Spreading messages

Abstract

We model a network in which messages spread by a simple directed graph G = ( V , E ) and a function α : V → N mapping each v ∈ 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 ∈ V will be convinced of a message when at least α ( v ) 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 , α ) of seeds needed to convince all individuals at the end. In particular, we prove a lower bound on min-seed ( G , α ) and the NP-completeness of computing min-seed ( G , α ) . 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 ) , Ω ( 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 | E | + 2 | V | ) | V | individuals at the end. Third, in a digraph G = ( V , E ) with a positive minimum indegree, one can find in polynomial (in | V | ) time a set of at most (23/27) | V | seeds convincing all individuals.