Contagious sets in random graphs

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 Assuming r > 1 to be a constant that does not depend on n, we prove that with high probability. We also show that the threshold probability for m(G,r) = r to hold.