On reversible cascades in scale-free and Erdős-Rényi random graphs

Consider the following cascading process on a simple undirected graph $G(V,E)$ with diameter $Δ$. In round zero, a set $S\subseteq V$ of vertices, called the seeds, are active. In round $i+1,$ $i\in\mathbb{N},$ a non-isolated vertex is activated if at least a $ρ\in(\,0,1\,]$ fraction of its neighbors are active in round $i$; it is deactivated otherwise. For $k\in\mathbb{N},$ let $\text{min-seed}^{(k)}(G,ρ)$ be the minimum number of seeds needed to activate all vertices in or before round $k$. This paper derives upper bounds on $\text{min-seed}^{(k)}(G,ρ)$. 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 $C/k^γ$ for all $k\in\mathbb{Z}^+,$ then $\text{min-seed}^{(Δ)}(G,ρ)=O(\lceilρ^{γ-1}\,|\,V\,|\rceil)$. Furthermore, for $n\in\mathbb{Z}^+,$ $p=Ω((\ln{(e/ρ)})/(ρn))$ and with probability $1-\exp{(-n^{Ω(1)})}$ over the Erdős-Rényi random graphs $G(n,p),$ $\text{min-seed}^{(1)}(G(n,p),ρ)=O(ρn)$.

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