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Detection of structurally homogeneous subsets in graphs

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Abstract

The analysis of complex networks is a rapidly growing topic with many applications in different domains. The analysis of large graphs is often made via unsupervised classification of vertices of the graph. Community detection is the main way to divide a large graph into smaller ones that can be studied separately. However another definition of a cluster is possible, which is based on the structural distance between vertices. This definition includes the case of community clusters but is more general in the sense that two vertices may be in the same group even if they are not connected. Methods for detecting communities in undirected graphs have been recently reviewed by Fortunato. In this paper we expand Fortunato’s work and make a review of methods and algorithms for detecting essentially structurally homogeneous subsets of vertices in binary or weighted and directed and undirected graphs.

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References

  • Airoldi, E., Blei, D., Fienberg, S., Xing, E.: Mixed membership stochastic blockmodels. J. Mach. Learn. Res. 9, 1981–2014 (2008)

    MATH  Google Scholar 

  • Ambroise, C., Matias, C.: New consistent and asymptotically normal parameter estimates for random-graph mixture models. J. R. Stat. Soc., Ser. B, Stat. Methodol. 74, 3–35 (2011)

    Article  MathSciNet  Google Scholar 

  • Arabie, P., Boorman, S., Levitt, P.: Constructing blockmodels: how and why. J. Math. Psychol. 17, 21–63 (1978). doi:10.1073/pnas.0907096106

    Article  MATH  Google Scholar 

  • Benzecri, J.: L Analyse des Donnees. Volume II. L Analyse des Correspondances. Dunod, Paris (1973)

    Google Scholar 

  • Bickel, P., Chen, A.: A nonparametric view of network models and Newman-Girvan and other modularities. Proc. Natl. Acad. Sci. USA, 1–6 (2010)

  • Brohee, S., Van Helden, J.: Evaluation of clustering algorithms for protein-protein interaction networks. BMC Bioinform. 7(1), 488 (2006)

    Article  Google Scholar 

  • Burt, R.: Cohesion versus structural equivalence as a basis for network subgroups. Sociol. Methods Res. 7(2), 189–212 (1978)

    Article  Google Scholar 

  • Celisse, A., Daudin, J., Pierre, L.: Consistency of maximum likelihood and variational estimators in mixture models for random graphs. Electron. J. Stat. 6, 1847–1899 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  • Choi, D., Wolfe, P., Airoldi, E.: Stochastic blockmodels with growing number of classes. Biometrika 99(2), 273–284 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  • Clauset, A., Newman, M., Moore, C.: Finding community structure in very large networks. Phys. Rev. E 70(6), 066111 (2004)

    Article  Google Scholar 

  • Csardi, G., Nepusz, T.: The igraph software package for complex network research. InterJournal, Complex Syst. 1695, 38 (2006). http://igraph.sf.net

    Google Scholar 

  • Daudin, J.: A review of statistical models for clustering networks with an application to a PPI network. J. Soc. Fr. Stat. 152(2), 111–125 (2011)

    MathSciNet  Google Scholar 

  • Daudin, J., Picard, F., Robin, S.: A mixture model for random graphs. Stat. Comput. 18(2), 173–183 (2008)

    Article  MathSciNet  Google Scholar 

  • Daudin, J.J., Pierre, L., Vacher, C.: Model for heterogeneous random networks using continuous latent variables and an application to a tree-fungus network. Biometrics 66(4), 1043–1051 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  • Decelle, A., Krzakala, F., Moore, C., Zdeborová, L.: Asymptotic analysis of the stochastic block model for modular networks and its algorithmic applications. Phys. Rev. E 84(6), 066106 (2011)

    Article  Google Scholar 

  • Donath, W.E., Hoffman, A.J.: Lower bounds for the partitioning of graphs. IBM J. Res. Dev. 17(5), 420–425 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  • Erosheva, E.: Comparing latent structures of the grade of membership, Rasch and latent class model. Psychometrika 70(4), 619–628 (2005)

    Article  MathSciNet  Google Scholar 

  • Fortunato, S.: Community detection in graphs. Phys. Rep. 486(3–5), 75–174 (2010). http://www.sciencedirect.com/science/article/pii/S0370157309002841. doi:10.1016/j.physrep.2009.11.002

    Article  MathSciNet  Google Scholar 

  • Girvan, M., Newman, M.: Community structure in social and biological networks. Proc. Natl. Acad. Sci. USA 99(12), 7821 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  • Guimera, R., Stouffer, D., Sales-Pardo, M., Leicht, E., Newman, M., Nunes Amaral, L.: Origin of compartmentalization in food webs. Ecology (2010). http://www.esajournals.org/doi/abs/10.1890/09-1175.1. doi:10.1890/09-1175.1

    MATH  Google Scholar 

  • Handcock, M.S., Raftery, A.E., Tantrum, J.: Model-based clustering for social networks. J. R. Stat. Soc. A 170(2), 301–354 (2007)

    Article  MathSciNet  Google Scholar 

  • Harshman, R.: Models for analysis of asymmetrical relationships among N objects or stimuli. In: First Joint Meeting of the Psychometric Society and the Society for Mathematical Psychology. McMaster University, Hamilton, Ontario, August (1978)

    Google Scholar 

  • Hartigan, J.: Clustering Algorithms. Wiley, New York (1975)

    MATH  Google Scholar 

  • Hirschfeld, H.: A connection between correlation and contingency. Proc. Camb. Philos. Soc. 31, 520–524 (1935)

    Article  Google Scholar 

  • Hofman, J.M., Wiggins, C.H.: Bayesian approach to network modularity. Phys. Rev. Lett. 100, 258701 (2008). http://link.aps.org/doi/10.1103/PhysRevLett.100.258701. doi:10.1103/PhysRevLett.100.258701

    Article  Google Scholar 

  • Holland, P., Laskey, K., Leinhardt, K.: Stochastic blockmodels: some first steps. Soc. Netw. 5, 109–137 (1983)

    Article  MathSciNet  Google Scholar 

  • Kiers, H., ten Berge, J., Takane, Y., de Leeuw, J.: A generalization of Takane’s algorithm for DEDICOM. Psychometrika 55(1), 151–158 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  • Latouche, P., Birmelé, E., Ambroise, C.: Overlapping stochastic block models with application to the French political blogosphere. Ann. Appl. Stat. 5(1), 309–336 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  • Lorrain, F., White, H.: Structural equivalence of individuals in social networks. J. Math. Sociol. 1, 49–80 (1971)

    Article  Google Scholar 

  • Manton, K., Woodbury, M., Tolley, H.: In: Statistical Applications Using Fuzzy Sets (1994)

    Google Scholar 

  • Marchette, D., Priebe, C.: Predicting unobserved links in incompletely observed networks. Comput. Stat. Data Anal. 52(3), 1373–1386 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  • Mariadassou, M., Robin, S., Vacher, C.: Uncovering latent structure in valued graphs: a variational approach. Ann. Appl. Stat. 4, 715–742 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  • Newman, M., Girvan, M.: Finding and evaluating community structure in networks. Phys. Rev. E 69(2), 026113 (2004)

    Article  Google Scholar 

  • Picard, F., Miele, V., Daudin, J.J., Cottret, L., Robin, S.: Deciphering the connectivity structure of biological networks using MixNet. BMC Bioinform. 10, S7 (2009)

    Article  Google Scholar 

  • Pons, P., Latapy, M.: Computing communities in large networks using random walks. J. Graph Algorithms Appl. 10(2), 191–218 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  • Raj, A., Wiggins, C.H.: An information-theoretic derivation of min-cut based clustering. IEEE Trans. Pattern Anal. Mach. Intell. 32, 988–995 (2010). doi:10.1109/TPAMI.2009.124

    Article  Google Scholar 

  • Rohe, K., Chatterjee, S., Yu, B.: Spectral clustering and the high-dimensional stochastic block model. Ann. Stat. 39(4), 1878–1915 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  • Sinkkonen, J., Aukia, J., Kaski, S.: Component models for large networks (2008a). arXiv:0803.1628

  • Sinkkonen, J., Aukia, J., Kaski, S.: Inferring vertex properties from topology in large networks (2008b). arXiv:0803.1628v1 [stat.ML]

  • Snijders, T., Nowicki, K.: Estimation and prediction for stochastic blockmodels for graphs with latent block structure. J. Classif. 14(1), 75–100 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  • Trendafilov, N.: GIPSCAL revisited. A projected gradient approach. Stat. Comput. 12(2), 135–145 (2002)

    Article  MathSciNet  Google Scholar 

  • Van Dongen, S.: Graph clustering by flow simulation. University of Utrecht 275 (2000)

  • Von Luxburg, U.: A tutorial on spectral clustering. Stat. Comput. 17(4), 395–416 (2007)

    Article  MathSciNet  Google Scholar 

  • White, H.C., Boorman, S.A., Breiger, R.L.: Social structure from multiple networks. Am. J. Sociol. 81, 730–780 (1976)

    Article  Google Scholar 

  • Winship, C., Mandel, M.: Roles and positions: a critique and extension of the blockmodeling approach. In: Sociological Methodology (1983)

    Google Scholar 

  • Zachary, W.: An information flow model for conflict and fission in small groups. J. Anthropol. Res. 33(4), 452–473 (1977)

    Google Scholar 

Download references

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Correspondence to Jean-Benoist Leger.

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Leger, JB., Vacher, C. & Daudin, JJ. Detection of structurally homogeneous subsets in graphs. Stat Comput 24, 675–692 (2014). https://doi.org/10.1007/s11222-013-9395-3

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