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Small-world graphs: characterization and alternative constructions

Part of: Graph theory

Published online by Cambridge University Press:  01 July 2016

Rama Cont  and
Emily Tanimura
Rama Cont*
Affiliation:
Columbia University
Emily Tanimura*
Affiliation:
Ecole des Hautes Etudes en Sciences Sociales
*
∗ Postal address: IEOR Department, Columbia University, 500 West 120th Street, New York, NY 10027, USA. Email address: rama.cont@columbia.edu
∗∗ Postal address: Centre d'Analyse et de Mathématique Sociales, L'Ecole des Hautes Etudes en Sciences Sociales, 54 Boulevard Raspail, 7520 Paris Cedex 06, France.
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Abstract

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Small-world graphs are examples of random graphs which mimic empirically observed features of social networks. We propose an intrinsic definition of small-world graphs, based on a probabilistic formulation of scaling properties of the graph, which does not rely on any particular construction. Our definition is shown to encompass existing models of small-world graphs, proposed by Watts (1999) and studied by Barbour and Reinert (2001), which are based on random perturbations of a regular lattice. We also propose alternative constructions of small-world graphs which are not based on lattices and study their scaling properties.

Keywords

Random graphsmall-world model

MSC classification

Primary: 05C80: Random graphs
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 40 , Issue 4 , December 2008 , pp. 939 - 965
Copyright
Copyright © Applied Probability Trust 2008 

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