Contagious sets in random graphs

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 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 $1\ll d\ll (\frac{n\log\log n}{\log^{2}n})^{\frac{r-1}{r}}$. Assuming $r>1$ to be a constant that does not depend on $n$, we prove that

\[m(G,r)=\Theta (\frac{n}{d^{\frac{r}{r-1}}\log d}),\] with high probability. We also show that the threshold probability for $m(G,r)=r$ to hold is $p^{*}=\Theta (\frac{1}{(n\log^{r-1}n)^{1/r}})$.