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

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 contagious. We prove that if G = (V,E) is an undirected graph, the size of a contagious set is bounded by ∑ v∈V min{1, 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.