Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Communities, modules and large-scale structure in networks

Nature Physics volume 8, pages 25–31 (2012)Cite this article

Abstract

Networks, also called graphs by mathematicians, provide a useful abstraction of the structure of many complex systems, ranging from social systems and computer networks to biological networks and the state spaces of physical systems. In the past decade there have been significant advances in experiments to determine the topological structure of networked systems, but there remain substantial challenges in extracting scientific understanding from the large quantities of data produced by the experiments. A variety of basic measures and metrics are available that can tell us about small-scale structure in networks, such as correlations, connections and recurrent patterns, but it is considerably more difficult to quantify structure on medium and large scales, to understand the ‘big picture’. Important progress has been made, however, within the past few years, a selection of which is reviewed here.

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

Relevant articles

Open Access articles citing this article.

Access options

Rent or buy this article

Get just this article for as long as you need it

$39.95

Learn more

Prices may be subject to local taxes which are calculated during checkout

Figure 1: Example network showing community structure.
Figure 2: A network of collaborations among scientists at a research institute.
Figure 3: Average-linkage clustering of a small social network.
Figure 4: Analysis of a network of links between web sites about US politics.
Figure 5: Hierarchical divisions in a food web of grassland species.

References

  1. Albert, R. & Barabási, A-L. Statistical mechanics of complex networks. Rev. Mod. Phys. 74, 47–97 (2002).

    Article  ADS  MathSciNet  Google Scholar 

  2. Dorogovtsev, S. N. & Mendes, J. F. F. Evolution of networks. Adv. Phys. 51, 1079–1187 (2002).

    Article  ADS  Google Scholar 

  3. Newman, M. E. J. The structure and function of complex networks. SIAM Rev. 45, 167–256 (2003).

    Article  ADS  MathSciNet  Google Scholar 

  4. Boccaletti, S., Latora, V., Moreno, Y., Chavez, M. & Hwang, D-U. Complex networks: Structure and dynamics. Phys. Rep. 424, 175–308 (2006).

    Article  ADS  MathSciNet  Google Scholar 

  5. Newman, M. E. J. Networks: An Introduction (Oxford Univ. Press, 2010).

    Book  Google Scholar 

  6. Cohen, R. & Havlin, S. Complex Networks: Structure, Stability and Function (Cambridge Univ. Press, 2010).

    Book  Google Scholar 

  7. Faloutsos, M., Faloutsos, P. & Faloutsos, C. On power-law relationships of the internet topology. Comput. Commun. Rev. 29, 251–262 (1999).

    Article  Google Scholar 

  8. Pastor-Satorras, R. & Vespignani, A. Evolution and Structure of the Internet (Cambridge Univ. Press, 2004).

    Book  Google Scholar 

  9. Pimm, S. L. Food Webs 2nd edn (Univ. Chicago Press, 2002).

    Google Scholar 

  10. Pascual, M. & Dunne, J. A. (eds) Ecological Networks: Linking Structure to Dynamics in Food Webs (Oxford Univ. Press, 2006).

  11. Wasserman, S. & Faust, K. Social Network Analysis (Cambridge Univ. Press, 1994).

    Book  Google Scholar 

  12. Scott, J. Social Network Analysis: A Handbook 2nd edn (Sage, 2000).

    Google Scholar 

  13. Costa, L. da F., Rodrigues, F. A., Travieso, G. & Boas, P. R. V. Characterization of complex networks: A survey of measurements. Adv. Phys. 56, 167–242 (2007).

    Article  ADS  Google Scholar 

  14. Girvan, M. & Newman, M. E. J. Community structure in social and biological networks. Proc. Natl Acad. Sci. USA 99, 7821–7826 (2002).

    Article  ADS  MathSciNet  Google Scholar 

  15. Fortunato, S. Community detection in graphs. Phys. Rep. 486, 75–174 (2010).

    Article  ADS  MathSciNet  Google Scholar 

  16. Jeong, H., Tombor, B., Albert, R., Oltvai, Z. N. & Barabási, A-L. The large-scale organization of metabolic networks. Nature 407, 651–654 (2000).

    Article  ADS  Google Scholar 

  17. Guimerà, R. & Amaral, L. A. N. Functional cartography of complex metabolic networks. Nature 433, 895–900 (2005).

    Article  ADS  Google Scholar 

  18. Newman, M. E. J. & Girvan, M. Finding and evaluating community structure in networks. Phys. Rev. E 69, 026113 (2004).

    Article  ADS  Google Scholar 

  19. Flake, G. W., Lawrence, S. R., Giles, C. L. & Coetzee, F. M. Self-organization and identification of Web communities. IEEE Comput. 35, 66–71 (2002).

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  22. Palla, G., Derényi, I., Farkas, I. & Vicsek, T. Uncovering the overlapping community structure of complex networks in nature and society. Nature 435, 814–818 (2005).

    Article  ADS  Google Scholar 

  23. Bagrow, J. P. & Bollt, E. M. Local method for detecting communities. Phys. Rev. E 72, 046108 (2005).

    Article  ADS  Google Scholar 

  24. Clauset, A. Finding local community structure in networks. Phys. Rev. E 72, 026132 (2005).

    Article  ADS  Google Scholar 

  25. Hastings, M. B. Community detection as an inference problem. Phys. Rev. E 74, 035102 (2006).

    Article  ADS  Google Scholar 

  26. Rosvall, M. & Bergstrom, C. T. An information-theoretic framework for resolving community structure in complex networks. Proc. Natl Acad. Sci. USA 104, 7327–7331 (2007).

    Article  ADS  Google Scholar 

  27. Blondel, V. D., Guillaume, J-L., Lambiotte, R. & Lefebvre, E. Fast unfolding of communities in large networks. J. Stat. Mech. 2008, P10008 (2008).

    Article  Google Scholar 

  28. Agrawal, G. & Kempe, D. Modularity-maximizing network communities via mathematical programming. Eur. Phys. J. B 66, 409–418 (2008).

    Article  ADS  MathSciNet  Google Scholar 

  29. Hofman, J. M. & Wiggins, C. H. Bayesian approach to network modularity. Phys. Rev. Lett. 100, 258701 (2008).

    Article  ADS  Google Scholar 

  30. Leskovec, J., Lang, K., Dasgupta, A. & Mahoney, M. Community structure in large networks: Natural cluster sizes and the absence of large well-defined clusters. Internet Math. 6, 29–123 (2009).

    Article  MathSciNet  Google Scholar 

  31. Ahn, Y-Y., Bagrow, J. P. & Lehmann, S. Link communities reveal multiscale complexity in networks. Nature 466, 761–764 (2010).

    Article  ADS  Google Scholar 

  32. Lancichinetti, A., Fortunato, S. & Radicchi, F. Benchmark graphs for testing community detection algorithms. Phys. Rev. E 78, 046110 (2008).

    Article  ADS  Google Scholar 

  33. Danon, L., Duch, J., Diaz-Guilera, A. & Arenas, A. Comparing community structure identification. J. Stat. Mech. P09008 (2005).

  34. Lancichinetti, A. & Fortunato, S. Community detection algorithms: A comparative analysis. Phys. Rev. E 80, 056117 (2009).

    Article  ADS  Google Scholar 

  35. Schaeffer, S. E. Graph clustering. Comput. Sci. Rev. 1, 27–64 (2007).

    Article  Google Scholar 

  36. Pothen, A., Simon, H. & Liou, K-P. Partitioning sparse matrices with eigenvectors of graphs. SIAM J. Matrix Anal. Appl. 11, 430–452 (1990).

    Article  MathSciNet  Google Scholar 

  37. Kernighan, B. W. & Lin, S. An efficient heuristic procedure for partitioning graphs. Bell Syst. Tech. J. 49, 291–307 (1970).

    Article  Google Scholar 

  38. Zachary, W. W. An information flow model for conflict and fission in small groups. J. Anthropol. Res. 33, 452–473 (1977).

    Article  Google Scholar 

  39. White, D. R. & Harary, F. The cohesiveness of blocks in social networks: Connectivity and conditional density. Sociol. Methodol. 31, 305–359 (2001).

    Article  Google Scholar 

  40. Duch, J. & Arenas, A. Community detection in complex networks using extremal optimization. Phys. Rev. E 72, 027104 (2005).

    Article  ADS  Google Scholar 

  41. Wilkinson, D. M. & Huberman, B. A. A method for finding communities of related genes. Proc. Natl Acad. Sci. USA 101, 5241–5248 (2004).

    Article  ADS  Google Scholar 

  42. Wu, F. & Huberman, B. A. Finding communities in linear time: A physics approach. Eur. Phys. J. B 38, 331–338 (2004).

    Article  ADS  Google Scholar 

  43. Rosvall, M. & Bergstrom, C. T. Multilevel compression of random walks on networks reveals hierarchical organization in large integrated systems. PLoS One 6, e18209 (2011).

    Article  ADS  Google Scholar 

  44. Zhou, H. & Lipowsky, R. Network Brownian Motion: A New Method to Measure Vertex–Vertex Proximity and to Identify Communities and Subcommunities 1062–1069 (Lecture Notes in Computer Science, Vol. 3038, Springer, 2004).

    Google Scholar 

  45. Pons, P. & Latapy, M. Proc. 20th International Symposium on Computer and Information Sciences 284–293 (Lecture Notes in Computer Science, Vol. 3733, Springer, 2005).

    Google Scholar 

  46. Reichardt, J. & Bornholdt, S. Detecting fuzzy community structures in complex networks with a Potts model. Phys. Rev. Lett. 93, 218701 (2004).

    Article  ADS  Google Scholar 

  47. Boccaletti, S., Ivanchenko, M., Latora, V., Pluchino, A. & Rapisarda, A. Detection of complex networks modularity by dynamical clustering. Phys. Rev. E 75, 045102 (2007).

    Article  ADS  Google Scholar 

  48. Karckhardt, D. & Stern, R. Informal networks and organizational crises: An experimental simulation. Soc. Psychol. Q. 51, 123–140 (1988).

    Article  Google Scholar 

  49. Karrer, B. & Newman, M. E. J. Stochastic blockmodels and community structure in networks. Phys. Rev. E 83, 016107 (2011).

    Article  ADS  MathSciNet  Google Scholar 

  50. Li, Z., Zhang, S., Wang, R-S., Zhang, X-S. & Chen, L. Quantitative function for community detection. Phys. Rev. E 77, 036109 (2008).

    Article  ADS  Google Scholar 

  51. Newman, M. E. J. Mixing patterns in networks. Phys. Rev. E 67, 026126 (2003).

    Article  ADS  MathSciNet  Google Scholar 

  52. Brandes, U. et al. Proc. 33rd International Workshop on Graph-Theoretic Concepts in Computer Science (Lecture Notes in Computer Science,Vol. 4769, Springer, 2007).

    Google Scholar 

  53. Medus, A., Acuña, G. & Dorso, C. O. Detection of community structures in networks via global optimization. Physica A 358, 593–604 (2005).

    Article  ADS  Google Scholar 

  54. Clauset, A., Newman, M. E. J. & Moore, C. Finding community structure in very large networks. Phys. Rev. E 70, 066111 (2004).

    Article  ADS  Google Scholar 

  55. Wakita, K. & Tsurumi, T. in Proc. IADIS International Conference, WWW/Internet 2007 (eds Isaı´as, P., Nunes, M. B. & Barroso, J.) 153–162 (IADIS Press, 2007).

    Google Scholar 

  56. Newman, M. E. J. Modularity and community structure in networks. Proc. Natl Acad. Sci. USA 103, 8577–8582 (2006).

    ADS  Google Scholar 

  57. Shuzhuo, L., Yinghui, C., Haifeng, D. & Feldman, M. W. A genetic algorithm with local search strategy for improved detection of community structure. Complexity 15, 53–60 (2010).

    MathSciNet  Google Scholar 

  58. Fortunato, S. & Barthélémy, M. Resolution limit in community detection. Proc. Natl Acad. Sci. USA 104, 36–41 (2007).

    Article  ADS  Google Scholar 

  59. Reichardt, J. & Bornholdt, S. Statistical mechanics of community detection. Phys. Rev. E 74, 016110 (2006).

    Article  ADS  MathSciNet  Google Scholar 

  60. Arenas, A., Fernandez, A. & Gomez, S. Analysis of the structure of complex networks at different resolution levels. New J. Phys. 10, 053039 (2008).

    Article  ADS  Google Scholar 

  61. Breiger, R. L., Boorman, S. A. & Arabie, P. An algorithm for clustering relations data with applications to social network analysis and comparison with multidimensional scaling. J. Math. Psychol. 12, 328–383 (1975).

    Article  Google Scholar 

  62. Holland, P. W., Laskey, K. B. & Leinhardt, S. Stochastic blockmodels: Some first steps. Soc. Networks 5, 109–137 (1983).

    Article  Google Scholar 

  63. Snijders, T. A. B. & Nowicki, K. Estimation and prediction for stochastic blockmodels for graphs with latent block structure. J. Classification 14, 75–100 (1997).

    Article  MathSciNet  Google Scholar 

  64. Nowicki, K. & Snijders, T. A. B. Estimation and prediction for stochastic blockstructures. J. Am. Stat. Assoc. 96, 1077–1087 (2001).

    Article  MathSciNet  Google Scholar 

  65. Airoldi, E. M., Blei, D. M., Fienberg, S. E. & Xing, E. P. Mixed membership stochastic blockmodels. J. Mach. Learning Res. 9, 1981–2014 (2008).

    MATH  Google Scholar 

  66. Goldenberg, A., Zheng, A. X., Feinberg, S. E. & Airoldi, E. M. A survey of statistical network structures. Found. Trends Mach. Learning 2, 1–117 (2009).

    Article  Google Scholar 

  67. Bickel, P. J. & Chen, A. A nonparametric view of network models and Newman–Girvan and other modularities. Proc. Natl Acad. Sci. USA 106, 21068–21073 (2009).

    Article  ADS  Google Scholar 

  68. Adamic, L. A. & Glance, N. Proc. WWW-2005 Workshop on the Weblogging Ecosystem (2005).

  69. Guimerà, R. & Sales-Pardo, M. Missing and spurious interactions and the reconstruction of complex networks. Proc. Natl Acad. Sci. USA 106, 22073–22078 (2009).

    Article  ADS  Google Scholar 

  70. Yan, X., Zhu, Y., Rouquier, J-B. & Moore, C. in Proc. 17th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (Association of Computing Machinery, 2011).

    Google Scholar 

  71. Clauset, A., Moore, C. & Newman, M. E. J. Hierarchical structure and the prediction of missing links in networks. Nature 453, 98–101 (2008).

    Article  ADS  Google Scholar 

Download references

Acknowledgements

Some of the work described here was financially supported by the US National Science Foundation under grants DMS–0405348 and DMS–0804778.

Author information

Authors and Affiliations

  1. Department of Physics and Center for the Study of Complex Systems, University of Michigan, Ann Arbor, Michigan 48109, USA

    M. E. J. Newman

Authors
  1. M. E. J. Newman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. E. J. Newman.

Ethics declarations

Competing interests

The author declares no competing financial interests.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Newman, M. Communities, modules and large-scale structure in networks. Nature Phys 8, 25–31 (2012). https://doi.org/10.1038/nphys2162

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/nphys2162

This article is cited by

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing