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On reversible cascades in scale-free and Erdős-Rényi random graphs (1011.0653v1)

Published 2 Nov 2010 in cs.DM

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 $\text{min-seed}{(k)}(G,\rho)$ 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 $G(n,p),$ $\text{min-seed}{(1)}(G(n,p),\rho)=O(\rho n)$.

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Authors (1)
  1. Ching-Lueh Chang (11 papers)
Citations (4)

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