Contagious Sets in Expanders

We consider the following activation process in d-regular 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 1 < r ≤ d is the activation threshold. Such processes have been studied extensively in several fields such as combinatorics, computer science, probability and statistical physics. 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 present upper bounds on m(G, r) on d-regular graphs with expansion properties (parameterized by the spectral gap and/or the girth of the graphs). In some cases we also provide nearly matching lower bounds. The general flavor of our results is that sufficiently strong expansion (i.e. λ(G) = O( √ d)) or sufficiently large girth (that is, girth Ω(log log d)) implies that in n-vertex graphs, m(G, 2) ≤ O( n d2 ). Furthermore, we show that in the absence of 4-cycles, λ(G) < (1 − )d ensures that m(G, 2) = O( log d 2d2 ). Time permitting, we shall discuss several open problems arising from our work. ∗Goethe University. [email protected]. Supported by ERC Starting Grant 278857PTCC (FP7). †The Weizmann Institute. [email protected]. Supported in part by The Israel Science Foundation (grant No. 621/12) and by the Citi Foundation ‡Tel-Aviv University. [email protected] . Research supported in part by: USAIsrael BSF Grant 2010115 and by grant 912/12 from the Israel Science Foundation. §The Weizmann Institute. [email protected]. Supported in part by The Israel Science Foundation (grant No. 621/12) and by the Citi Foundation