Journal of Statistical Physics

, Volume 151, Issue 3–4, pp 689–706 | Cite as

Overlapping Modularity at the Critical Point of k-Clique Percolation

  • Bálint TóthEmail author
  • Tamás Vicsek
  • Gergely Palla


One of the most remarkable social phenomena is the formation of communities in social networks corresponding to families, friendship circles, work teams, etc. Since people usually belong to several different communities at the same time, the induced overlaps result in an extremely complicated web of the communities themselves. Thus, uncovering the intricate community structure of social networks is a non-trivial task with great potential for practical applications, gaining a notable interest in the recent years. The Clique Percolation Method (CPM) is one of the earliest overlapping community finding methods, which was already used in the analysis of several different social networks. In this approach the communities correspond to k-clique percolation clusters, and the general heuristic for setting the parameters of the method is to tune the system just below the critical point of k-clique percolation. However, this rule is based on simple physical principles and its validity was never subject to quantitative analysis. Here we examine the quality of the partitioning in the vicinity of the critical point using recently introduced overlapping modularity measures. According to our results on real social and other networks, the overlapping modularities show a maximum close to the critical point, justifying the original criteria for the optimal parameter settings.


Community finding Clique percolation Modularity Critical point 



This work was supported by the European Union and co-financed by the European Social Fund (grant agreement no. TAMOP 4.2.1/B-09/1/KMR-2010-0003) and by the Hungarian National Science Fund (OTKA K105447).


  1. 1.
    Adamcsek, B., Palla, G., Farkas, I.J., Derényi, I., Vicsek, T.: CFinder: locating cliques and overlapping modules in biological networks. Bioinformatics 22, 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) ADSCrossRefGoogle Scholar
  3. 3.
    Aiello, L.M., Barrat, A., Cattuto, C., Ruffo, G., Schifanella, R.: Link creation and profile alignment in the aNobii social network. In: Proceedings of the Second IEEE International Conference on Social Computing SocialCom 2010, pp. 249–256 (2010) CrossRefGoogle Scholar
  4. 4.
    Albert, R., Barabási, A.L.: Statistical mechanics of complex networks. Rev. Mod. Phys. 74, 47–97 (2002) ADSzbMATHCrossRefGoogle Scholar
  5. 5.
    Avetisov, V.A., Chertovich, A.V., Nechaev, S.K., Vasilyev, O.A.: On scale-free and poly-scale behaviors of random hierarchical network. J. Stat. Mech. 2009, P07008 (2009) CrossRefGoogle Scholar
  6. 6.
    Barabási, A.L., Albert, R.: Emergence of scaling in random networks. Science 286, 509–512 (1999) MathSciNetADSCrossRefGoogle Scholar
  7. 7.
    Barabási, A.L., Jeong, H., Néda, Z., Ravasz, E., Schubert, A., Vicsek, T.: Evolution of the social network of scientific collaborations. Phys. A, Stat. Mech. Appl. 311, 590–614 (2002) zbMATHCrossRefGoogle Scholar
  8. 8.
    Barabási, A.L., Ravasz, E., Vicsek, T.: Deterministic scale-free networks. Physica A 299, 559–564 (2001) ADSzbMATHCrossRefGoogle Scholar
  9. 9.
    Barrat, A., Barthelemy, M., Vespignani, A.: Weighted evolving networks: coupling topology and weight dynamics. Phys. Rev. Lett. 92, 228701 (2004) ADSCrossRefGoogle Scholar
  10. 10.
    Blondel, V., Guillaume, J.L., Lambiotte, R., Lefebvre, E.: Fast unfolding of community hierarchies in large networks. J. Stat. Mech. 2008, P10008 (2008) CrossRefGoogle Scholar
  11. 11.
    Blondel, V., Krings, G., Thomas, I.: Regions and borders of mobile telephony in Belgium and in the Brussels Metropolitan zone. Bruss. Stud. (2010). doi: 2078.1/95261. ISSN 2031-0293 Google Scholar
  12. 12.
    Börner, K., Maru, J.T., Goldstone, R.L.: The simultaneous evolution of author and paper networks. Proc. Natl. Acad. Sci. USA 101(Suppl. 1), 5266–5273 (2004) CrossRefGoogle Scholar
  13. 13.
    Chau, M., Xu, J.: Mining communities and their relationships in blogs: a study of online hate groups. Int. J. Hum.-Comput. Stud. 65, 57–70 (2007) CrossRefGoogle Scholar
  14. 14.
    Chen, D., Shang, M., Lv, Z., Fu, Y.: Detecting overlapping communities of weighted networks via a local algorithm. Physica A 389, 4177–4187 (2010) ADSCrossRefGoogle Scholar
  15. 15.
    Clauset, A., Moore, C., Newman, M.E.J.: Hierarchical structure and the prediction of missing links in networks. Nature 453, 98–101 (2008) ADSCrossRefGoogle Scholar
  16. 16.
    Derényi, I., Palla, G., Vicsek, T.: Clique percolation in random networks. Phys. Rev. Lett. 94, 160202 (2005) ADSCrossRefGoogle Scholar
  17. 17.
    Donetti, L., Muñoz, M.A.: Detecting network communities: a new systematic and efficient algorithm. J. Stat. Mech. 2004, P10012 (2004) CrossRefGoogle Scholar
  18. 18.
    Ebel, H., Mielsch, L.I., Bornholdt, S.: Scale-free topology of e-mail networks. Phys. Rev. E 66, 035103 (2002) ADSCrossRefGoogle Scholar
  19. 19.
    Eriksen, K.A., Simonsen, I., Maslov, S., Sneppen, K.: Modularity and extreme edges of the internet. Phys. Rev. Lett. 90, 148701 (2003) ADSCrossRefGoogle Scholar
  20. 20.
    Fortunato, S.: Community detection in graphs. Phys. Rep. 486, 75–174 (2010) MathSciNetADSCrossRefGoogle Scholar
  21. 21.
    Fortunato, S., Barthelemy, M.: Resolution limit in community detection. Proc. Natl. Acad. Sci. USA 104, 36–41 (2007) ADSCrossRefGoogle Scholar
  22. 22.
    Girvan, M., Newman, M.E.J.: Community structure in social and biological networks. Proc. Natl. Acad. Sci. USA 99, 7821–7826 (2002) MathSciNetADSzbMATHCrossRefGoogle Scholar
  23. 23.
    González, M.C., Herrmann, H.J., Kertész, J., Vicsek, T.: Community structure and ethnic preferences in school friendship networks. Phys. A, Stat. Mech. Appl. 379, 307–316 (2007) CrossRefGoogle Scholar
  24. 24.
    Guimerá, R., Danon, L., Díaz-Guilera, A., Giralt, F., Arenas, A.: Self-similar community structure in a network of human interactions. Phys. Rev. E 68, 065103 (2003) ADSCrossRefGoogle Scholar
  25. 25.
    Kossinets, G., Watts, D.J.: Empirical analysis of an evolving social network. Science 311, 88–90 (2006) MathSciNetADSzbMATHCrossRefGoogle Scholar
  26. 26.
    Kumar, R., Novak, J., Tomkins, A.: Structure and evolution of online social networks. In: Yu, P.S.S., Han, J., Faloutsos, C. (eds.) Link Mining: Models, Algorithms, and Applications, pp. 337–357. Springer, New York (2010) CrossRefGoogle Scholar
  27. 27.
    Lambiotte, R., Blondel, V.D., de Kerchove, C., Huens, E., Prieur, C., Smoreda, Z., Dooren, P.V.: Geographical dispersal of mobile communication networks. Phys. A, Stat. Mech. Appl. 387, 5317–5325 (2008) CrossRefGoogle Scholar
  28. 28.
    Lancichinetti, A., Fortunato, S., Kertész, J.: Detecting the overlapping and hierarchical community structure in complex networks. New J. Phys. 11, 033015 (2009) ADSCrossRefGoogle Scholar
  29. 29.
    Lázár, A., Ábel, D., Vicsek, T.: Modularity measure of networks with overlapping communities. Europhys. Lett. 90, 18001 (2010) ADSCrossRefGoogle Scholar
  30. 30.
    Mendes, J.F.F., Dorogovtsev, S.N.: Evolution of Networks: From Biological Nets to the Internet and WWW. Oxford University Press, Oxford (2003) zbMATHGoogle Scholar
  31. 31.
    Mislove, A., Marcon, M., Gummadi, K.P., Druschel, P., Bhattacharjee, B.: Measurement and analysis of online social networks. In: Proceedings of the 7th ACM SIGCOMM Conference on Internet Measurement, IMC’07, pp. 29–42. ACM, New York (2007) CrossRefGoogle Scholar
  32. 32.
    Molloy, M., Reed, B.: A critical point for random graphs with a given degree sequence. Random Struct. Algorithms 6, 161–180 (1995) MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Nazir, A., Raza, S., Chuah, C.N.: Unveiling Facebook: a measurement study of social network based applications. In: Proceedings of the 8th ACM SIGCOMM Conference on Internet Measurement, IMC’08, pp. 43–56. ACM, New York (2008) CrossRefGoogle Scholar
  34. 34.
    Nelson, D.L., McEvoy, C.L., Schreiber, T.A.: The university of south Florida word association rhyme, and word fragment norms (1998). URL
  35. 35.
    Nepusz, T., Petróczi, A., Négyessy, L., Bazsó, F.: Fuzzy communities and the concept of bridgeness in complex networks. Phys. Rev. E 77, 016107 (2008) MathSciNetADSCrossRefGoogle Scholar
  36. 36.
    Newman, M.E.J.: The structure of scientific collaboration networks. Proc. Natl. Acad. Sci. USA 98, 404–409 (2001) MathSciNetADSzbMATHCrossRefGoogle Scholar
  37. 37.
    Newman, M.E.J.: Coauthorship networks and patterns of scientific collaboration. Proc. Natl. Acad. Sci. USA 101(Suppl. 1), 5200–5205 (2004) ADSCrossRefGoogle Scholar
  38. 38.
    Newman, M.E.J., Forrest, S., Balthrop, J.: Email networks and the spread of computer viruses. Phys. Rev. E 66, 035101 (2002) ADSCrossRefGoogle Scholar
  39. 39.
    Newman, M.E.J., Girvan, M.: Finding and evaluating community structure in networks. Phys. Rev. E 69, 026113 (2004) ADSCrossRefGoogle Scholar
  40. 40.
    Nicosia, V., Mangioni, G., Carchiolo, V., Malgeri, M.: Extending the definition of modularity to directed graphs with overlapping communities. J. Stat. Mech. 2009, P03024 (2009) CrossRefGoogle Scholar
  41. 41.
    Onnela, J.P., Chakraborti, A., Kaski, K., Kertész, J., Kanto, A.: Dynamics of market correlations: taxonomy and portfolio analysis. Phys. Rev. E 68, 056110 (2003) ADSCrossRefGoogle Scholar
  42. 42.
    Onnela, J.P., Saramäki, J., Hyvönen, J., Szabó, G., Lazer, D., Kaski, K., Kertész, J., Barabási, A.L.: Structure and tie strengths in mobile communication networks. Proc. Natl. Acad. Sci. USA 104, 7332–7336 (2007) ADSCrossRefGoogle Scholar
  43. 43.
    Onnela, J.P., Saramäki, J., Hyvönen, J., Szabó, G., de Menezes, M.A., Kaski, K., Barabási, A.L., Kertész, J.: Analysis of a large-scale weighted network of one-to-one human communication. New J. Phys. 9, 179 (2007) ADSCrossRefGoogle Scholar
  44. 44.
    Opsahl, T., Panzarasa, P.: Clustering in weighted networks. Soc. Netw. 31, 155–163 (2009) CrossRefGoogle Scholar
  45. 45.
    Palla, G., Barabási, A.L., Vicsek, T.: Quantifying social group evolution. Nature 446, 664–667 (2007) ADSCrossRefGoogle Scholar
  46. 46.
    Palla, G., Derényi, I., Farkas, I., Vicsek, T.: Uncovering the overlapping community structure of complex networks in nature and society. Nature 435, 814–818 (2005) ADSCrossRefGoogle Scholar
  47. 47.
    Palla, G., Derényi, I., Vicsek, T.: The critical point of k-clique percolation in the Erdõs-Rényi graph. J. Stat. Phys. 128, 219–227 (2007) MathSciNetADSzbMATHCrossRefGoogle Scholar
  48. 48.
    Palla, G., Lovász, L., Vicsek, T.: Multifractal network generator. Proc. Natl. Acad. Sci. USA 107, 7640–7645 (2010) ADSCrossRefGoogle Scholar
  49. 49.
    Palla, G., Pollner, P., Vicsek, T.: Rotated multifractal network generator. J. Stat. Mech. 2011, P02003 (2011) CrossRefGoogle Scholar
  50. 50.
    Ramasco, J.J., Dorogovtsev, S.N., Pastor-Satorras, R.: Self-organization of collaboration networks. Phys. Rev. E 70, 036106 (2004) ADSCrossRefGoogle Scholar
  51. 51.
    Ravasz, E., Somera, A.L., Mongru, D.A., Oltvai, Z., Barabási, A.L.: Hierarchical organization of modularity in metabolic networks. Science 297, 1551 (2002) ADSCrossRefGoogle Scholar
  52. 52.
    Reichardt, J., Bornholdt, S.: Detecting fuzzy community structures in complex networks with a Potts model. Phys. Rev. Lett. 93, 218701 (2004) ADSCrossRefGoogle Scholar
  53. 53.
    Ronhovde, P., Nussinov, Z.: Multiresolution community detection for megascale networks by information-based replica correlations. Phys. Rev. E 80, 016109 (2009) ADSCrossRefGoogle Scholar
  54. 54.
    Rosvall, M., Bergstrom, C.T.: An information-theoretic framework for resolving community structure in complex networks. Proc. Natl. Acad. Sci. USA 104, 7327 (2007) ADSCrossRefGoogle Scholar
  55. 55.
    Schifanella, R., Barrat, A., Cattuto, C., Markines, B., Menczer, F.: Folks in folksonomies: social link prediction from shared metadata. In: WSDM’10 Proceedings of the Third ACM International Conference on Web Search and Data Mining, pp. 271–280 (2010) CrossRefGoogle Scholar
  56. 56.
    Scott, J.: Social Network Analysis: A Handbook, 2nd edn. Sage, London (2000) Google Scholar
  57. 57.
    Seshadri, M., Machiraju, S., Sridharan, A., Bolot, J., Faloutsos, C., Leskovec, J.: Mobile call graphs: beyond power-law and lognormal distributions. In: Proceedings of the 14th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD’08, pp. 596–604. ACM, New York (2008) CrossRefGoogle Scholar
  58. 58.
    Shen, H., Cheng, X., Cai, K., Hu, M.B.: Detect overlapping and hierarchical community structure in networks. Physica A 388, 1706–1712 (2009) ADSCrossRefGoogle Scholar
  59. 59.
    Shen, H.W., Cheng, X.Q., Guo, J.F.: Quantifying and identifying the overlapping community structure in networks. J. Stat. Mech. 2009, P07042 (2009) CrossRefGoogle Scholar
  60. 60.
    Song, C.M., Havlin, S., Makse, H.A.: Self-similarity of complex networks. Nature 433, 392–395 (2005) ADSCrossRefGoogle Scholar
  61. 61.
    Spirin, V., Mirny, L.A.: Protein complexes and functional modules in molecular networks. Proc. Natl. Acad. Sci. USA 100, 12123 (2003) ADSCrossRefGoogle Scholar
  62. 62.
    Gavin, A.-C., et al.: Functional organization of the yeast proteome by systematic analysis of protein complexes. Nature 415, 141 (2002) ADSCrossRefGoogle Scholar
  63. 63.
    Vicsek, T.: The bigger picture. Nature 418, 131 (2002) ADSCrossRefGoogle Scholar
  64. 64.
    Vicsek, T., Zafeiris, A.: Collective motion. Phys. Rep. 517, 71–140 (2012) ADSCrossRefGoogle Scholar
  65. 65.
    Wagner, C.S.: Six case studies of international collaboration in science. Scientometrics 62, 3–26 (2005) CrossRefGoogle Scholar
  66. 66.
    Wagner, C.S., Leydesdorff, L.: Network structure, self-organization, and the growth of international collaboration in science. Res. Policy 34, 1608–1618 (2005) CrossRefGoogle Scholar
  67. 67.
    Watts, D.J., Dodds, P.S., Newman, M.E.J.: Identity and search in social networks. Science 296, 1302 (2002) ADSCrossRefGoogle Scholar
  68. 68.
    Watts, D.J., Strogatz, S.H.: Collective dynamics of ‘small-world’ networks. Nature 393, 440–442 (1998) ADSCrossRefGoogle Scholar
  69. 69.
    Wilkinson, D.M., Huberman, B.A.: A method for finding communities of related genes. Proc. Natl. Acad. Sci. USA 101, 5241 (2004) ADSCrossRefGoogle Scholar
  70. 70.
    Zhou, Y., Davis, J.: Discovering web communities in the blogspace. In: Proceedings of the 40th Annual Hawaii International Conference on System Sciences, HICSS’07, p. 85c. IEEE Comput. Soc., Los Alamitos (2007). doi: 10.1109/HICSS.2007.177 Google Scholar
  71. 71.
    Zhu, C., Kuh, A., Wang, J., Wilde, P.D.: Analysis of an evolving email network. Phys. Rev. E 74, 046109 (2006) ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Dept. of Biological PhysicsEötvös Univ.BudapestHungary
  2. 2.Statistical and Biological Physics Research Group of HASBudapestHungary

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