Triggering Cascades on Strongly Connected Directed Graphs

Consider the following process of activation on a directed graph G(V,E). In round zero, a set of vertices, called the seeds, are active. Thereafter, a vertex is activated in a round if at least a ρ ∈ (0,1] fraction of its in-neighbors are active in the previous round. Once a vertex is activated, it remains active. Assuming the strong connectivity of G, this paper proves the existence of O([ρ|V|]) seeds that will activate all vertices after a finite number of rounds.