## Asymptotic behavior of Aldous' gossip process

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 - 2α/3)Nα/3 logN. 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/3 log(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. © Institute of Mathematical Statistics, 2011.

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- Statistics & Probability
- 0104 Statistics
- 0102 Applied Mathematics

### Citation

*Annals of Applied Probability*,

*21*(6), 2447–2482. https://doi.org/10.1214/10-AAP750

*Annals of Applied Probability*21, no. 6 (December 1, 2011): 2447–82. https://doi.org/10.1214/10-AAP750.

*Annals of Applied Probability*, vol. 21, no. 6, Dec. 2011, pp. 2447–82.

*Scopus*, doi:10.1214/10-AAP750.

## Published In

## DOI

## ISSN

## Publication Date

## Volume

## Issue

## Start / End Page

## Related Subject Headings

- Statistics & Probability
- 0104 Statistics
- 0102 Applied Mathematics