Contagious Sets in Expanders

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 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. It is known that for every d-regular or nearly d-regular graph on n vertices, m(G, r) ≤ O(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 properties imply that m(G, 2) ≤ O(n/d2) (and more generally, m(G, r) ≤ O(n/dr/(r-1))). In addition, we demonstrate that rather weak assumptions on the girth and/or the spectral gap suffice in order to imply that m(G, 2) ≤ O(n log d/d2). For example, we show this for graphs of girth at least 7, and for graphs with λ(G) < (1 − e)d, provided the graph has no 4-cycles. Our results are algorithmic, entailing simple and efficient algorithms for selecting contagious sets.