Influence is a Matter of Degree: New Algorithms for Activation Problems

TL;DR

This paper introduces new algorithms for the target set selection problem, providing bounds on contagious set sizes and efficient methods for dense graphs, advancing understanding of activation processes in networks.

Contribution

The paper establishes a bound on contagious set size for undirected graphs and presents an efficient algorithm to find such sets, improving previous approaches.

Findings

Bound on contagious set size for undirected graphs.

Efficient algorithm for finding contagious sets.

Application to dense graphs.

Abstract

We consider the target set selection problem. In this problem, a vertex is active either if it belongs to a set of initially activated vertices or if at some point it has at least $k$ active neighbors ($k$ is identical for all vertices of the graph). Our goal is to find a set of minimum size whose activation will result with the entire graph being activated. Call such a set \emph{contagious}. We prove that if $G=(V,E)$ is an undirected graph, the size of a contagious set is bounded by $\sum_{v\in V}{\min \{1,\frac{k}{d(v)+1}\}}$ (where $d(v)$ is the degree of $v$). We present a simple and efficient algorithm that finds a contagious set that is not larger than the aforementioned bound and discuss algorithmic applications of this algorithm to finding contagious sets in dense graphs.