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Small-world graphs: characterization and alternative constructions

  • Rama Cont (a1) and Emily Tanimura (a2)
Abstract

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.

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Copyright
Corresponding author
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.
References
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[1] 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. 5187.
[2] Barabasi, A., Newman, M. E. J. and Watts, D. (2005). The Structure and Dynamics of Networks. Princeton University Press.
[3] Barabasi, A.L. and Albert, R. (1999). Emergence of scaling in random networks. Science 286, 509512.
[4] Barbour, A. D. and Reinert, G. (2006). Discrete small world networks. Electron. J. Prob. 11, 12341283.
[5] Bollobás, B. (2001). Random Graphs. Academic Press, New York.
[6] Bollobás, B. and Riordan, O. (2004). The diameter of a scale-free random graph. Combinatorica 24, 534.
[7] 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. 134
[8] Bollobás, B., Janson, S. and Riordan, O. (2007). The phase transition in inhomogeneous random graphs. Random Structures Algorithms 31, 3122.
[9] Bollobás, B., Riordan, O., Spencer, J. and Tusnády, G. (2001). The degree sequence of a scale-free random graph process. Random Structures Algorithms 18, 279290.
[10] Chung, F. and Lu, L. (2006). Complex Graphs and Networks. American Mathematical Society, Providence, RI.
[11] Cont, R. and Loewe, M. (2008). Social distance and social interactions: a discrete choice model. To appear in J. Math. Econom.
[12] Degenne, A. and Forsé, M. (1999). Introducing Social Networks. Sage Publications, London.
[13] Erdős, P. and Rényi, A. (1960). On the evolution of random graphs. Magyar Tud. Akad. Mat. Kutato Int. Közl. 5, 1761.
[14] Fernholz, D. and Ramachandran, V. (2007). The diameter of sparse random graphs. Random Structures Algorithms 31, 482516.
[15] Girvan, M. and Newman, M. E. J. (1999). Community structure in social and biological networks. Proc. Nat. Acad. Sci. USA 99, 78217826.
[16] Granovetter, M. (1973). The strength of weak ties. Amer. J. Sociology 78, 12871303.
[17] Hofman, J. and Wiggins, C. (2008). A Bayesian approach to network modularity. Phys. Rev. Lett. 100, 258701.
[18] Jackson, M. and Rogers, B. (2008). The economics of small worlds. J. Europ. Econom. Assoc. 3, 617627.
[19] Johnson, S. (1967). Hierarchical clustering schemes. Psychometrika 2, 241254.
[20] Newman, M. (2003). The structure and function of complex networks. SIAM Rev. 45, 167256.
[21] Newman, M., Barabasi, A. and Watts, D. (2006). The Structure and Dynamics of Networks. Princeton University Press.
[22] Newman, M. E. J., Moore, C. and Watts, D. (2000). Mean-field solution of small world networks. Phys. Rev. Lett. 84, 32013204.
[23] Newman, M. E. J., Watts, D. and Strogatz, S. H. (2002). Random graph models of social networks, Proc. Nat. Acad. Sci. USA 99, 25662572.
[24] Travers, J. and Milgram, S. (1969). An experimental study of the small world problem. Sociometry 32, 425443.
[25] Watts, D. J. (1999). Small Worlds. Princeton University Press.
[26] Watts, D. J. and Strogatz, S. H. (1998). Collective dynamics of small world networks. Nature 393, 440442.
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Advances in Applied Probability
  • ISSN: 0001-8678
  • EISSN: 1475-6064
  • URL: /core/journals/advances-in-applied-probability
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