Spreading messages

Abstract

We model a network in which messages spread by a simple directed graph $G=(V,E)$ and a function $\alpha :V\to \mathbb{N}$ mapping each $v\in V$ to a positive integer less than or equal to the indegree of $v$. The graph $G$ represents the individuals in the network and the communication channels between them. An individual $v\in V$ will be convinced of a message when at least $\alpha \left(v\right)$ of its in-neighbors are convinced. Suppose we are to convince a message to the individuals by first convincing a subset of individuals, called the seeds, and then let the message spread. We study the minimum number min-seed $(G,\alpha )$ of seeds needed to convince all individuals at the end. In particular, we prove a lower bound on min-seed $(G,\alpha )$ and the NP-completeness of computing min-seed $(G,\alpha )$. We also analyze the special case, called the strict-majority scenario, where each individual is convinced of a message when more than half of its in-neighbors are convinced. For the strict-majority scenario, we prove three results. First, we show that with high probability over the Erdős–Rényi random graphs $G(n,p)$, $\Omega (min\{n,1/p\})$ seeds are needed to convince all individuals at the end. Second, if $G=(V,E)$ is undirected, then a set of $s$ uniformly random samples from $V$ convinces no more than an expected $s(2\left|E\right|+2\left|V\right|)\left|V\right|$ individuals at the end. Third, in a digraph $G=(V,E)$ with a positive minimum indegree, one can find in polynomial (in $\left|V\right|$) time a set of at most (23/27)$\left|V\right|$ seeds convincing all individuals.