Analysis of the Propagation Time of a Rumour in Large-scale Distributed Systems

This article has received the Best Student Paper AwardInternational audienceThe context of this work is the well studied dissemination of information in large scale distributed networks through pairwise interactions. This problem, originally called rumor mongering, and then rumor spreading has mainly been investigated in the synchronous model. This model relies on the assumption that all the nodes of the network act in synchrony, that is, at each round of the protocol, each node is allowed to contact a random neighbor. In this paper, we drop this assumption under the argument that it is not realistic in large scale systems. We thus consider the asynchronous variant, where at time unit, a single node interacts with a randomly chosen neighbor. We perform a thorough study of Tn the total number of interactions needed for all the n nodes of the network to discover the rumor. While most of the existing results involve huge constants that do not allow for comparing different protocols, we prove that in a complete graph of size n ≥ 2, the probability that Tn > k for all k ≥ 1 is less than 1 + 2k(n−2) 2 n 1 − 2 n (k−1). We also study the behavior of the complementary distribution of Tn at point cE(Tn) when n tends to infinity for c = 1. We end our analysis by conjecturing that when n tends to infinity, Tn > E(Tn) with probability close to 0.4484