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 $\Delta$. 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 $\rho\in(\,0,1\,]$ fraction of its neighbors are active in round $i$; it is deactivated otherwise. For $k\in\mathbb{N},$ let $\text{min-seed}^{{(k)}(G,\rho)$} be the minimum number of seeds needed to activate all vertices in or before round $k$. This paper derives upper bounds on $\text{min-seed}^{{(k)}(G,\rho)$.} In particular, if $G$ is connected and there exist constants $C>0$ and $\gamma>2$ such that the fraction of degree-$k$ vertices in $G$ is at most $C/k^\gamma$ for all $k\in\mathbb{Z}^+,$ then $\text{min-seed}^{{(\Delta)}(G,\rho)=O(\lceil\rho{\gamma-1}\,|\,V\,|\rceil)$.} Furthermore, for $n\in\mathbb{Z}^+,$ $p=\Omega((\ln{(e/\rho)})/(\rho n))$ and with probability $1-\exp{(-n^{{\Omega(1)})}$} over the Erd\H{o}s-R\'enyi random graphs $G(n,p),$ $\text{min-seed}^{{(1)}(G(n,p),\rho)=O(\rho} n)$.