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Navigation in small-world networks: a scale-free continuum model

Part of: Geometric probability and stochastic geometry Special processes

Published online by Cambridge University Press:  14 July 2016

Massimo Franceschetti  and
Ronald Meester
Massimo Franceschetti*
Affiliation:
University of California, San Diego
Ronald Meester*
Affiliation:
Vrije Universiteit Amsterdam
*
∗Postal address: Electrical and Computer Engineering Department, University of California, San Diego, 9500 Gillman Drive, MC 0407, La Jolla, CA 92093-0407, USA.
∗∗Postal address: Divisie Wiskunde en Informatica, Faculteit der Exacte Wetenshappen, Vrije Universiteit Amsterdam, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands. Email address: [email protected]
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Abstract

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The small-world phenomenon, the principle that we are all linked by a short chain of intermediate acquaintances, has been investigated in mathematics and social sciences. It has been shown to be pervasive both in nature and in engineering systems like the World Wide Web. Work of Jon Kleinberg has shown that people, using only local information, are very effective at finding short paths in a network of social contacts. In this paper we argue that the underlying key to finding short paths is scale invariance. In order to appreciate scale invariance we suggest a continuum setting, since true scale invariance happens at all scales, something which cannot be observed in a discrete model. We introduce a random-connection model that is related to continuum percolation, and we prove the existence of a unique scale-free model, among a large class of models, that allows the construction, with high probability, of short paths between pairs of points separated by any distance.

Keywords

Small-world networkscale invariancerandom-connection modelPoisson process

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60K37: Processes in random environments
Type
Short Communications
Information
Journal of Applied Probability , Volume 43 , Issue 4 , December 2006 , pp. 1173 - 1180
Copyright
© Applied Probability Trust 2006 

Footnotes

Presented at the ICMS Workshop on Spatial Stochastic Modelling with Applications to Communications Networks (Edinburgh, June 2004).

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