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Asymptotic behaviour of gossip processes and small-world networks. (English) Zbl 1308.60107
In [“Collective dynamics of ‘small-world’ networks”, Nature 393, 440–442 (1998; doi:10.1038/30918)], D. J. Watts and S. H. Strogatz have provided a “small-world” model, where a random number of chords are superimposed as shortcuts on a circle \(C\) of circumference \(L\). The chords have endpoints uniformly and independently distributed on \(C\), and the number of chords follows a Poisson distribution.
Further, C. Moore and M. E. J. Newman [“Epidemics and percolation in small-world networks”, Phys. Rev. E 61, 5678–5682 (2000; doi:10.1103/PhysRevE.61.5678)] generalized the Watts and Strogatz’ model to a continuous analogue. A closely related model, the so-called “great circle model”, was earlier introduced by F. Ball et al. [Ann. Appl. Probab. 7, No. 1, 46–89 (1997; Zbl 0909.92028)].
In the present paper, the authors consider the Poisson process above on a smooth, closed, homogeneous Riemannian manifold \(C\) of dimension \(d\), such as a sphere or a torus, having large finite volume \(|C|=L\) with respect to its intrinsic metric.
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 authors show that their common behaviour can be interpreted as a product of the locally branching nature of the models.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J85 Applications of branching processes
05C82 Small world graphs, complex networks (graph-theoretic aspects)
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References:
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[9] Watts, D. J. and Strogatz, S. H. (1998). Collective dynamics of ‘small-world’ networks. Nature 393 , 440-442. · Zbl 1368.05139
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