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Searching for Network Modules

  • Giovanni RossiEmail author
Conference paper
Part of the Lecture Notes in Networks and Systems book series (LNNS, volume 70)

Abstract

When analyzing complex networks, a key target is to uncover their modular structure, which means searching for a family of node subsets spanning each an exceptionally dense subnetwork. Objective function-based graph clustering procedures such as modularity maximization output a partition of nodes, i.e. a family of pair-wise disjoint subsets, although single nodes are likely to be included in multiple or overlapping modules. Thus in fuzzy clustering approaches each node may be included in different modules with different [0, 1]-ranged memberships. This work proposes a novel type of objective function for graph clustering, in the form of a multilinear polynomial extension whose coefficients are determined by network topology. It may be seen as a potential, taking values on fuzzy clusterings or families of fuzzy subsets of nodes over which every node distributes a unit membership. If suitably parameterized, this potential attains its maximum when every node concentrates its all unit membership on some module. Maximizers thus remain partitions, while the original discrete optimization problem is turned into a continuous version allowing to conceive alternative search strategies. The instance of the problem being a pseudo-Boolean function assigning real-valued cluster scores to node subsets, modularity maximization is employed to exemplify a so-called quadratic form, in that the scores of singletons and pairs also fully determine the scores of larger clusters, while the resulting multilinear polynomial potential function has degree 2. After considering further quadratic instances, different from modularity and obtained by interpreting network topology in alternative manners, a greedy local-search strategy for the continuous framework is analytically compared with an existing greedy agglomerative procedure for the discrete case. Overlapping is finally discussed in terms of multiple runs, i.e. several local searches with different initializations.

Keywords

Modularity Fuzzy clustering Pseudo-Boolean function 

References

  1. 1.
    Adamcsek, B., Palla, G., Farkas, I.J., Derényi, I., Vicsek, T.: CFinder: locating cliques and overlapping modules in biological networks. Bioinformatics 22(8), 1021–1023 (2006)CrossRefGoogle Scholar
  2. 2.
    Ahn, Y.Y., Bagrow, J.P., Lehmann, S.: Link communities reveal multiscale complexity in networks. Nature 466, 761–764 (2010)CrossRefGoogle Scholar
  3. 3.
    Aigner, M.: Combinatorial Theory. Springer, Berlin (1997)CrossRefGoogle Scholar
  4. 4.
    Altaf-Ul-Amin, M., Shinbo, Y., Mihara, K., Kurokawa, K., Kanaya, S.: Development and implementation of an algorithm for detection of protein complexes in large interaction networks. BMC Bioinform. 7(207) (2006)Google Scholar
  5. 5.
    Asur, S., Ucar, D., Parthasarathy, S.: An ensemble framework for clustering protein-protein interaction networks. Bioinformatics 23, i29–i40 (2007)CrossRefGoogle Scholar
  6. 6.
    Bollobás, B., Riordan, O.M.: Mathematical results on scale-free random graphs. In: Bornholdt, S., Schuster, H.G. (eds.) Handbook of Graphs and Networks: from the Genome to the Internet, pp. 1–34. Wiley, Berlin (2003)zbMATHGoogle Scholar
  7. 7.
    Boros, E., Hammer, P.: Pseudo-Boolean optimization. Discrete Appl. Math. 123, 155–225 (2002)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Brandes, U., Delling, D., Gaertler, M., Görke, R., Hoefer, M., Nikoloski, Z., Wagner, D.: On modularity clustering. IEEE Trans. Knowl. Data Eng. 20(2), 172–188 (2007)CrossRefGoogle Scholar
  9. 9.
    Brower, A.E., Haemers, W.H.: Spectra of Graphs. Springer, New York (2011)Google Scholar
  10. 10.
    Chakrabarti, M., Heath, L., Ramakrishnan, N.: New methods to generate massive synthetic networks. cs. SI, arXiv:1705.08473 v1 (2017)
  11. 11.
    Diestel, R.: Graph Theory. Springer, New York (2010)CrossRefGoogle Scholar
  12. 12.
    Fortunato, S.: Community detection in graphs. Phys. Rep. 486(3–5), 75–174 (2010)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Freeman, T.C., Goldovsky, L., Brosch, M., van Dongen, S., Mazire, P., Grocock, R.J., Freilich, S., Thornton, J., Enright, A.J.: Construction, visualisation, and clustering of transcription networks from microarray expression data. PLOS Comp. Biol. 3(10–e206), 2032–2042 (2007)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Gilboa, I., Lehrer, E.: Global games. Int. J. Game Theory 20, 120–147 (1990)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Gilboa, I., Lehrer, E.: The value of information—an axiomatic approach. J. Math. Econ. 20(5), 443–459 (1991)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics—A Foundation for Computer Science, 2nd edn. Addison-Wesley, Reading (1994)Google Scholar
  17. 17.
    Lancichinetti, A., Fortunato, S., Kertész, J.: Detecting the overlapping and hierarchical community structure in complex networks. New J. Phys. 11(3), 033015 (2009)CrossRefGoogle Scholar
  18. 18.
    Lancichinetti, A., Fortunato, S., Radicchi, F.: Benchmark graphs for testing community detection algorithms. Phys. Rev. E 78(4), 046110 (2008)CrossRefGoogle Scholar
  19. 19.
    Lei, X., Wu, S., Ge, L., Zhang, A.: Clustering and overlapping modules detection in PPI network based on IBFO. Proteomics 13(2), 278–290 (2013)CrossRefGoogle Scholar
  20. 20.
    Li, Y., Shang, Y., Yang, Y.: Clustering coefficients of large networks. Inf. Sci. 382–383, 350–358 (2017)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Miyamoto, S., Ichihashi, H., Honda, K.: Algorithms for Fuzzy Clustering. Springer, Berlin (2008)zbMATHGoogle Scholar
  22. 22.
    Nepusz, T., Petróczi, A., Négyessy, L., Baszó, F.: Fuzzy communities and the concept of bridgeness in complex networks. Phys. Rev. E 77(1), 016107 (2008)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Newman, M.E.J.: The structure and function of complex networks. SIAM Rev. 45(2), 167–256 (2003)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Newman, M.E.J.: Fast algorithm for detecting communities in networks. Phys. Rev. E 69(6), 066133 (2004)CrossRefGoogle Scholar
  25. 25.
    Newman, M.E.J.: Modularity and community structure in networks. PNAS 103, 8577–8582 (2006)CrossRefGoogle Scholar
  26. 26.
    Newman, M.E.J.: Random graphs with clustering. Phys. Rev. Lett. 103(5), 058701(4) (2009)Google Scholar
  27. 27.
    Newman, M.E.J., Barabási, A.L., Watts, D.J.: The Structure and Dynamics of Networks. Princeton University Press, Princeton (2006)zbMATHGoogle Scholar
  28. 28.
    Newman, M.E.J., Park, J.: Why social networks are different from other types of networks. Phys. Rev. E 68(3), 036122 (2003)CrossRefGoogle Scholar
  29. 29.
    Pereira-Leal, J.B., Enright, A.J., Ouzounis, C.A.: Detection of functional modules from protein interaction networks. PROTEINS: Struct. Funct. Bioinform. 54, 49–57 (2004)CrossRefGoogle Scholar
  30. 30.
    Reichardt, J., Bornholdt, S.: Detecting fuzzy community structures in complex networks with a Potts model. Phys. Rev. Lett. 93(21), 218701 (2004)CrossRefGoogle Scholar
  31. 31.
    Rossi, G.: Multilinear objective function-based clustering. In: Proceedings of 7th IJCCI, vol. 2. Fuzzy Computation Theory and Applications, pp. 141–149 (2015)Google Scholar
  32. 32.
    Rossi, G.: Near-Boolean optimization—a continuous approach to set packing and partitioning. In: LNCS 10163 Pattern Recognition Applications and Methods, pp. 60–87. Springer (2017)Google Scholar
  33. 33.
    Rota, G.C.: The number of partitions of a set. Am. Math. Monthly 71, 499–504 (1964)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Rota, G.C.: On the foundations of combinatorial theory I: theory of Möbius functions. Z. Wahrscheinlichkeitsrechnung u. verw. Geb. 2, 340–368 (1964)Google Scholar
  35. 35.
    Rotta, R., Noack, A.: Multilevel local search clustering algorithms for modularity clustering. ACM J. Exp. Algorithmics 16(2), 2.3:1–27 (2011)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Schaeffer, S.E.: Graph clustering. Comput. Sci. Rev. 1, 27–64 (2007)CrossRefGoogle Scholar
  37. 37.
    Schmidt, M.C., Samatova, N.F., Thomas, K., Park, B.H.: A scalable, parallel algorithm for maximal clique enumeration. J. Parallel Distrib. Comput. 69(4), 417–428 (2009)CrossRefGoogle Scholar
  38. 38.
    Sharan, R., Ulitsky, I., Shamir, R.: Network-based prediction of protein function. Mol. Syst. Biol. 3, 88 (2007)CrossRefGoogle Scholar
  39. 39.
    Stanley, R.: Modular elements of geometric lattices. Algebra Universalis 1, 214–217 (1971)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Szalay-Bekő, M., Palotai, R., Szappanos, B., Kovás, I.A., Papp, B., Csermely, P.: Hierarchical layers of overlapping network modules and community centrality. Bioinformatics 28(16), 2202–2204 (2012)CrossRefGoogle Scholar
  41. 41.
    Vlasblom, J., Wodak, S.J.: Markov clustering versus affinity propagation for the partitioning of protein interaction graphs. BMC Bioinform. 10, 99 (2009)CrossRefGoogle Scholar
  42. 42.
    Wang, J., Run, J., Li, M., Wu, F.X.: Identification of hierarchical and overlapping functional modules in PPI networks. IEEE Trans. Nanobiosci. 11(4), 386–393 (2012)CrossRefGoogle Scholar
  43. 43.
    Wu, H., Gao, L., Dong, J., Jang, X.: Detecting overlapping protein complexes by rough-fuzzy clustering in protein-protein networks. Plos ONE 9(3–e91856) (2014)CrossRefGoogle Scholar
  44. 44.
    Xie, J., Kelley, S., Szymanski, B.K.: Overlapping community detection in networks: the state of the art and a comparative study. ACM Comput. Surv. 45(43), 43:1–43:35 (2012)CrossRefGoogle Scholar
  45. 45.
    Yu, T., Liu, M.: A linear time algorithm for maximal clique enumeration in large sparse graphs. Inf. Process. Lett. 125, 35–40 (2017)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Zhang, S., Wang, R.S., Zhang, X.S.: Identification of overlapping community structure in complex networks using fuzzy c-means clustering. Phisica A 374, 483–490 (2007)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of Computer Science and Engineering DISIUniversity of BolognaBolognaItaly

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