Skip to main content
SpringerLink
Log in

Fuzzy modularity and fuzzy community structure in networks

  • Published:
The European Physical Journal B Aims and scope Submit manuscript

Abstract.

To find the fuzzy community structure in a complex network, in which each node has a certain probability of belonging to a certain community, is a hard problem and not yet satisfactorily solved over the past years. In this paper, an extension of modularity, the fuzzy modularity is proposed, which can provide a measure of goodness for the fuzzy community structure in networks. The simulated annealing strategy is used to maximize the fuzzy modularity function, associating with an alternating iteration based on our previous work. The proposed algorithm can efficiently identify the probabilities of each node belonging to different communities with random initial fuzzy partition during the cooling process. An appropriate number of communities can be automatically determined without any prior knowledge about the community structure. The computational results on several artificial and real-world networks confirm the capability of the algorithm.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R. Albert, A.-L. Barabási, Rev. Mod. Phys. 74, 47 (2002)

    Article  ADS  Google Scholar 

  2. M.E.J. Newman, SIAM Rev. 45, 167 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  3. M.E.J. Newman, A.L. Barabási, D.J. Watts, The Structure and Dynamics of Networks (Princeton University Press, Princeton, 2005)

  4. A.L. Barabási, H. Jeong, Z. Neda, E. Ravasz, A. Schubert, T. Vicsek, Physica A 311, 590 (2002)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  5. E. Ravasz, A.L. Somera, D.A. Mongru, Z.N. Oltvai, A.L. Barabási, Science 297, 1551 (2002)

    Article  ADS  Google Scholar 

  6. G.W. Flake, S. Lawrence, C.L. Giles, F.M. Coetzee, IEEE Comput. 35, 66 (2002)

    Google Scholar 

  7. J. Shi, J. Malik, IEEE Trans. Pattern Anal. Mach. Intel. 22, 888 (2000)

    Article  Google Scholar 

  8. M. Meilǎ, J. Shi, A random walks view of spectral segmentation, in Proceedings of the Eighth International Workshop on Artificial Intelligence and Statistics (Kaufmann, San Francisco, 2001), pp. 92–97

  9. S. Lafon, A.B. Lee, IEEE Trans. Pattern Anal. Mach. Intel. 28, 1393 (2006)

    Article  Google Scholar 

  10. M. Girvan, M.E.J. Newman, Proc. Natl. Acad. Sci. USA 99, 7821 (2002)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  11. M.E.J. Newman, M. Girvan, Phys. Rev. E 69, 026113 (2004)

    Article  ADS  Google Scholar 

  12. A. Clauset, M.E.J. Newman, C. Moore, Phys. Rev. E 70, 066111 (2004)

    Article  ADS  Google Scholar 

  13. M.E.J. Newman, Phys. Rev. E 69, 066133 (2004)

    Article  ADS  Google Scholar 

  14. M.E.J. Newman, Eur. Phys. J. B 38, 321 (2004)

    Article  ADS  Google Scholar 

  15. M.E.J. Newman, Phys. Rev. E 74, 036104 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  16. M.E.J. Newman, Proc. Natl. Acad. Sci. USA 103, 8577 (2006)

    Article  ADS  Google Scholar 

  17. M.E.J. Newman, E.A. Leicht, Proc. Natl. Acad. Sci. USA 104, 9564 (2007)

    Article  MATH  ADS  Google Scholar 

  18. H. Zhou, Phys. Rev. E 67, 041908 (2003)

    Article  ADS  Google Scholar 

  19. H. Zhou, Phys. Rev. E 67, 061901 (2003)

    Article  ADS  Google Scholar 

  20. F. Wu, B.A. Huberman, Eur. Phys. J. B 38, 331 (2004)

    Article  ADS  Google Scholar 

  21. J. Duch, A. Arenas, Phys. Rev. E 72, 027104 (2005)

    Article  ADS  Google Scholar 

  22. R. Guimera, L.A.N. Amaral, Nature 433, 895 (2005)

    Article  ADS  Google Scholar 

  23. G. Palla, I. Derényi, I. Farkas, T. Vicsek, Nature 435, 814 (2005)

    Article  ADS  Google Scholar 

  24. L. Danon, A. Diaz-Guilera, J. Duch, A. Arenas, J. Stat. Mech. 9, P09008 (2005)

    Article  Google Scholar 

  25. S. Zhang, R.S. Wang, X.S. Zhang, Physica A 374, 483 (2007)

    Article  ADS  Google Scholar 

  26. S. Zhang, X.M. Ning, X.S. Zhang, Eur. Phys. J. B 57, 67 (2007)

    Article  ADS  Google Scholar 

  27. J.M. Hofman, C.H. Wiggins, Phys. Rev. Lett. 100, 258701 (2008)

    Article  ADS  Google Scholar 

  28. M. Rosvall, C.T. Bergstrom, Proc. Natl. Acad. Sci. USA 105, 1118 (2008)

    Article  ADS  Google Scholar 

  29. W. E, T. Li, E. Vanden-Eijnden, Proc. Natl. Acad. Sci. USA 105, 7907 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  30. T. Li, J. Liu, W. E, Phys. Rev. E 80, 026106 (2009)

    Article  ADS  Google Scholar 

  31. A. Lancichinetti, S. Fortunato, J. Kertész, New J. Phys. 11, 033015 (2009)

    Article  ADS  Google Scholar 

  32. A. Lancichinetti, S. Fortunato, F. Radicchi, Phys. Rev. E 78, 46110 (2008)

    Article  ADS  Google Scholar 

  33. A. Lancichinetti, S. Fortunato, Phys. Rev. E 80, 16118 (2009)

    Article  ADS  Google Scholar 

  34. A. Lancichinetti, S. Fortunato, Phys. Rev. E 80, 56117 (2009)

    Article  ADS  Google Scholar 

  35. S. Fortunato, Phys. Rep. 486, 75 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  36. W.H.A. Schilders, H.A. Van der Vorst, J. Rommes, Model Order Reduction: Theory, Research Aspects and Applications (Springer, Berlin, Heidelberg, 2008)

  37. T. Hastie, R. Tibshirani, J. Friedman, The Elements of statistical Learning: Data Mining, Inference, and Prediction (Springer, New York, 2001)

  38. N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, E. Teller, J. Chem. Phys. 21, 1087 (1953)

    Article  ADS  Google Scholar 

  39. S. Kirkpatrick, C.D. Gelatt, M.P. Vecchi, Science 220, 671 (1983)

    Article  MathSciNet  ADS  Google Scholar 

  40. X.L. Xie, G. Beni, IEEE Tran. Pattern Anal. Mach. Intel. 13, 841 (1991)

    Article  Google Scholar 

  41. J. Noh, H. Rieger, Phys. Rev. Lett. 92, 118701 (2004)

    Article  ADS  Google Scholar 

  42. L. Lovasz, Combinatorics, Paul Erdös is Eighty 2, 1 (1993)

    MathSciNet  Google Scholar 

  43. F.R.K. Chung, Spectral Graph Theory (American Mathematical Society, Rhode Island, 1997)

  44. W.W. Zachary, J. Anthrop. Res. 33, 452 (1977)

    Google Scholar 

  45. D. Lusseau, Proc. R. Soc. B: Biol. Sci. 270, 186 (2003)

    Article  Google Scholar 

  46. D. Lusseau, K. Schneider, O.J Boisseau, P. Haase, E. Slooten, S.M. Dawson, Behav. Ecol. Sociobiol. 54, 396 (2003)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. LMAM and School of Mathematical Sciences, Peking University, Beijing, 100871, P.R. China

    Jian Liu

Authors
  1. Jian Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jian Liu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Liu, J. Fuzzy modularity and fuzzy community structure in networks. Eur. Phys. J. B 77, 547–557 (2010). https://doi.org/10.1140/epjb/e2010-00290-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1140/epjb/e2010-00290-3

Keywords

Search

Navigation