On reversible cascades in scale-free and Erd\H{o}s-R\'enyi random graphs

Consider the following cascading process on a simple undirected graphG(V,E) with diameter ∆. In round zero, a set S ⊆ V of vertices, called the seeds, are active. In round i+1, i ∈ N, a non-isolated vertex is activated if at least a ρ ∈ ( 0, 1 ] fraction of its neighbors are active in round i; it is deactivated otherwise. For k ∈ N, let min-seed(G, ρ) be the minimum number of seeds needed to activate all vertices in or before round k. This paper derives upper bounds on min-seed(G, ρ). In particular, if G is connected and there exist constants C > 0 and γ > 2 such that the fraction of degree-k vertices in G is at most C/kγ for all k ∈ Z+, then min-seed(G, ρ) = O(⌈ργ−1 |V |⌉). Furthermore, for n ∈ Z+, p = Ω((ln (e/ρ))/(ρn)) and with probability 1 − exp (−nΩ(1)) over the Erdős-Rényi random graphs G(n, p), min-seed(G(n, p), ρ) = O(ρn).