Application of Semidefinite Programming to Maximize the Spectral Gap Produced by Node Removal

  • Naoki MasudaEmail author
  • Tetsuya Fujie
  • Kazuo Murota
Conference paper
Part of the Studies in Computational Intelligence book series (SCI, volume 476)


The smallest positive eigenvalue of the Laplacian of a network is called the spectral gap and characterizes various dynamics on networks. We propose mathematical programming methods to maximize the spectral gap of a given network by removing a fixed number of nodes. We formulate relaxed versions of the original problem using semidefinite programming and apply them to example networks.


combinatorial optimization network synchronization random walk opinion formation Laplacian eigenvalue 


  1. 1.
  2. 2.
  3. 3.
    Almendral, J.A., Díaz-Guilera, A.: Dynamical and spectral properties of complex networks. New J. Phys. 9, 187 (2007)CrossRefGoogle Scholar
  4. 4.
    Arenas, A., Díaz-Guilera, A., Kurths, J., Moreno, Y., Zhou, C.: Synchronization in complex networks. Phys. Rep. 469, 93–153 (2008)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Barabási, A.L., Albert, R.: Emergence of scaling in random networks. Science 286, 509–512 (1999)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bendsøe, M.P., Sigmund, O.: Topology Optimization. Springer (2003)Google Scholar
  7. 7.
    Chen, B.L., Hall, D.H., Chklovskii, D.B.: Wiring optimization can relate neuronal structure and function. Proc. Natl. Acad. Sci. USA 103, 4723–4728 (2006)CrossRefGoogle Scholar
  8. 8.
    Cvetković, D., Rowlinson, P., Simić, S.: An Introduction to the Theory of Graph Spectra. CMU (2010)Google Scholar
  9. 9.
    Cvetkovic, D., Cangalovic, M., Kovacevic-Vujcic, V.: Semidefinite Programming Methods for the Symmetric Traveling Salesman Problem. In: Cornuéjols, G., Burkard, R.E., Woeginger, G.J. (eds.) IPCO 1999. LNCS, vol. 1610, pp. 126–136. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  10. 10.
    Donetti, L., Neri, F., Munoz, M.A.: Optimal network topologies: expanders, cages, Ramanujan graphs, entangled networks and all that. J. Stat. Mech., P08007 (2006)Google Scholar
  11. 11.
    Fukuda, M., Kojima, M., Murota, K., Nakata, K.: Exploiting sparsity in semidefinite programming via matrix completion I: general framework. SIAM J. Optim. 11, 647–674 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Goemans, M.X.: Semidefinite programming in combinatorial optimization. Math. Programming 79, 143–161 (1997)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Grötschel, M., Lovász, L., Schrijver, A.: Relaxations of vertex packing. J. Comb. Theory B 40, 330–343 (1986)zbMATHCrossRefGoogle Scholar
  14. 14.
    Lovász, L.: On the Shannon capacity of a graph. IEEE Trans. on Info. Th. 25, 1–7 (1979)zbMATHCrossRefGoogle Scholar
  15. 15.
    Lovász, L., Schrijver, A.: Cones of matrices and set-functions and 0–1 optimization. Siam J. Optimiz. 1, 166–190 (1991)zbMATHCrossRefGoogle Scholar
  16. 16.
    Nakata, K., Fujisawa, K., Fukuda, M., Kojima, M., Murota, K.: Exploiting sparsity in semidefinite programming via matrix completion II: implementation and numerical results. Math. Program. Ser. B 95, 305–327 (2003)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Olfati-Saber, R., Fax, J., Murray, R.: Consensus and cooperation in networked multi-agent systems. Proceedings of the IEEE 95, 215–233 (2007)CrossRefGoogle Scholar
  18. 18.
    Padberg, M.: The Boolean quadric polytope—some characteristics, facets and relatives. Math. Programming 45(1), 139–172 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Sporns, O., Zwi, J.D.: The small world of the cerebral cortex. Neuroinformatics 4, 145–162 (2004)CrossRefGoogle Scholar
  20. 20.
    Watanabe, T., Masuda, N.: Enhancing the spectral gap of networks by node removal. Phys. Rev. E 82, 46102 (2010)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Zachary, W.W.: An information flow model for conflict and fission in small groups. J. Anthropological Res. 33, 452–473 (1977)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of Mathematical InformaticsThe University of TokyoBunkyoJapan
  2. 2.PRESTOJapan Science and Technology AgencyKawaguchiJapan
  3. 3.Graduate School of BusinessUniversity of HyogoNishi-kuJapan

Personalised recommendations