Contagious Sets in Expanders

TL;DR

This paper investigates the size of contagious sets in expander graphs with various spectral and girth properties, providing bounds and efficient algorithms for activation processes with threshold r.

Contribution

It establishes new bounds on minimal contagious set sizes in expanders with spectral and girth conditions, and introduces efficient algorithms for their selection.

Findings

Strong expansion implies small contagious sets, e.g., O(n/d^2) for certain spectral/girth conditions.

Weaker expansion still yields bounds like O((n log d)/d^2).

Random graphs G(n,p) with p ~ d/n have contagious sets of size between Omega(n/d^2 log d) and O(n log log d / d^2 log d).

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, where $r>1$ is the activation threshold. 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. Computing $m(G,r)$ is NP-hard. It is known that for every $d$-regular or nearly $d$-regular graph on $n$ vertices, $m(G,r) \le O(\frac{nr}{d})$. We consider such graphs that additionally have expansion properties, parameterized by the spectral gap and/or the girth of the graphs. The general flavor of our results is that sufficiently strong expansion (e.g., $\lambda(G)=O(\sqrt{d})$, or girth $\Omega(\log \log d)$) implies that $m(G,2) \le O(\frac{n}{d^2})$ (and more generally, $m(G,r) \le O(\frac{n}{d^{r/(r-1)}})$). Significantly weaker expansion properties suffice in order to imply that $m(G,2)\le O(\frac{n \log d}{d^2})$. For example, we show this for graphs of girth at least~7, and for graphs with $\lambda(G)<(1-\epsilon)d$, provided the graph has no 4-cycles. Nearly $d$-regular expander graphs can be obtained by considering the binomial random graph $G(n,p)$ with $p \simeq \frac{d}{n}$ and $d > \log n$. For such graphs we prove that $\Omega(\frac{n}{d^2 \log d}) \le m(G,2) \le O(\frac{n\log\log d}{d^2\log d})$ almost surely. Our results are algorithmic, entailing simple and efficient algorithms for selecting contagious sets.