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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 

References

Aldous, D. (2004). A tractable complex network model based on the stochastic mean-field model of distance. In Complex Networks (Lecture Notes Phys. 650), Springer, Berlin, pp. 51–87.CrossRefGoogle Scholar
Barabasi, A., Newman, M. E. J. and Watts, D. (2005). The Structure and Dynamics of Networks. Princeton University Press.Google Scholar
Barabasi, A.L. and Albert, R. (1999). Emergence of scaling in random networks. Science 286, 509–512.CrossRefGoogle ScholarPubMed
Barbour, A. D. and Reinert, G. (2006). Discrete small world networks. Electron. J. Prob. 11, 1234–1283.CrossRefGoogle Scholar
Bollobás, B. (2001). Random Graphs. Academic Press, New York.CrossRefGoogle Scholar
Bollobás, B. and Riordan, O. (2004). The diameter of a scale-free random graph. Combinatorica 24, 5–34.CrossRefGoogle Scholar
Bollobás, B., Janson, S. and Riordan, O. (2002). Mathematical results on scale-free random graphs. Handbook of Graphs and Networks – From the Genome to the Internet, eds Bornholdt, S. and Schuster, H., John Wiley, New York, pp. 1–34 Google Scholar
Bollobás, B., Janson, S. and Riordan, O. (2007). The phase transition in inhomogeneous random graphs. Random Structures Algorithms 31, 3–122.CrossRefGoogle Scholar
Bollobás, B., Riordan, O., Spencer, J. and Tusnády, G. (2001). The degree sequence of a scale-free random graph process. Random Structures Algorithms 18, 279–290.CrossRefGoogle Scholar
Chung, F. and Lu, L. (2006). Complex Graphs and Networks. American Mathematical Society, Providence, RI.CrossRefGoogle Scholar
Cont, R. and Loewe, M. (2008). Social distance and social interactions: a discrete choice model. To appear in J. Math. Econom. Google Scholar
Degenne, A. and Forsé, M. (1999). Introducing Social Networks. Sage Publications, London.CrossRefGoogle Scholar
Erdős, P. and Rényi, A. (1960). On the evolution of random graphs. Magyar Tud. Akad. Mat. Kutato Int. Közl. 5, 17–61.Google Scholar
Fernholz, D. and Ramachandran, V. (2007). The diameter of sparse random graphs. Random Structures Algorithms 31, 482–516.CrossRefGoogle Scholar
Girvan, M. and Newman, M. E. J. (1999). Community structure in social and biological networks. Proc. Nat. Acad. Sci. USA 99, 7821–7826.CrossRefGoogle Scholar
Granovetter, M. (1973). The strength of weak ties. Amer. J. Sociology 78, 1287–1303.CrossRefGoogle Scholar
Hofman, J. and Wiggins, C. (2008). A Bayesian approach to network modularity. Phys. Rev. Lett. 100, 258701.CrossRefGoogle ScholarPubMed
Jackson, M. and Rogers, B. (2008). The economics of small worlds. J. Europ. Econom. Assoc. 3, 617–627.CrossRefGoogle Scholar
Johnson, S. (1967). Hierarchical clustering schemes. Psychometrika 2, 241–254.CrossRefGoogle Scholar
Newman, M. (2003). The structure and function of complex networks. SIAM Rev. 45, 167–256.CrossRefGoogle Scholar
Newman, M., Barabasi, A. and Watts, D. (2006). The Structure and Dynamics of Networks. Princeton University Press.Google Scholar
Newman, M. E. J., Moore, C. and Watts, D. (2000). Mean-field solution of small world networks. Phys. Rev. Lett. 84, 3201–3204.CrossRefGoogle Scholar
Newman, M. E. J., Watts, D. and Strogatz, S. H. (2002). Random graph models of social networks, Proc. Nat. Acad. Sci. USA 99, 2566–2572.CrossRefGoogle ScholarPubMed
Travers, J. and Milgram, S. (1969). An experimental study of the small world problem. Sociometry 32, 425–443.CrossRefGoogle Scholar
Watts, D. J. (1999). Small Worlds. Princeton University Press.CrossRefGoogle Scholar
Watts, D. J. and Strogatz, S. H. (1998). Collective dynamics of small world networks. Nature 393, 440–442.CrossRefGoogle ScholarPubMed