Journal of Systems Science and Complexity

, Volume 23, Issue 5, pp 1024–1036 | Cite as

Detecting community structure: From parsimony to weighted parsimony

Article

Abstract

Community detection has attracted a great deal of attention in recent years. A parsimony criterion for detecting this structure means that as minimal as possible number of inserted and deleted edges is needed when we make the network considered become a disjoint union of cliques. However, many small groups of nodes are obtained by directly using this criterion to some networks especially for sparse ones. In this paper we propose a weighted parsimony model in which a weight coefficient is introduced to balance the inserted and deleted edges to ensure the obtained subgraphs to be reasonable communities. Some benchmark testing examples are used to validate the effectiveness of the proposed method. It is interesting that the weight here can be determined only by the topological features of the network. Meanwhile we make some comparison of our model with maximizing modularity Q and modularity density D on some of the benchmark networks, although sometimes too many or a little less numbers of communities are obtained with Q or D, a proper number of communities are detected with the weighted model. All the computational results confirm its capability for community detection for the small or middle size networks.

Key words

Cliques community detection complex networks parsimony 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    M. Faloutsos, P. Faloutsos, and C. Faloutsos, On power-law relationships of the Internet topology, Comput. Commun. Rev., 1999, 29: 251–262.CrossRefGoogle Scholar
  2. [2]
    M. E. J. Newman and J. Park, Why social networks are different from other types of networks, Phys. Rev. E, 2003, 68: 036122.CrossRefGoogle Scholar
  3. [3]
    J. Scott, Social Network Analysis: A Handbook, 2nd ed., Sage Publications, London, 2000.Google Scholar
  4. [4]
    E. Almaas, B. Kovács, T. Vicsek, et al., Global organization of metabolic fluxes in the bacterium Escherichia coli, Nature, 2004, 427: 839–843.CrossRefGoogle Scholar
  5. [5]
    F. Rao and A. Caflau]isch, The protein folding network, J. Mol. Biol., 2004, 342: 299–306.CrossRefGoogle Scholar
  6. [6]
    J. A. Dunne, R. J. Williams, and N. D. Martinez, Food-web structure and network theory: The role of connectance and size, Proc. Natl. Acad. Sci., 2002, 99: 12917–12922.CrossRefGoogle Scholar
  7. [7]
    M. Girvan and M. E. J. Newman, Community structure in social and biological networks, Proc. Natl. Acad. Sci., 2002, 99: 7821–7826.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    M. E. J. Newman, Finding community structure in networks using the eigenvectors of matrices, Phys. Rev. E, 2006, 74: 036104.CrossRefMathSciNetGoogle Scholar
  9. [9]
    M. E. J. Newman, Modularity and community structure in networks, Proc. Natl Acad. Sci., 2006, 103: 8577–8582.CrossRefGoogle Scholar
  10. [10]
    M. E. J. Newman and M. Girvan, Finding and evaluating community structure in networks, Phys. Rev. E, 2004, 69: 026113.CrossRefGoogle Scholar
  11. [11]
    S. Fortunatoa and C. Castellano, Community structure in graphs, arXiv: 0712.2716 [physics.socph], 2007.Google Scholar
  12. [12]
    M. A. Porter, J. P. Onnela, and P. J. Mucha, Communities in networks, arXiv: 0902.3788 [physics.socph], 2009.Google Scholar
  13. [13]
    W. Y. C. Chen, A. W. M. Dress, and W. Q. Yu, Checking the reliability of a linear-programming based approach towards detecting community structures in networks, IET Syst. Biol., 2007, 1: 286–291.CrossRefGoogle Scholar
  14. [14]
    W. Y. C. Chen, A.W. M. Dress, and W. Q. Yu, Community structures of networks, Math. Comput. Sci., 2008, 1: 441–457.MATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    A. Clauset, C. Moore, and M. E. J. Newman, Hierarchical structure and the prediction of missing links in networks, Nature, 2008, 453: 98–101.CrossRefGoogle Scholar
  16. [16]
    W. E, T. Li, and E. Vanden-Eijnden, Optimal partition and effective dynamics of complex networks, Proc. Natl. Acad. Sci., 2008, 105: 7907–7912.CrossRefMathSciNetGoogle Scholar
  17. [17]
    M. Rosvall and C. T. Bergstrom, Maps of random walks on complex networks reveal community structure, Proc. Natl. Acad. Sci., 2008, 105: 1118–1123.CrossRefGoogle Scholar
  18. [18]
    J. Zhang, S. Zhang, and X. S. Zhang, Detecting community structure in complex networks based on a measure of information discrepancy, Physica A, 2008, 387: 1675–1682.CrossRefGoogle Scholar
  19. [19]
    Z. Li, S. Zhang, R. S. Wang, X. S. Zhang, and L. Chen, Quantitative function for community detection, Phys. Rev. E, 2008, 77: 036109.CrossRefGoogle Scholar
  20. [20]
    J. Reichardt and S. Bornholdt, Detecting fuzzy community structures in complex networks with a Potts model, Phys. Rev. Lett., 2004, 93: 218701.CrossRefGoogle Scholar
  21. [21]
    J. Reichardt and S. Bornholdt, Statistical mechanics of community detection, Phys. Rev. E, 2006, 74: 016110.CrossRefMathSciNetGoogle Scholar
  22. [22]
    P. Ronhovde and Z. Nussinov, Local resolution-limit-free Potts model for community detection, arXiv: 0803.2548 [physics.soc-ph], 2008.Google Scholar
  23. [23]
    J. M. Kumpula, M. Kivelä, K. Kaski, and J. Saramäki, A sequential algorithm for fast clique percolation, arXiv: 0805.1449 [physics.soc-ph], 2008.Google Scholar
  24. [24]
    G. Palla, I. Derényi, I. Farkas, and T. Vicsek, Uncovering the overlapping community structure of complex networks in nature and society, Nature, 2005, 435: 814–818.CrossRefGoogle Scholar
  25. [25]
    J. Zhang, S. Zhang, and X. S. Zhang, Comparative study on a class of evaluation indices for community detection, Lecture notes in operations research, Optimization and Systems Biology, 2008, 9: 294–303.Google Scholar
  26. [26]
    S. Fortunato and M. Barthlemy, Resolution limit in community detection, Proc. Natl. Acad. Sci., 2007, 104: 36–41.CrossRefGoogle Scholar
  27. [27]
    M. Rosvall and C. T. Bergstrom, An information-theoretic framework for resolving community structure in complex networks, Proc. Natl. Acad. Sci., 2007, 104: 7327–7331.CrossRefGoogle Scholar
  28. [28]
    M. Grötschel and Y. Wakabayashi, A cutting plane algorithm for a clustering problem, Mathematical Programming, 1989, 45: 59–96.MATHCrossRefMathSciNetGoogle Scholar
  29. [29]
    M. Grötschel and Y. Wakabayashi, Facets of the clique partitioning polytope, Mathematical Programming, 1990, 47: 367–387.MATHCrossRefMathSciNetGoogle Scholar
  30. [30]
    R. Guimerà and L. A. N. Amaral, Functional cartography of complex metabolic networks, Nature, 2005, 433: 895–900.CrossRefGoogle Scholar
  31. [31]
    W. W. Zachary, An information flow model for conflict and fission in small groups, J. Anthropol. Res., 1977, 33: 452–473.Google Scholar
  32. [32]
    M. E. J. Newman, Detecting community structure in networks, Eur. Phys. J. B, 2004, 38: 321–330.CrossRefGoogle Scholar
  33. [33]
    F. Radicchi, C. Castellano, F. Cecconi, et al., Defining and identifying communities in networks, Proc. Natl Acad. Sci., 2004, 101: 2658–2663.CrossRefGoogle Scholar
  34. [34]
    F. Wu and B. A. Huberman, Finding communities in linear time: a physics approach, Eur. Phys. J. B., 2004, 38: 331–338.CrossRefGoogle Scholar
  35. [35]
    D. J. Watts and S. Strogatz, Collective dynamics of ’small-world’ networks, Nature, 1998, 393: 440–442.CrossRefGoogle Scholar
  36. [36]
    E. Ravasz, A. L. Somera, D. A. Mongru, et al., Hierarchical organization of modularity in metabolic networks, Science, 2002, 297: 1551–1555.CrossRefGoogle Scholar
  37. [37]
    A. Lancichinetti, S. Fortunato, and J. Kertész, Detecting the overlapping and hierarchical community structure in complex networks, New J. Phys., 2009, 11: 033015.CrossRefGoogle Scholar
  38. [38]
    S. Zhang, R. S. Wang, and X. S. Zhang, Identification of overlapping community structure in complex networks using fuzzy c-means clustering, Physica A, 2007, 374: 483–490.CrossRefGoogle Scholar
  39. [39]
    S. Zhang, R. S. Wang, and X. S. Zhang, Uncovering fuzzy community structure in complex networks, Phys. Rev. E, 2007, 76: 046103.CrossRefGoogle Scholar
  40. [40]
    D. Lusseau, K. Schneider, O. J. Boisseau, et al., The bottlenose dolphin community of Doubtful Sound features a large proportion of long-lasting associations, Behav. Ecol. Sociobiol., 2003, 54: 396–405.CrossRefGoogle Scholar

Copyright information

© Institute of Systems Science, Academy of Mathematics and Systems Science, CAS and Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.Key Laboratory of Random Complex Structures and Data Science, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina
  2. 2.Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina

Personalised recommendations