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

The 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 T-n 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 T-n > k for all k >= 1 is less than (1 vertical bar 2k(n-2)(2)/n) (1 2/n)((k- l)). We also study the behavior of the complementary distribution of T-n at point cE(T-n) when n tends to infinity for c not equal 1. We end our analysis by conjecturing that when n tends to infinity, T-n > E(T-n) with probability dose to 0.4484.