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

TL;DR

This paper analyzes the minimum seed sets needed to activate all nodes in scale-free and Erdős-Rényi graphs under a threshold cascade model, providing bounds based on graph degree distributions and probabilistic properties.

Contribution

It derives new upper bounds on the seed set size for complete activation within a fixed number of rounds in both scale-free and Erdős-Rényi random graphs.

Findings

For scale-free graphs with degree distribution decay, seed size is proportional to the number of vertices.

In Erdős-Rényi graphs, the seed size for immediate activation is proportional to the product of the threshold and total nodes.

Provides probabilistic bounds for seed sizes in Erdős-Rényi graphs with high probability.

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