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A mixture model for random graphs

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The Erdös–Rényi model of a network is simple and possesses many explicit expressions for average and asymptotic properties, but it does not fit well to real-world networks. The vertices of those networks are often structured in unknown classes (functionally related proteins or social communities) with different connectivity properties. The stochastic block structures model was proposed for this purpose in the context of social sciences, using a Bayesian approach. We consider the same model in a frequentest statistical framework. We give the degree distribution and the clustering coefficient associated with this model, a variational method to estimate its parameters and a model selection criterion to select the number of classes. This estimation procedure allows us to deal with large networks containing thousands of vertices. The method is used to uncover the modular structure of a network of enzymatic reactions.

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  • Albert, R., Barabási, A.L.: Statistical mechanics of complex networks. Rev. Mod. Phys. 74, 47–97 (2002)

    Article  Google Scholar 

  • Alm, E., Arkin, A.P.: Biological networks. Cur. Op. Struct. Biol. 13, 193–202 (2002)

    Article  Google Scholar 

  • Arita, M.: The metabolic world of Escherichia coli is not small. PNAS 101, 1543–1547 (2004)

    Article  Google Scholar 

  • Barabási, A.L., Albert, R.: Emergence of scaling in random networks. Science 286, 509–512 (1999)

    Article  MathSciNet  Google Scholar 

  • Biernacki, C., Celeux, G., Govaert, G.: Assessing a mixture model for clustering with the integrated completed likelihood. IEEE Trans. Pattern Anal. Mach. Intell. 22, 719–725 (2000)

    Article  Google Scholar 

  • Dempster, A.P., Laird, N.M., Rubin, D.B.: Maximum likelihood from incomplete data via the EM algorithm. J. Roy. Stat. Soc. B 39, 1–38 (1977)

    MATH  MathSciNet  Google Scholar 

  • Govaert, G., Nadif, M.: An EM algorithm for the block mixture model. IEEE Trans. Pattern Anal. Mach. Intell. 27, 643–647 (2005)

    Article  Google Scholar 

  • Jaakkola, T.: Advanced Mean Field Methods: Theory and Practice. MIT Press, Cambridge (2000). Chapter: Tutorial on variational approximation methods

    Google Scholar 

  • Jones, J., Handcock, M.: Likelihood-based inference for stochastic models of sexual network formation. Theor. Pop. Biol. 65, 413–422 (2004)

    Article  MATH  Google Scholar 

  • Jordan, M.I., Ghahramani, Z., Jaakkola, T., Saul, L.K.: An introduction to variational methods for graphical models. Mach. Learn. 37, 183–233 (1999)

    Article  MATH  Google Scholar 

  • Molloy, M., Reed, B.: A critical point for random graphs with a given degree sequence. Rand. Struct. Algorithms 6, 161–179 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  • Newman, M.E.J.: Fast algorithm for detecting community structure in networks. Phys. Rev. E 69, 066133 (2004)

    Article  Google Scholar 

  • Newman, M.E.J., Girvan, M.: Statistical Mechanics of Complex Networks. Springer, Berlin (2003). Chapter: Mixing patterns and community structure in networks

    Google Scholar 

  • Newman, M.E.J., Watts, D.J., Strogatz, S.H.: Random graph models of social networks. PNAS 99, 2566–2572 (2002)

    Article  MATH  Google Scholar 

  • Nowicki, K., Snijders, T.: Estimation and prediction for stochastic block-structures. J. Am. Stat. Assoc. 96, 1077–1087 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  • Pattison, P.E., Robins, G.L.: Handbook of Probability Theory with Applications. Sage, Beverley Hills (2007). Chapter: Probabilistic network theory

    Google Scholar 

  • Shen-Orr, S.S., Milo, R., Mangan, S., Alon, U.: Networks motifs in the transcriptional regulation network of Escherichia coli. Nat. Genet. 31, 64–68 (2002)

    Article  Google Scholar 

  • Stumpf, M., Wiuf, C., May, R.: Subnets of scale-free networks are not scale-free: sampling properties of networks. Proc. Natl. Acad. Sci. USA 102, 4221–4224 (2005)

    Article  Google Scholar 

  • Tanaka, R., Doyle, J.: Some protein interaction data do not exhibit power law statistics. FEBS Lett. 579, 5140–5144 (2005)

    Article  Google Scholar 

  • Zhang, V.L., King, O.D., Wong, S.L., Goldberg, D.S., Tong, A.H.Y., Lesage, G., Andrews, B., Bussey, H., Boone, C., Roth, F.P.: Motifs, themes and thematic maps of an integrated Saccharomyces cerevisiae interaction network. J. Biol. 4, 1–13 (2005)

    Article  Google Scholar 

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Correspondence to S. Robin.

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Daudin, JJ., Picard, F. & Robin, S. A mixture model for random graphs. Stat Comput 18, 173–183 (2008).

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