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Maximum modular graphs

  • S. TrajanovskiEmail author
  • H. Wang
  • P. Van Mieghem
Regular Article

Abstract

Modularity has been explored as an important quantitative metric for community and cluster detection in networks. Finding the maximum modularity of a given graph has been proven to be NP-complete and therefore, several heuristic algorithms have been proposed. We investigate the problem of finding the maximum modularity of classes of graphs that have the same number of links and/or nodes and determine analytical upper bounds. Moreover, from the set of all connected graphs with a fixed number of links and/or number of nodes, we construct graphs that can attain maximum modularity, named maximum modular graphs. The maximum modularity is shown to depend on the residue obtained when the number of links is divided by the number of communities. Two applications in transportation networks and data-centers design that can benefit of maximum modular partitioning are proposed.

Keywords

Statistical and Nonlinear Physics 

References

  1. 1.
    M.E.J. Newman, M. Girvan, Phys. Rev. E 69, 026113 (2004)ADSCrossRefGoogle Scholar
  2. 2.
    U. Brandes, D. Delling, M. Gaertler, R. Görke, M. Hoefer, Z. Nikoloski, D. Wagner, in Graph-Theoretic Concepts in Computer Science, Lecture Notes in Computer Science (Springer, Berlin/Heidelberg, 2007), Vol. 4769, Chap. 12, pp. 121–132.Google Scholar
  3. 3.
    M.E.J. Newman, Phys. Rev. E 69, 066133 (2004)ADSCrossRefGoogle Scholar
  4. 4.
    R. Guimerà, L.A.N. Amaral, Nature 433, 895 (2005)ADSCrossRefGoogle Scholar
  5. 5.
    R. Guimerà, M. Sales-Pardo, L.A.N. Amaral, Phys. Rev. E 70, 025101 (2004)ADSCrossRefGoogle Scholar
  6. 6.
    J. Duch, A. Arenas, Phys. Rev. E 72, 027104 (2005)ADSCrossRefGoogle Scholar
  7. 7.
    S. Fortunato, Phys. Rep. 486, 75 (2010)MathSciNetADSCrossRefGoogle Scholar
  8. 8.
    P. Van Mieghem, X. Ge, P. Schumm, S. Trajanovski, H. Wang, Phys. Rev. E 82, 056113 (2010)ADSCrossRefGoogle Scholar
  9. 9.
    S. Fortunato, M. Barthélemy, Proc. Natl. Acad. Sci. 104, 36 (2007)ADSCrossRefGoogle Scholar
  10. 10.
    M. Fiedler, Czechoslovak Math. J. 23, 298 (1973)MathSciNetGoogle Scholar
  11. 11.
    M.E.J. Newman, Proc. Natl. Acad. Sci.103, 8577 (2006)ADSCrossRefGoogle Scholar
  12. 12.
    P. Van Mieghem, Graph Spectra for Complex Networks (Cambridge University Press, Cambridge, 2010)Google Scholar
  13. 13.
    A. Clauset, M.E.J. Newman, C. Moore, Phys. Rev. E70, 066111 (2004)ADSCrossRefGoogle Scholar
  14. 14.
    G. Agarwal, D. Kempe, Eur. Phys. J. B 66, 409 (2008)MathSciNetADSzbMATHCrossRefGoogle Scholar
  15. 15.
    J.-C. Delvenne, S.N. Yaliraki, M. Barahona, Proc. Natl. Acad. Sci. 107, 12755 (2010)ADSCrossRefGoogle Scholar
  16. 16.
    A. Cauchy, Cours d’Analyse de l’Ecole Royale Polytechnique: Analyse Algébrique. Debure (reissued by Cambridge University Press, Cambridge, 2009), p. 1821Google Scholar
  17. 17.
    P. Van Mieghem, H. Wang, X. Ge, S. Tang, F.A. Kuipers, Eur. Phys. J. B 76, 643 (2010)ADSzbMATHCrossRefGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Faculty of EEMCSDelft University of TechnologyDelftThe Netherlands

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