The Annals of Applied Probability

Asymptotic behavior of Aldous’ gossip process

Shirshendu Chatterjee and Rick Durrett

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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 Nα 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)Nα/3log N. If ρs is the fraction of the population who know the information at time s and ε is small then, for large N, the time until ρs reaches ε is T(ε) ≈ T + Nα/3log(3ε/M), where M is a random variable determined by the early spread of the information. The value of ρs at time s = T(1/3) + tNα/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.

Article information

Source
Ann. Appl. Probab., Volume 21, Number 6 (2011), 2447-2482.

Dates
First available in Project Euclid: 23 November 2011

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1322057327

Digital Object Identifier
doi:10.1214/10-AAP750

Mathematical Reviews number (MathSciNet)
MR2895421

Zentralblatt MATH identifier
1246.60117

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
Gossip branching process first-passage percolation integro-differential equation

Citation

Chatterjee, Shirshendu; Durrett, Rick. Asymptotic behavior of Aldous’ gossip process. Ann. Appl. Probab. 21 (2011), no. 6, 2447--2482. doi:10.1214/10-AAP750. https://projecteuclid.org/euclid.aoap/1322057327


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References

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