Skip to main content
SpringerLink
Log in

A locally optimal hierarchical divisive heuristic for bipartite modularity maximization

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

Given a set of entities, cluster analysis aims at finding subsets, also called clusters or communities or modules, entities of which are homogeneous and well separated. In the last ten years clustering on networks, or graphs, has been a subject of intense study. Edges between pairs of vertices within the same cluster should be relatively dense, while edges between pairs of vertices in different clusters should be relatively sparse. This led Newman to define the modularity of a cluster as the difference between the number of internal edges and the expected number of such edges in a random graph with the same degree distribution. The modularity of a partition of the vertices is the sum of the modularities of its clusters. Modularity has been extended recently to the case of bipartite graphs. In this paper we propose a hierarchical divisive heuristic for approximate modularity maximization in bipartite graphs. The subproblem of bipartitioning a cluster is solved exactly; hence the heuristic is locally optimal. Several formulations of this subproblem are presented and compared. Some are much better than others, and this illustrates the importance of reformulations. Computational experiences on a series of ten test problems from the literature are reported.

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

Fig. 1

Similar content being viewed by others

References

  1. Aloise, D., Cafieri, S., Caporossi, G., Hansen, P., Perron, S., Liberti, L.: Column generation algorithms for exact modularity maximization in networks. Phys. Rev. E 82(4), 046112 (2010)

    Article  Google Scholar 

  2. Arenas, A., Fernández, A., Gómez, S.: Analysis of the structure of complex networks at different resolution levels. New J. Phys. 10(5), 053039 (2008)

    Article  Google Scholar 

  3. Barber, M.J.: Modularity and community detection in bipartite networks. Phys. Rev. E 76(6), 066102 (2007)

    Article  MathSciNet  Google Scholar 

  4. Barber, M.J., Clark, J.W.: Detecting network communities by propagating labels under constraints. Phys. Rev. E 80(2), 026129 (2009)

    Article  Google Scholar 

  5. Batagelj, V., Mrvar, A.: Pajek datasets. http://vlado.fmf.uni-lj.si/pub/networks/data (2006)

  6. Belotti, P., Lee, J., Liberti, L., Margot, F., Wächter, A.: Branching and bounds tightening techniques for non-convex MINLP. Optim. Methods Softw. 24(4–5), 597–634 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  7. Boulle, M.: Compact mathematical formulation for graph partitioning. Optim. Eng. 5(3), 315–333 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  8. Brandes, U., Delling, D., Gaertler, M., Görke, R., Hoefer, M., Nikoloski, Z., Wagner, D.: On modularity clustering. IEEE Trans. Knowl. Data Eng. 20(2), 172–188 (2008)

    Article  Google Scholar 

  9. Cafieri, S., Costa, A., Hansen P.: Reformulation of a model for hierarchical divisive graph modularity maximization. Ann. Operat. Res. (accepted)

  10. Cafieri, S., Hansen, P., Liberti, L.: Loops and multiple edges in modularity maximization of networks. Phys. Rev. E 81(4), 046102 (2010)

    Article  MathSciNet  Google Scholar 

  11. Cafieri, S., Hansen, P., Liberti, L.: Locally optimal heuristic for modularity maximization of networks. Phys. Rev. E 83(5), 056105 (2011)

    Article  Google Scholar 

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

    Article  Google Scholar 

  13. Costa A.: Applications of reformulation in mathematical programming. PhD thesis, École Polytechnique (2012)

  14. Costa, A., Hansen, P.: Comment on “Evolutionary method for finding communities in bipartite networks”. Phys. Rev. E 84(5), 058101 (2011)

    Article  Google Scholar 

  15. Fan, N., Pardalos, P.M.: Linear and quadratic programming approaches for the general graph partitioning problem. J. Glob. Optim. 48(1), 57–71 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  16. Fortet, R.: Applications de l’algèbre de Boole en recherche opérationelle. Revue Française de Recherche Opérationelle 4(14), 17–26 (1960)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  18. Fortunato, S., Barthélemi, M.: Resolution limit in community detection. Proc. Nat. Acad. Sci. USA 104(1), 36–41 (2007)

    Article  Google Scholar 

  19. Girvan, M., Newman, M.E.J.: Community structure in social and biological networks. Proc. Nat. Acad. Sci. USA 99(12), 7821–7826 (2007)

    Article  MathSciNet  Google Scholar 

  20. Good, B.H., de Montjoye, Y.-A., Clauset, A.: Performance of modularity maximization in practical contexts. Phys. Rev. E 81(4), 046106 (2010)

    Google Scholar 

  21. IBM. ILOG CPLEX 12.2 User’s Manual. IBM (2010)

  22. Kumpula, J.M., Saramäki, J., Kaski, K., Kertész, J.: Limited resolution and multiresolution methods in complex network community detection. Fluctuations Noise Lett. 7(3), 209 (2007)

    Article  Google Scholar 

  23. Liberti, L.: Reformulations in mathematical programming: definitions and systematics. RAIRO-OR 43(1), 55–86 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  24. Liu, X., Murata, T.: Community detection in large-scale bipartite networks. In: IEEE/WIC/ACM international conference on web intelligence and Intelligent Agent Technologies, pp. 50–57 (2009)

  25. Liu, X., Murata, T.: An efficient algorithm for optimizing bipartite modularity in bipartite networks. J. Adv. Comput. Intell. Intell. Inform. 14(4), 408–415 (2010)

    Google Scholar 

  26. Newman, M.E.J.: Fast algorithm for detecting community structure in networks. Phys. Rev. E 69(6), 066133 (2004)

    Article  Google Scholar 

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

    Article  Google Scholar 

  28. Raghavan, U.N., Albert, R., Kumara, S.: Near linear time algorithm to detect community structures in large-scale networks. Phys. Rev. E 76(3), 036106 (2007)

    Article  Google Scholar 

  29. Reichardt, J., Bornholdt, S.: Statistical mechanics of community detection. Phys. Rev. E 74(1), 016110 (2006)

    Article  MathSciNet  Google Scholar 

  30. Sales-Pardo, M., Guimerà, R., Moreira, A.A., Amaral, L.A.N.: Extracting the hierarchical organization of complex systems. Proc. Nat. Acad. Sci. USA 104(39), 15224–15229 (2007)

    Google Scholar 

  31. Schuetz, P., Caflisch, A.: Efficient modularity optimization by multistep greedy algorithm and vertex mover refinement. Phys. Rev. E 77(4), 046112 (2008)

    Article  Google Scholar 

  32. Schuetz, P., Caflisch, A.: Multistep greedy algorithm identifies community structure in real-world and computer-generated networks. Phys. Rev. E 78(2), 026112 (2008)

    Article  Google Scholar 

  33. Xu, G., Tsoka, S., Papageorgiou, L.G.: Finding community structures in complex networks using mixed integer optimisation. Eur. Phys. J. B 60(2), 231–239 (2007)

    Article  MATH  Google Scholar 

  34. Zhan, W., Zhang, Z., Guan, J., Zhou, S.: Evolutionary method for finding communities in bipartite networks. Phys. Rev. E 83(6), 066120 (2011)

    Article  Google Scholar 

Download references

Acknowledgments

The authors would like to thank Sonia Cafieri and Leo Liberti for the precious suggestions and comments. Financial support by grants: Digiteo 2009-14D “RMNCCO”, Digiteo 2009-55D “ARM” is gratefully acknowledged.

Author information

Authors and Affiliations

  1. LIX, École Polytechnique, 91128 , Palaiseau, France

    Alberto Costa & Pierre Hansen

  2. GERAD, HEC Montréal, 3000 Chemin de la Côte-Sainte-Catherine, Montréal, H3T 2A7, Canada

    Pierre Hansen

Authors
  1. Alberto Costa
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pierre Hansen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alberto Costa.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Costa, A., Hansen, P. A locally optimal hierarchical divisive heuristic for bipartite modularity maximization. Optim Lett 8, 903–917 (2014). https://doi.org/10.1007/s11590-013-0621-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-013-0621-x

Keywords

Search

Navigation