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

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