph $G$, let $m(G,r)$ be the minimal size of a contagious set. We study this process on the binomial random graph $G:=G(n,p)$ with $p: = \frac{d}{n}$ and $1 \ll d \ll \left(\frac{n \log \log n}{\log^2 n}\right)^{\frac{r-1}{r}}$. Assuming $r > 1$ to be a constant that does not depend on $n$, we prove that $$m(G,r) = \Theta\left(\frac{n}{d^{\frac{r}{r-1}}\log d}\right),$$ with high probability. We also show that the threshold probability for $m(G,r)=r$ to hold is $p^*=\Theta\left(\frac{1}{(n \log^{r-1} n)^{1/r}}\right)$.