Knowledge and Information Systems

, Volume 60, Issue 2, pp 757–779 | Cite as

Community detection using multilayer edge mixture model

  • Han Zhang
  • Chang-Dong WangEmail author
  • Jian-Huang Lai
  • Philip S. Yu
Regular Paper


Multilayer networks are networks where edges exist in multiple layers that encode different types or sources of interactions. As one of the most important problems in network science, discovering the underlying community structure in multilayer networks has received an increasing amount of attention in recent years. One of the challenging issues is to develop effective community structure quality functions for characterizing the structural or functional properties of the expected community structure. Although several quality functions have been developed for evaluating the detected community structure, little has been explored about how to explicitly bring our knowledge of the desired community structure into such quality functions, in particular for the multilayer networks. To address this issue, we propose the multilayer edge mixture model (MEMM), which is positioned as a general framework that enables us to design a quality function that reflects our knowledge about the desired community structure. The proposed model is based on a mixture of the edges, and the weights reflect their role in the detection process. By decomposing a community structure quality function into the form of MEMM, it becomes clear which type of community structure will be discovered by such quality function. Similarly, after such decomposition we can also modify the weights of the edges to find the desired community structure. In this paper, we apply the quality functions modified with the knowledge of MEMM to different multilayer benchmark networks as well as real-world multilayer networks and the detection results confirm the feasibility of MEMM.


Community detection Multilayer network General quality function Edge mixture 



This work was supported by NSFC (61502543 and 61672313), National Key Research and Development Program of China (2016YFB1001003), Guangdong Natural Science Funds for Distinguished Young Scholar (2016A030306014), Tip-top Scientific and Technical Innovative Youth Talents of Guangdong special support program (2016TQ03X542), NSF through Grants IIS-1526499, IIS-1763325, and CNS-1626432.


  1. 1.
    Bang-Jensen J, Gutin GZ (2008) Digraphs: theory, algorithms and applications. Springer, BerlinzbMATHGoogle Scholar
  2. 2.
    Barrat A, Barthelemy M, Pastor-Satorras R, Vespignani A (2004) The architecture of complex weighted networks. Proc Natl Acad Sci USA 101(11):3747–3752CrossRefGoogle Scholar
  3. 3.
    Bazzi M, Porter MA, Williams S, McDonald M, Fenn DJ, Howison SD (2014) Community detection in temporal multilayer networks, and its application to correlation networks, arXiv preprint arXiv:1501.00040
  4. 4.
    Boccaletti S, Bianconi G, Criado R, Del Genio CI, Gómez-Gardeñes J, Romance M, Sendina-Nadal I, Wang Z, Zanin M (2014) The structure and dynamics of multilayer networks. Phys Rep 544(1):1–122MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bollobás B (1998) Modern graph theory, vol 184. Springer, BerlinzbMATHCrossRefGoogle Scholar
  6. 6.
    Cardillo A, Gómez-Gardenes J, Zanin M, Romance M, Papo D, del Pozo F, Boccaletti S (2013) Emergence of network features from multiplexity. Sci Rep 3
  7. 7.
    Clauset A, Newman ME, Moore C (2004) Finding community structure in very large networks. Phys Rev E 70(6):066111CrossRefGoogle Scholar
  8. 8.
    Condon A, Karp RM (2001) Algorithms for graph partitioning on the planted partition model. Random Struct Algorithms 18(2):116–140MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Danon L, Diaz-Guilera A, Duch J, Arenas A (2005) Comparing community structure identification. J Stat Mech: Theory Exp 09:P09008zbMATHGoogle Scholar
  10. 10.
    De Domenico M, Lancichinetti A, Arenas A, Rosvall M (2015) Identifying modular flows on multilayer networks reveals highly overlapping organization in interconnected systems. Phys Rev X 5(1):011027Google Scholar
  11. 11.
    Delvenne J-C, Yaliraki SN, Barahona M (2010) Stability of graph communities across time scales. Proc Natl Acad Sci 107(29):12755–12760CrossRefGoogle Scholar
  12. 12.
    Djidjev HN (2006) A scalable multilevel algorithm for graph clustering and community structure detection. Algorithms and models for the web-graph. Springer, Berlin pp 117–128Google Scholar
  13. 13.
    Doreian P, Mrvar A (2009) Partitioning signed social networks. Soc Netw 31(1):1–11zbMATHCrossRefGoogle Scholar
  14. 14.
    Fortunato S (2010) Community detection in graphs. Phys Rep 486(3):75–174MathSciNetCrossRefGoogle Scholar
  15. 15.
    Gao J, Buldyrev SV, Havlin S, Stanley HE (2012) Robustness of a network formed by n interdependent networks with a one-to-one correspondence of dependent nodes. Phys Rev E 85(6):066134CrossRefGoogle Scholar
  16. 16.
    Girvan M, Newman ME (2002) Community structure in social and biological networks. Proc Natl Acad Sci 99(12):7821–7826MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Guimerà R, Sales-Pardo M (2009) Missing and spurious interactions and the reconstruction of complex networks. Proc Natl Acad Sci 106(52):22073–22078CrossRefGoogle Scholar
  18. 18.
    Holland PW, Laskey KB, Leinhardt S (1983) Stochastic blockmodels: first steps. Soc Netw 5(2):109–137MathSciNetCrossRefGoogle Scholar
  19. 19.
    Holme P, Saramäki J (2012) Temporal networks. Phys Rep 519(3):97–125CrossRefGoogle Scholar
  20. 20.
    Hu H-B, Wang X-F (2009) Disassortative mixing in online social networks. Europhys Lett 86(1):18003CrossRefGoogle Scholar
  21. 21.
    Hu Y, Li M, Zhang P, Fan Y, Di Z (2008) Community detection by signaling on complex networks. Phys Rev E 78(1):016115CrossRefGoogle Scholar
  22. 22.
    Jeub LGS, Bazzi M, Jutla IS, Mucha PJ (2011-2017) A generalized louvain method for community detection implemented in matlab,
  23. 23.
    Karrer B, Newman ME (2011) Stochastic blockmodels and community structure in networks. Phys Rev E 83(1):016107MathSciNetCrossRefGoogle Scholar
  24. 24.
    Kivelä M, Arenas A, Barthelemy M, Gleeson JP, Moreno Y, Porter MA (2014) Multilayer networks. J Complex Netw 2(3):203–271CrossRefGoogle Scholar
  25. 25.
    Lancichinetti A, Fortunato S, Radicchi F (2008) Benchmark graphs for testing community detection algorithms. Phys Rev E 78(4):046110CrossRefGoogle Scholar
  26. 26.
    Li J-H, Wang C-D, Li P-Z, Lai J-H (2018) Discriminative metric learning for multi-view graph partitioning. Pattern Recognit 75:199–213CrossRefGoogle Scholar
  27. 27.
    Li W, Liu C-C, Zhang T, Li H, Waterman MS, Zhou XJ (2011) Integrative analysis of many weighted co-expression networks using tensor computation. PLoS Comput Biol 7(6):e1001106MathSciNetCrossRefGoogle Scholar
  28. 28.
    Meilă M (2007) Comparing clusterings–an information based distance. J Multivar Anal 98(5):873–895MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Mieghem PV, Ge X, Schumm P, Trajanovski S, Wang H (2010) Spectral graph analysis of modularity and assortativity. Phys Rev E 82:056113CrossRefGoogle Scholar
  30. 30.
    Moosavi SA, Jalali M, Misaghian N, Shamshirband S, Anisi MH (2016) Community detection in social networks using user frequent pattern mining. Knowl Inf Syst. CrossRefGoogle Scholar
  31. 31.
    Mucha PJ, Richardson T, Macon K, Porter MA, Onnela J-P (2010) Community structure in time-dependent, multiscale, and multiplex networks. Science 328(5980):876–878MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Newman M (2010) Networks: an introduction. Oxford University Press, OxfordzbMATHCrossRefGoogle Scholar
  33. 33.
    Newman ME (2004) Analysis of weighted networks. Phys Rev E 70(5):056131CrossRefGoogle Scholar
  34. 34.
    Newman ME (2006) Modularity and community structure in networks. Proc Natl Acad Sci 103(23):8577–8582CrossRefGoogle Scholar
  35. 35.
    Newman ME, Girvan M (2004) Finding and evaluating community structure in networks. Phys Rev E 69(2):026113CrossRefGoogle Scholar
  36. 36.
    Peixoto TP (2015) Inferring the mesoscale structure of layered, edge-valued, and time-varying networks. Phys Rev E 92(4):042807MathSciNetCrossRefGoogle Scholar
  37. 37.
    Radicchi F, Castellano C, Cecconi F, Loreto V, Parisi D (2004) Defining and identifying communities in networks. Proc Natl Acad Sci USA 101(9):2658–2663CrossRefGoogle Scholar
  38. 38.
    Reichardt J, Bornholdt S (2006) Statistical mechanics of community detection. Phys Rev E 74(1):016110MathSciNetCrossRefGoogle Scholar
  39. 39.
    Reichardt J, White DR (2007) Role models for complex networks. Eur Phys J B 60(2):217–224zbMATHCrossRefGoogle Scholar
  40. 40.
    Rocklin M, Pinar A (2013) On clustering on graphs with multiple edge types. Internet Math 9(1):82–112MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Rosvall M, Bergstrom CT (2007) An information-theoretic framework for resolving community structure in complex networks. Proc Natl Acad Sci 104(18):7327–7331CrossRefGoogle Scholar
  42. 42.
    Rosvall M, Bergstrom CT (2008) Maps of random walks on complex networks reveal community structure. Proc Natl Acad Sci 105(4):1118–1123CrossRefGoogle Scholar
  43. 43.
    Strogatz SH (2001) Exploring complex networks. Nature 410(6825):268–276zbMATHCrossRefGoogle Scholar
  44. 44.
    Szell M, Lambiotte R, Thurner S (2010) Multirelational organization of large-scale social networks in an online world. Proc Natl Acad Sci 107(31):13636–13641CrossRefGoogle Scholar
  45. 45.
    Verbrugge LM (1979) Multiplexity in adult friendships. Soc Forces 57(4):1286–1309CrossRefGoogle Scholar
  46. 46.
    Wang CD, Lai JH, Yu PS (2014) NEIWalk: community discovery in dynamic content-based networks. IEEE Trans Knowl Data Eng 26(7):1734–1748CrossRefGoogle Scholar
  47. 47.
    Wang Y, Xun J, Yang Z, Li J (2017) Query optimal k-plex based community in graphs. Data Sci Eng 2:257CrossRefGoogle Scholar
  48. 48.
    Wasserman S, Faust K (1994) Social network analysis: methods and applications, vol 8. Cambridge University Press, CambridgezbMATHCrossRefGoogle Scholar
  49. 49.
    Yang B, Cheung WK, Liu J (2007) Community mining from signed social networks. IEEE Trans Knowl Data Eng 19(10):1333–1348CrossRefGoogle Scholar
  50. 50.
    Yang J, Leskovec J (2015) Defining and evaluating network communities based on ground-truth. Knowl Inf Syst 42(1):181–213. CrossRefGoogle Scholar
  51. 51.
    Zhang S, Wang R-S, Zhang X-S (2007) Uncovering fuzzy community structure in complex networks. Phys Rev E 76(4):046103CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2018

Authors and Affiliations

  • Han Zhang
    • 1
  • Chang-Dong Wang
    • 1
    Email author
  • Jian-Huang Lai
    • 1
  • Philip S. Yu
    • 2
  1. 1.School of Data and Computer ScienceSun Yat-sen UniversityGuangzhouPeople’s Republic of China
  2. 2.University of Illinois at ChicagoChicagoUSA

Personalised recommendations