VGHC: a variable granularity hierarchical clustering for community detection

Abstract

Hierarchical clustering is an effective method for community detection. This kind of method usually selects clustering threshold for layering, which will affect the performance of community detection. Most of them select uniform clustering threshold. It means that the hierarchical structure of the community is obtained by merging synchronously. Actually, the communities are merged asynchronously . So, how to capture the adaptive hierarchical structure of the community is a challenge. In this paper, we propose a variable granularity method VGHC to construct the hierarchical structure based on quotient space theory (QST). QST uses cut set to form a hierarchical structure which represents multi-granular spaces. Firstly, initial granules are built with the important nodes as the granule center. Secondly, we choose the median clustering coefficient of the lower layer as the cut set, and then lower layer can be clustering to higher layer. Finally, the extended modularity (EQ) or modularity (Q) is taken as the evaluation criterion to select a certain layer in the hierarchical structure as the community detection result. Some overlapping nodes exist because of clustering mechanism. To get a non-overlapping community structure, a local modularity optimization approach (LMO) is used to deal with overlapping nodes. Experiments on seven real-world networks demonstrate that our method is effective for community detection in networks compared with the state-of-the-art algorithms.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3

References

  1. Ahn YY, Bagrow JP, Lehmann S (2010) Link communities reveal multiscale complexity in networks. Nature 466(7307):761

    Google Scholar 

  2. Bezdek JC (2013) Pattern recognition with fuzzy objective function algorithms. Springer Science & Business Media, Berlin

    Google Scholar 

  3. Blondel VD, Guillaume JL, Lambiotte R, Lefebvre E (2008) Fast unfolding of communities in large networks. J Stat Mech Theory Exp 10:P10008

    MATH  Google Scholar 

  4. Boccaletti S, Ivanchenko M, Latora V, Pluchino A, Rapisarda A (2007) Detecting complex network modularity by dynamical clustering. Phys Rev E 75(4):045102

    Google Scholar 

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

    Google Scholar 

  6. Cui Y, Wang X, Eustace J (2014) Detecting community structure via the maximal sub-graphs and belonging degrees in complex networks. Phys A Stat Mech Appl 416:198–207

    MATH  Google Scholar 

  7. Danon L, Diaz-Guilera A, Duch J, Arenas A (2005) Comparing community structure identification. J Stat Mech Theory Exp 09:P09008

    MATH  Google Scholar 

  8. Fang LD, Zhang YP, Chen J, Wang QQ, Liu F, Wang G (2017) Three-way decision based on non-overlapping community division. CAAI Trans Intell Syst 12(3):293–300

    Google Scholar 

  9. Fortunato S (2010) Community detection in graphs. Phys Rep 486(3–5):75–174

    MathSciNet  Google Scholar 

  10. Girvan M, Newman ME (2002) Community structure in social and biological networks. Proc Natl Acad Sci 99(12):7821–7826

    MathSciNet  MATH  Google Scholar 

  11. Guimera R, Danon L, Diaz-Guilera A, Giralt F, Arenas A (2003) Self-similar community structure in a network of human interactions. Phys Rev E 68(6):065103

    Google Scholar 

  12. Kernighan BW, Lin S (1970) An efficient heuristic procedure for partitioning graphs. Bell Syst Tech J 49(2):291–307

    MATH  Google Scholar 

  13. Liu Z, Ma Y (2019) A divide and agglomerate algorithm for community detection in social networks. Inf Sci 482:321–333

    Google Scholar 

  14. Lloyd S (1982) Least squares quantization in pcm. IEEE Trans Inf Theory 28(2):129–137

    MathSciNet  MATH  Google Scholar 

  15. Lusseau D (2003) The emergent properties of a dolphin social network. Proc R Soc Lond Ser B Biol Sci 270(suppl–2):S186–S188

    Google Scholar 

  16. Newman ME (2004) Fast algorithm for detecting community structure in networks. Phys Rev E 69(6):066133

    Google Scholar 

  17. Newman ME (2006a) Finding community structure in networks using the eigenvectors of matrices. Phys Rev E 74(3):036104

    MathSciNet  Google Scholar 

  18. Newman ME (2006b) Modularity and community structure in networks. Proc Natl Acad Sci 103(23):8577–8582

    Google Scholar 

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

    Google Scholar 

  20. Ng AY, Jordan MI, Weiss Y (2002) On spectral clustering: analysis and an algorithm. In: Advances in neural information processing systems, pp 849–856

  21. Papadopoulos S, Kompatsiaris Y, Vakali A, Spyridonos P (2012) Community detection in social media. Data Min Knowl Discov 24(3):515–554

    Google Scholar 

  22. Pothen A (1997) Graph partitioning algorithms with applications to scientific computing. In: Parallel numerical algorithms. Springer, pp 323–368

  23. Radicchi F, Castellano C, Cecconi F, Loreto V, Parisi D (2004) Defining and identifying communities in networks. Proc Natl Acad Sci 101(9):2658–2663

    Google Scholar 

  24. Roux M (2015) A comparative study of divisive hierarchical clustering algorithms. arXiv:1506.08977

  25. Semertzidis T, Rafailidis D, Strintzis MG, Daras P (2015) Large-scale spectral clustering based on pairwise constraints. Inf Process Manag 51(5):616–624

    Google Scholar 

  26. Shen H, Cheng X, Cai K, Hu MB (2009) Detect overlapping and hierarchical community structure in networks. Phys A Stat Mech Appl 388(8):1706–1712

    Google Scholar 

  27. Tian B, Li W (2018) Community detection method based on mixed-norm sparse subspace clustering. Neurocomputing 275:2150–2161

    Google Scholar 

  28. Vladimir B, Andrej M (2006) Pajek datasets. http://vlado.fmf.uni-lj.si/pub/networks/data/

  29. Wang G, Yang J, Xu J (2017) Granular computing: from granularity optimization to multi-granularity joint problem solving. Granul Comput 2(3):105–120

    Google Scholar 

  30. Watts DJ, Strogatz SH (1998) Collective dynamics of ‘small-world’ networks. Nature 393(6684):440

    MATH  Google Scholar 

  31. Yao Y (2016) A triarchic theory of granular computing. Granul Comput 1(2):145–157

    Google Scholar 

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

    Google Scholar 

  33. Zhang B, Zhang L (1992) Theory and applications of problem solving, vol 9. North-Holland, Amsterdam

    Google Scholar 

  34. Zhang L, Zhang B (2003) Theory of fuzzy quotient space (methods of fuzzy granular computing). J Softw 14(4):770–776

    MATH  Google Scholar 

  35. Zhang L, Ye Q, Shao Y, Li C, Gao H (2014) An efficient hierarchy algorithm for community detection in complex networks. Math Probl Eng 2014:874217. https://doi.org/10.1155/2014/874217

    Article  Google Scholar 

  36. Zhang W, Kong F, Yang L, Chen Y, Zhang M (2018) Hierarchical community detection based on partial matrix convergence using random walks. Tsinghua Sci Technol 23(1):35–46

    Google Scholar 

  37. Zhao S, Wang KE, Chen J, Zhang Y, University A (2014) Community detection algorithm based on clustering granulation. J Comput Appl 34(10):2812–2815

    Google Scholar 

  38. Zhao S, Ke W, Chen J, Liu F, Huang M, Zhang Y, Tang J (2015) Tolerance granulation based community detection algorithm. Tsinghua Sci Technol 20(6):620–626

    Google Scholar 

  39. Zhou H (2003) Distance, dissimilarity index, and network community structure. Phys Rev E 67(6):061901

    Google Scholar 

  40. Zhou X, Yang K, Xie Y, Yang C, Huang T (2019) A novel modularity-based discrete state transition algorithm for community detection in networks. Neurocomputing 334:89–99

    Google Scholar 

  41. Zou F, Chen D, Li S, Lu R, Lin M (2017) Community detection in complex networks: multi-objective discrete backtracking search optimization algorithm with decomposition. Appl Soft Comput 53:285–295

    Google Scholar 

Download references

Acknowledgements

This work is supported by the National Natural Science Foundation of China [Grants Numbers 61602003, #61673020, #61876001]; the Provincial Natural Science Foundation of Anhui Province [Grants Number 1708085QF156]; the Innovation Zone Project Program for Science and Technology of China’s National Defense [Grant Number 2017-0001-863015-0009]; and the National Key Research and Development Program of China [Grant Number 2017YFB1401903].

Author information

Affiliations

Authors

Corresponding author

Correspondence to Shu Zhao.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Chen, J., Li, Y., Yang, X. et al. VGHC: a variable granularity hierarchical clustering for community detection. Granul. Comput. 6, 37–46 (2021). https://doi.org/10.1007/s41066-019-00195-1

Download citation

Keywords

  • Community detection
  • Hierarchical clustering
  • Variable granularity
  • Local modularity optimization approach