Computing an upper bound of modularity

  • Atsushi MiyauchiEmail author
  • Yuichiro Miyamoto
Regular Article


Modularity proposed by Newman and Girvan is a quality function for community detection. Numerous heuristics for modularity maximization have been proposed because the problem is NP-hard. However, the accuracy of these heuristics has yet to be properly evaluated because computational experiments typically use large networks whose optimal modularity is unknown. In this study, we propose two powerful methods for computing a nontrivial upper bound of modularity. More precisely, our methods can obtain the optimal value of a linear programming relaxation of the standard integer linear programming for modularity maximization. The first method modifies the traditional row generation approach proposed by Grötschel and Wakabayashi to shorten the computation time. The second method is based on a row and column generation. In this method, we first solve a significantly small subproblem of the linear programming and iteratively add rows and columns. Owing to the speed and memory efficiency of these proposed methods, they are suitable for large networks. In particular, the second method consumes exceedingly small memory capacity, enabling us to compute the optimal value of the linear programming for the Power Grid network (consisting of 4941 vertices and 6594 edges) on a standard desktop computer.


Statistical and Nonlinear Physics 


  1. 1.
    S. Fortunato, Phys. Rep. 486, 75 (2010)MathSciNetADSCrossRefGoogle Scholar
  2. 2.
    M.E.J. Newman, M. Girvan, Phys. Rev. E 69, 026113 (2004)ADSCrossRefGoogle Scholar
  3. 3.
    S. Fortunato, M. Barthélemy, Proc. Natl. Acad. Sci. USA 104, 36 (2007)ADSCrossRefGoogle Scholar
  4. 4.
    U. Brandes, D. Delling, M. Gaertler, R. Görke, M. Hoefer, Z. Nikoloski, D. Wagner, IEEE Trans. Knowl. Data Eng. 20, 172 (2008)CrossRefGoogle Scholar
  5. 5.
    A. Clauset, M.E.J. Newman, C. Moore, Phys. Rev. E 70, 066111 (2004)ADSCrossRefGoogle Scholar
  6. 6.
    V.D. Blondel, J.-L. Guillaume, R. Lambiotte, E. Lefebvre, J. Stat. Mech. 2008, P10008 (2008)CrossRefGoogle Scholar
  7. 7.
    R. Guimerà, L.A.N. Amaral, Nature 433, 895 (2005)ADSCrossRefGoogle Scholar
  8. 8.
    C.P. Massen, J.P.K. Doye, Phys. Rev. E 71, 046101 (2005)MathSciNetADSCrossRefGoogle Scholar
  9. 9.
    A.D. Medus, G. Acuña, C.O. Dorso, Physica A 358, 593 (2005)ADSCrossRefGoogle Scholar
  10. 10.
    J. Duch, A. Arenas, Phys. Rev. E 72, 027104 (2005)ADSCrossRefGoogle Scholar
  11. 11.
    M.E.J. Newman, Proc. Natl. Acad. Sci. USA 103, 8577 (2006)ADSCrossRefGoogle Scholar
  12. 12.
    T. Richardson, P.J. Mucha, M.A. Porter, Phys. Rev. E 80, 036111 (2009)ADSCrossRefGoogle Scholar
  13. 13.
    S. Boccaletti, M. Ivanchenko, V. Latora, A. Pluchino, A. Rapisarda, Phys. Rev. E 75, 045102 (2007)ADSCrossRefGoogle Scholar
  14. 14.
    H.N. Djidjev, Lect. Notes Comput. Sci. 4936, 117 (2008)CrossRefGoogle Scholar
  15. 15.
    Y. Niu, B. Hu, W. Zhang, M. Wang, Physica A 387, 6215 (2008)MathSciNetADSCrossRefGoogle Scholar
  16. 16.
    G. Agarwal, D. Kempe, Eur. Phys. J. B 66, 409 (2008)MathSciNetADSCrossRefzbMATHGoogle Scholar
  17. 17.
    S. Cafieri, P. Hansen, L. Liberti, Discrete Appl. Math., in press, DOI: 10.1016/j.dam.2012.03.030
  18. 18.
    M. Grötschel, Y. Wakabayashi, Math. Prog. 45, 59 (1989)CrossRefzbMATHGoogle Scholar
  19. 19.
    M. Grötschel, Y. Wakabayashi, Math. Prog. 47, 367 (1990)CrossRefzbMATHGoogle Scholar
  20. 20.
    G. Xu, S. Tsoka, L.G. Papageorgiou, Eur. Phys. J. B 60, 231 (2007)ADSCrossRefzbMATHGoogle Scholar
  21. 21.
    D. Aloise, S. Cafieri, G. Caporossi, P. Hansen, S. Perron, L. Liberti, Phys. Rev. E 82, 046112 (2010)ADSCrossRefGoogle Scholar
  22. 22.
    B. DasGupta, D. Desai, J. Comput. System Sci. 79, 50 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    D.J. Watts, S.H. Strogatz, Nature 393, 440 (1998)ADSCrossRefGoogle Scholar
  24. 24.
    T.N. Dinh, M.T. Thai, arXiv:1108.4034 (2011)Google Scholar
  25. 25.
    W. Zachary, J. Anthropol. Res. 33, 452 (1977)Google Scholar
  26. 26.
    D. Lusseau, K. Schneider, O.J. Boisseau, P. Haase, E. Slooten, S.M. Dawson, Behav. Ecol. Sociobiol. 54, 396 (2003)CrossRefGoogle Scholar
  27. 27.
    D.E. Knuth, The Stanford GraphBase: A Platform for Combinatorial Computing (Addison-Wesley, Reading, MA, 1993)Google Scholar
  28. 28.
  29. 29.
    M. Girvan, M.E.J. Newman, Proc. Natl. Acad. Sci. USA 99, 7821 (2002)MathSciNetADSCrossRefzbMATHGoogle Scholar
  30. 30.
  31. 31.
  32. 32.
    M.E.J. Newman, Phys. Rev. E 74, 036104 (2006)MathSciNetADSCrossRefGoogle Scholar
  33. 33.
    R. Guimerà, L.A.N. Amaral, J. Stat. Mech. 2005, P02001 (2005)CrossRefGoogle Scholar
  34. 34.
    S. Cafieri, P. Hansen, L. Liberti, Phys. Rev. E 83, 056105 (2011)ADSCrossRefGoogle Scholar
  35. 35.
    W. Li, D. Schuurmans, in Proceedings of the Twenty-Second international joint conference on Artificial Intelligence, IJCAI’11 (AAAI Press, 2011), Vol. 2, pp. 1366–1371Google Scholar
  36. 36.
    S. Cafieri, A. Costa, P. Hansen, Ann. Oper. Res., in press, DOI: 10.1007/s10479-012-1286-z

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Graduate School of Decision Science and Technology, Tokyo Institute of TechnologyTokyoJapan
  2. 2.Faculty of Science and Technology, Sophia UniversityTokyoJapan

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