On Reversible Cascades in Scale-Free and Erdos-Renyi Random Graphs

Consider the following cascading process on a simple undirected graph G(V, E) with diameter Delta. In round zero, a set S subset of V of vertices, called the seeds, are active. In round i + 1, i is an element of N, a non-isolated vertex is activated if at least a rho is an element of (0, 1] fraction of its neighbors are active in round i; it is deactivated otherwise. For k is an element of N, let min-seed((k))(G, rho) be the minimum number of seeds needed to activate all vertices in or before round k. This paper derives upper bounds on min-seed((k))(G, rho). In particular, if G is connected and there exist constants C > 0 and. > 2 such that the fraction of degree-k vertices in G is at most C/k(gamma) for all k is an element of Z(+), then min-seed((Delta))(G, rho) = O(inverted right perpendicular rho(gamma-1)vertical bar V vertical bar inverted left perpendicular). Furthermore, for n is an element of Z(+), p = Omega((ln (e/rho))/(rho n)) and with probability 1- exp (-n(Omega(1))) over the Erdos-Renyi random graphs G(n, p), min-seed((1))(G(n, p), rho) = O(rho n).