Computational Complexity

2012 Edition
| Editors: Robert A. Meyers (Editor-in-Chief)

Community Structure in Graphs

  • Santo Fortunato
  • Claudio Castellano
Reference work entry

Article Outline


Definition of the Subject


Elements of Community Detection

Computer Science: Graph Partitioning

Social Science: Hierarchical and k‑Means Clustering

New Methods

Testing Methods

The Mesoscopic Description of a Graph

Future Directions



Cluster Coefficient Community Detection Modularity Maximum Sparse Graph Community Detection Algorithm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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Primary Literature

  1. 1.
    Euler L (1736) Solutio problematis ad geometriam situs pertinentis. Commentarii Academiae Petropolitanae 8:128–140Google Scholar
  2. 2.
    Bollobás B (1998) Modern Graph Theory. Springer, New YorkGoogle Scholar
  3. 3.
    Wasserman S, Faust K (1994) Social Network Analysis: Methods and Applications. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  4. 4.
    Scott JP (2000) Social Network Analysis. Sage Publications Ltd, LondonGoogle Scholar
  5. 5.
    Barabási AL, Albert R (2002) Statistical mechanics of complex networks. Rev Mod Phys 74:47–97Google Scholar
  6. 6.
    Dorogovtsev SN, Mendes JFF (2003) Evolution of Networks: from biological nets to the Internet and WWW. Oxford University Press, OxfordMATHGoogle Scholar
  7. 7.
    Newman MEJ (2003) The structure and function of complex networks. SIAM Rev 45:167–256 MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Pastor-Satorras R, Vespignani A (2004) Evolution and structure of the Internet: A statistical physics approach. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  9. 9.
    Boccaletti S, Latora V, Moreno Y, Chavez M, Hwang DU (2006) Complex Networks: Structure and Dynamics. Phys Rep 424:175–308MathSciNetCrossRefGoogle Scholar
  10. 10.
    Erdös P, Rényi A (1959) On Random Graphs. Publicationes Mathematicae Debrecen 6:290–297Google Scholar
  11. 11.
    Flake GW, Lawrence S, Lee Giles C, Coetzee FM (2002) Self-Organization and Identification of Web Communities. IEEE Comput 35(3):66–71CrossRefGoogle Scholar
  12. 12.
    Guimerà R, Amaral LAN (2005) Functional cartography of complex metabolic networks. Nature 433:895–900Google Scholar
  13. 13.
    Palla G, Derényi I, Farkas I, Vicsek T (2005) Uncovering the overlapping community structure of complex networks in nature and society. Nature 435:814–818Google Scholar
  14. 14.
    Girvan M, Newman MEJ (2002) Community structure in social and biological networks. Proc Nat Acad Sci USA 99(12): 7821–7826MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Lusseau D, Newman MEJ (2004) Identifying the role that animals play in their social networks. Proc R Soc Lond B 271: S477–S481CrossRefGoogle Scholar
  16. 16.
    Pimm SL (1979) The structure of food webs. Theor Popul Biol 16:144–158CrossRefGoogle Scholar
  17. 17.
    Krause AE, Frank KA, Mason DM, Ulanowicz RE, Taylor WW (2003) Compartments exposed in food-web structure. Nature 426:282–285CrossRefGoogle Scholar
  18. 18.
    Granovetter M (1973) The Strength of Weak Ties. Am J Sociol 78:1360–1380CrossRefGoogle Scholar
  19. 19.
    Burt RS (1976) Positions in Networks. Soc Force 55(1):93–122MathSciNetGoogle Scholar
  20. 20.
    Freeman LC (1977) A Set of Measures of Centrality Based on Betweenness. Sociometry 40(1):35–41CrossRefGoogle Scholar
  21. 21.
    Pollner P, Palla G, Vicsek T (2006) Preferential attachment of communities: The same principle, but a higher level. Europhys Lett 73(3):478–484MathSciNetCrossRefGoogle Scholar
  22. 22.
    Newman MEJ (2004) Detecting community structure in networks. Eur Phys J B 38:321–330CrossRefGoogle Scholar
  23. 23.
    Danon L, Duch J, Arenas A, Díaz-Guilera A (2007) Community structure identification. In: Caldarelli G, Vespignani A (eds) Large Scale Structure and Dynamics of Complex Networks: From Information Technology to Finance and Natural Science. World Scientific, Singapore, pp 93–114Google Scholar
  24. 24.
    Bron C, Kerbosch J (1973) Finding all cliques on an undirected graph. Commun ACM 16:575–577MATHCrossRefGoogle Scholar
  25. 25.
    Radicchi F, Castellano C, Cecconi F, Loreto V, Parisi D (2004) Defining and identifying communities in networks. Proc Nat Acad Sci USA 101(9):2658–2663 CrossRefGoogle Scholar
  26. 26.
    Newman MEJ, Girvan M (2004) Finding and evaluating community structure in networks. Phys Rev E 69:026113CrossRefGoogle Scholar
  27. 27.
    Arenas A, Fernández A, Fortunato S, Gómez S (2007) Motif-based communities in complex networks. arXiv:0710.0059Google Scholar
  28. 28.
    Reichardt J, Bornholdt S (2006) Statistical mechanics of community detection. Phys Rev E 74:016110MathSciNetCrossRefGoogle Scholar
  29. 29.
    Massen CP, Doye JPK (2006) Thermodynamics of community structure. arXiv:cond-mat/0610077Google Scholar
  30. 30.
    Arenas A, Duch J, Fernándes A, Gómez S (2007) Size reduction of complex networks preserving modularity. New J Phys 9(6):176–180 Google Scholar
  31. 31.
    Guimerà R, Sales-Pardo M, Amaral LAN (2004) Modularity from fluctuations in random graphs and complex networks. Phys Rev E 70:025101(R)Google Scholar
  32. 32.
    Fortunato S, Barthélemy M (2007) Resolution limit in community detection. Proc Nat Acad Sci USA 104(1):36–41Google Scholar
  33. 33.
    Gfeller D, Chappelier J-C, De Los Rios P (2005) Finding instabilities in the community structure of complex networks. Phys Rev E 72:056135CrossRefGoogle Scholar
  34. 34.
    Pothen A (1997) Graph partitioning algorithms with applications to scientific computing. In: Keyes DE, Sameh A, Venkatakrishnan V (eds) Parallel Numerical Algorithms. Kluwer Academic Press, Boston, pp 323–368CrossRefGoogle Scholar
  35. 35.
    Kernighan BW, Lin S (1970) An efficient heuristic procedure for partitioning graphs. Bell Syst Tech J 49:291–307MATHGoogle Scholar
  36. 36.
    Golub GH, Van Loan CF (1989) Matrix computations. John Hopkins University Press, BaltimoreMATHGoogle Scholar
  37. 37.
    JB MacQueen (1967) Some methods for classification and analysis of multivariate observations. In: Proceedings of 5th Berkeley Symposium on Mathematical Statistics and Probability, Berkeley. University of California Press, pp 281–297Google Scholar
  38. 38.
    Brandes U (2001) A faster algorithm for betweenness centrality. J Math Sociol 25(2):163–177MATHCrossRefGoogle Scholar
  39. 39.
    Newman MEJ (2005) A measure of betweenness centrality based on random walks. Soc Netw 27:39–54CrossRefGoogle Scholar
  40. 40.
    Tyler JR, Wilkinson DM, Huberman BA (2003) Email as spectroscopy: automated discovery of community structure within organizations. In: Huysman M, Wenger E, Wulf V (eds) Proceeding of the First International Conference on Communities and Technologies. Kluwer Academic Press, AmsterdamGoogle Scholar
  41. 41.
    Wilkinson DM, Huberman BA (2004) A method for finding communities of related genes. Proc Nat Acad Sci USA 101(1): 5241–5248CrossRefGoogle Scholar
  42. 42.
    Latora V, Marchiori M (2001) Efficient behavior of small-world networks. Phys Rev Lett 87:198701CrossRefGoogle Scholar
  43. 43.
    Fortunato S, Latora V, Marchiori M (2004) A method to find community structures based on information centrality. Phys Rev E 70:056104CrossRefGoogle Scholar
  44. 44.
    Watts D, Strogatz SH (1998) Collective dynamics of “small‐world” networks. Nature 393:440–442CrossRefGoogle Scholar
  45. 45.
    Brandes U, Delling D, Gaertler M, Görke R, Hoefer M, Nikoloski Z, Wagner D (2007) On finding graph clusterings with maximum modularity. In: Proceedings of the 33rd International Workshop on Graph-Theoretical Concepts in Computer Science (WG’07). Springer, BerlinGoogle Scholar
  46. 46.
    Newman MEJ (2004) Fast algorithm for detecting community structure in networks. Phys Rev E 69:066133CrossRefGoogle Scholar
  47. 47.
    Clauset A, Newman MEJ, Moore C (2004) Finding community structure in very large networks. Phys Rev E 70:066111CrossRefGoogle Scholar
  48. 48.
    Danon L, Díaz-Guilera A, Arenas A (2006) The effect of size heterogeneity on community identification in complex networks. J Stat Mech Theory Exp 11:P11010Google Scholar
  49. 49.
    Pujol JM, Béjar J, Delgado J (2006) Clustering algorithm for determining community structure in large networks. Phys Rev E 74:016107Google Scholar
  50. 50.
    Kirkpatrick S, Gelatt CD, Vecchi MP (1983) Optimization by simulated annealing. Science 220(4598):671–680MathSciNetMATHCrossRefGoogle Scholar
  51. 51.
    Massen CP, Doye JPK (2005) Identifying communities within energy landscapes. Phys Rev E 71:046101MathSciNetCrossRefGoogle Scholar
  52. 52.
    Boettcher S, Percus AG (2001) Optimization with extremal dynamics. Phys Rev Lett 86:5211–5214CrossRefGoogle Scholar
  53. 53.
    Duch J, Arenas A (2005) Community detection in complex networks using extremal optimization. Phys Rev E 72:027104 CrossRefGoogle Scholar
  54. 54.
    Newman MEJ (2006) Modularity and community structure in networks. Proc Nat Acad Sci USA 103 (23):8577–8582CrossRefGoogle Scholar
  55. 55.
    Newman MEJ (2006) Finding community structure in networks using the eigenvectors of matrices. Phys Rev E 74:036104 MathSciNetCrossRefGoogle Scholar
  56. 56.
    Reichardt J, Bornholdt S (2007) Partitioning and modularity of graphs with arbitrary degree distribution. Phys Rev E 76:015102(R)CrossRefGoogle Scholar
  57. 57.
    Reichardt J, Bornholdt S (2006) When are networks truly modular? Phys D 224:20–26MathSciNetMATHCrossRefGoogle Scholar
  58. 58.
    Kumpula JM, Saramäki J, Kaski K, Kertész J (2007) Limited resolution in complex network community detection with Potts model approach. Eur Phys J B 56:41–45Google Scholar
  59. 59.
    Arenas A, Fernándes A, Gómez S (2007) Multiple resolution of the modular structure of complex networks. arXiv:physics/0703218Google Scholar
  60. 60.
    Ruan J, Zhang W (2007) Identifying network communities with high resolution. arXiv:0704.3759Google Scholar
  61. 61.
    Kumpula JM, Saramäki J, Kaski K, Kertész J (2007) Limited resolution and multiresolution methods in complex network community detection. In: Kertész J, Bornholdt S, Mantegna RN (eds) Noise and Stochastics in Complex Systems and Finance. Proc SPIE 6601:660116Google Scholar
  62. 62.
    Muff S, Rao F, Caflisch A (2005) Local modularity measure for network clusterizations. Phys Rev E 72:056107CrossRefGoogle Scholar
  63. 63.
    Donetti L, Muñoz MA (2004) Detecting network communities: a new systematic and efficient algorithm. J Stat Mech Theory Exp P10012Google Scholar
  64. 64.
    Capocci A, Servedio VDP, Caldarelli G, Colaiori F (2004) Detecting communities in large networks. Phys A 352(2–4):669–676Google Scholar
  65. 65.
    Wu F, Huberman BA (2004) Finding communities in linear time: a physics approach. Eur Phys J B 38:331–338CrossRefGoogle Scholar
  66. 66.
    Eriksen KA, Simonsen I, Maslov S, Sneppen K (2003) Modularity and extreme edges of the Internet. Phys Rev Lett 90(14):148701CrossRefGoogle Scholar
  67. 67.
    Simonsen I, Eriksen KA, Maslov S, Sneppen K (2004) Diffusion on complex networks: a way to probe their large-scale topological structure. Physica A 336:163–173CrossRefGoogle Scholar
  68. 68.
    Wu FY (1982) The Potts model. Rev Mod Phys 54:235–268CrossRefGoogle Scholar
  69. 69.
    Blatt M, Wiseman S, Domany E (1996) Superparamagnetic clustering of data. Phys Rev Lett 76(18):3251–3254CrossRefGoogle Scholar
  70. 70.
    Reichardt J, Bornholdt S (2004) Detecting fuzzy community structure in complex networks. Phys Rev Lett 93(21):218701CrossRefGoogle Scholar
  71. 71.
    Mezard M, Parisi G, Virasoro M (1987) Spin glass theory and beyond. World Scientific Publishing Company, SingaporeMATHGoogle Scholar
  72. 72.
    Zhou H (2003) Network landscape from a Brownian particle’s perspective. Phys Rev E 67:041908CrossRefGoogle Scholar
  73. 73.
    Zhou H (2003) Distance, dissimilarity index, and network community structure. Phys Rev E 67:061901CrossRefGoogle Scholar
  74. 74.
    Zhou H, Lipowsky R (2004) Network Brownian motion: A new method to measure vertex‐vertex proximity and to identify communities and subcommunities. Lect Notes Comput Sci 3038:1062–1069CrossRefGoogle Scholar
  75. 75.
    Latapy M, Pons P 92005) Computing communities in large networks using random walks. Lect Notes Comput Sci 3733: 284–293Google Scholar
  76. 76.
    Arenas A, Díaz-Guilera A, Pérez-Vicente CJ (2006) Synchronization reveals topological scales in complex networks. Phys Rev Lett 96:114102Google Scholar
  77. 77.
    Kuramoto Y (1984) Chemical Oscillations, Waves and Turbulence. Springer, BerlinMATHCrossRefGoogle Scholar
  78. 78.
    Arenas A, Díaz-Guilera A (2007) Synchronization and modularity in complex networks. Eur Phys J ST 143:19–25 Google Scholar
  79. 79.
    Boccaletti S, Ivanchenko M, Latora V, Pluchino A, Rapisarda A (2007) Detecting complex network modularity by dynamical clustering. Phys Rev E 76:045102(R)Google Scholar
  80. 80.
    Pluchino A, Latora V, Rapisarda A (2005) Changing opinions in a changing world: a new perspective in sociophysics. Int J Mod Phys C 16(4):505–522CrossRefGoogle Scholar
  81. 81.
    Farkas I, Ábel D, Palla G, Vicsek T (2007) Weighted network modules. New J Phys 9:180 Google Scholar
  82. 82.
    Palla G, Farkas IJ, Pollner P, Derényi I, Vicsek T (2007) Directed network modules. New J Phys 9:186 Google Scholar
  83. 83.
    Palla G, Barabási A-L, Vicsek T (2007) Quantifying social groups evolution. Nature 446:664–667Google Scholar
  84. 84.
    van Dongen S (2000) Graph Clustering by Flow Simulation. Ph D thesis, University of Utrecht, The NetherlandsGoogle Scholar
  85. 85.
    Newman MEJ, Leicht E (2007) Mixture models and exploratory analysis in networks. Proc Nat Acad Sci USA 104(23):9564–9569MATHCrossRefGoogle Scholar
  86. 86.
    Bagrow JP, Bollt EM (2005) Local method for detecting communities. Phys Rev E 72:046108CrossRefGoogle Scholar
  87. 87.
    Clauset A (2005) Finding local community structure in networks. Phys Rev E 72:026132CrossRefGoogle Scholar
  88. 88.
    Eckmann J-P, Moses E (2002) Curvature of co-links uncovers hidden thematic layers in the World Wide Web. Proc Nat Acad Sci USA 99(9):5825–5829MathSciNetCrossRefGoogle Scholar
  89. 89.
    Sales-Pardo M, Guimerá R, Amaral LAN (2007) Extracting the hierarchical organization of complex systems. arXiv:0705.1679Google Scholar
  90. 90.
    Rosvall M, Bergstrom CT (2007) An information‐theoretic framework for resolving community structure in complex networks. Proc Nat Acad Sci USA 104(18):7327–7331 CrossRefGoogle Scholar
  91. 91.
    Shannon CE, Weaver V (1949) The Mathematical Theory of Communication. University of Illinois Press, ChampaignMATHGoogle Scholar
  92. 92.
    Danon L, Díaz-Guilera A, Duch J, Arenas A (2005) Comparing community structure identification. J Stat Mech Theory Exp P09008Google Scholar
  93. 93.
    Gustafsson M, Hörnquist M, Lombardi A (2006) Comparison and validation of community structures in complex networks. Physica A 367:559–576Google Scholar
  94. 94.
    Zachary WW (1977) An information flow model for conflict and fission in small groups. J Anthr Res 33:452–473Google Scholar
  95. 95.
    Lusseau D (2003) The emergent properties of a dolphin social network. Proc R Soc Lond B 270(2):S186–188CrossRefGoogle Scholar
  96. 96.
    Guimerá R, Amaral LAN (2005) Cartography of complex networks: modules and universal roles. J Stat Mech Theory Exp P02001Google Scholar

Books and Reviews

  1. 97.
    Bollobás B (2001) Random Graphs. Cambridge University Press, CambridgeGoogle Scholar
  2. 98.
    Chung FRK (1997) Spectral Graph Theory. CBMS Regional Conference Series in Mathematics 92. American Mathematical Society, ProvidenceGoogle Scholar
  3. 99.
    Dorogovtsev SN, Mendes JFF (2003) Evolution of Networks: From Biological Nets to the Internet and WWW. Oxford University Press, OxfordMATHGoogle Scholar
  4. 100.
    Elsner U (1997) Graph Partitioning: a Survey. Technical Report 97-27, Technische Universität Chemnitz, ChemnitzGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Santo Fortunato
    • 1
  • Claudio Castellano
    • 2
  1. 1.Complex Networks Lagrange Laboratory (CNLL)ISI FoundationTorinoItaly
  2. 2.SMC, INFM-CNR and Dipartimento di Fisica“Sapienza” Università di RomaRomaItaly