Advances in Applied Probability

Asymptotic behaviour of gossip processes and small-world networks

A. D. Barbour and G. Reinert

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Abstract

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

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

Dates
First available in Project Euclid: 12 December 2013

Permanent link to this document
https://projecteuclid.org/euclid.aap/1386857854

Digital Object Identifier
doi:10.1239/aap/1386857854

Mathematical Reviews number (MathSciNet)
MR3161293

Zentralblatt MATH identifier
1308.60107

Subjects
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]

Keywords
Small-world graph gossip process branching process approximation

Citation

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. https://projecteuclid.org/euclid.aap/1386857854


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References

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