Asymptotic behavior of Aldous' gossip process

TL;DR

This paper analyzes the asymptotic spread of information in Aldous' gossip process on a torus, revealing precise scaling laws and deterministic approximations for the time until full information dissemination.

Contribution

It provides rigorous asymptotic results and a simplified model describing the spread of information, confirming Aldous's heuristic calculations.

Findings

Time until everyone knows: asymptotically $(2-2eta/3)N^{eta} ext{log}N$

Fraction of informed individuals follows a deterministic function near critical times

Early spread characterized by a random variable $M$ influencing subsequent dynamics

Abstract

Aldous [(2007) Preprint] defined a gossip process in which space is a discrete $N\times N$ 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 $N^{-\alpha}$ to a site chosen at random from the torus. We will be interested in the case in which $\alpha<3$, 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 $T=(2-2\alpha/3)N^{\alpha/3}\log N$. If $\rho_s$ is the fraction of the population who know the information at time $s$ and $\varepsilon$ is small then, for large $N$, the time until $\rho_s$ reaches $\varepsilon$ is $T(\varepsilon)\approx T+N^{\alpha/3}\log (3\varepsilon /M)$, where $M$ is a random variable determined by the early spread of the information. The value of $\rho_s$ at time $s=T(1/3)+tN^{\alpha/3}$ is almost a deterministic function $h(t)$ which satisfies an odd looking integro-differential equation. The last result confirms a heuristic calculation of Aldous.