Advances in Applied Probability

Asymptotic behaviour of gossip processes and small-world networks

A. D. Barbour and G. Reinert

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Both small-world models of random networks with occasional long-range connections and gossip processes with occasional long-range transmission of information have similar characteristic behaviour. The long-range elements appreciably reduce the effective distances, measured in space or in time, between pairs of typical points. In this paper we show that their common behaviour can be interpreted as a product of the locally branching nature of the models. In particular, it is shown that both typical distances between points and the proportion of space that can be reached within a given distance or time can be approximated by formulae involving the limit random variable of the branching process.

Article information

Adv. in Appl. Probab., Volume 45, Number 4 (2013), 981-1010.

First available in Project Euclid: 12 December 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 92H30
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60J85: Applications of branching processes [See also 92Dxx]

Small-world graph gossip process branching process approximation


Barbour, A. D.; Reinert, G. Asymptotic behaviour of gossip processes and small-world networks. Adv. in Appl. Probab. 45 (2013), no. 4, 981--1010. doi:10.1239/aap/1386857854.

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