Asymptotic behavior of Aldous’ gossip process

Abstract

Aldous [(2007) Preprint] defined a gossip process in which space is a discrete N × N torus, and the state of the process at time t is the set of individuals who know the information. Information spreads from a site to its nearest neighbors at rate 1/4 each and at rate N^−α to a site chosen at random from the torus. We will be interested in the case in which α < 3, where the long range transmission significantly accelerates the time at which everyone knows the information. We prove three results that precisely describe the spread of information in a slightly simplified model on the real torus. The time until everyone knows the information is asymptotically T = (2 − 2 α/3)N^α/3log N. If ρ_s is the fraction of the population who know the information at time s and ε is small then, for large N, the time until ρ_s reaches ε is T(ε) ≈ T + N^α/3log(3ε/M), where M is a random variable determined by the early spread of the information. The value of ρ_s at time s = T(1/3) + tN^α/3 is almost a deterministic function h(t) which satisfies an odd looking integro-differential equation. The last result confirms a heuristic calculation of Aldous.