Evolving networks with bimodal degree distribution

  • Abhijeet R. Sonawane
  • A. Bhattacharyay
  • M. S. Santhanam
  • G. Ambika
Regular Article

Abstract

Networks with bimodal degree distribution are most robust to targeted and random attacks. We present a model for constructing a network with bimodal degree distribution. The procedure adopted is to add nodes to the network with a probability p and delete the links between nodes with probability (1 − p). We introduce an additional constraint in the process through an immunity score, which controls the dynamics of the growth process based on the feedback value of the last few time steps. This results in bimodal nature for the degree distribution. We study the standard quantities which characterize the networks, like average path length and clustering coefficient in the context of our growth process and show that the resultant network is in the small world family. It is interesting to note that bimodality in degree distribution is an emergent phenomenon.

Keywords

Statistical and Nonlinear Physics 

References

  1. 1.
    M.E.J. Newman, Networks: an introduction (Oxford, UK, 2010)Google Scholar
  2. 2.
    F. Kepes, Biological Networks (World-Scientific, Singa- pore, 2007)Google Scholar
  3. 3.
    S.N. Dorogovtsev, J.F.F. Mendes, Evolution of networks: from biological nets to the Internet and WWW (Oxford University Press, UK, 2003)Google Scholar
  4. 4.
    R. Cohen, S. Havlin, Phys. Rev. Lett. 90, 058701 (2003)ADSCrossRefGoogle Scholar
  5. 5.
    D.J. Watts, S.H. Strogatz, Nature 393, 440 (1998)ADSCrossRefGoogle Scholar
  6. 6.
    M.E.J. Newman, D.J. Watts, Phys. Lett. A 263, 341 (1999)MathSciNetADSMATHCrossRefGoogle Scholar
  7. 7.
    M.E.J. Newman, D.J. Watts, Phys. Rev. E 60, 7332 (1999)ADSCrossRefGoogle Scholar
  8. 8.
    A.-L. Barábasi, R. Albert, Science 286, 509 (1999)MathSciNetCrossRefGoogle Scholar
  9. 9.
    R.M. D’Souza, P.L. Krapivsky, C. Moore, Eur. Phys. J. B 59, 535 (2007)MathSciNetADSMATHCrossRefGoogle Scholar
  10. 10.
    R.M. D’Souza, C. Borgs, J.T. Chayes, N. Berger, R.D. Klienberg, Proc. Natl. Acad. Sci. 104, 6112 (2007)ADSCrossRefGoogle Scholar
  11. 11.
    C.I. Del Genio, T. Gross, K.E. Bassler, Phys. Rev. Lett. 107, 178701 (2011)ADSCrossRefGoogle Scholar
  12. 12.
    R. Albert, A.-L. Barábasi, Rev. Mod. Phys. 74, 47 (2002)ADSMATHCrossRefGoogle Scholar
  13. 13.
    R. Cohen, S. Havlin, Complex Networks: Structure, Robustness and Functions (Cambridge University Press, UK, 2010)Google Scholar
  14. 14.
    G. Paul, T. Tanizawa, S. Havlin, H.E. Stanley, Eur. Phys. J. B 38, 187 (2004)ADSCrossRefGoogle Scholar
  15. 15.
    T. Tanizawa, G. Paul, S. Havlin, H.E. Stanley, Phys. Rev. E 74, 016215 (2006)ADSCrossRefGoogle Scholar
  16. 16.
    T. Tanizawa, G. Paul, R. Cohen, S. Havlin, H.E. Stanley, Phys. Rev. E 71, 047101 (2005)ADSCrossRefGoogle Scholar
  17. 17.
    A. Valente, A. Sarkar, H.A. Stone, Phys. Rev. Lett. 92, 118702 (2004)ADSCrossRefGoogle Scholar
  18. 18.
    M.E.J. Newman, S.H. Strogatz, D.J. Watts, Phys. Rev. E 64, 026118 (2001)ADSCrossRefGoogle Scholar
  19. 19.
    E.A. Bender, E.R. Canfield, J. Comb. Th. A 24, 296 (1978)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    B. Bollobas, Random Graphs (Academic Press, New York, 1985)Google Scholar
  21. 21.
    M. Molloy, B. Reed, Random. Struct. Algorithms 6, 161 (1995)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    M. Molloy, B. Reed, Combinatorics, Probability Computing 7, 295 (1998)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    T. Britton, M. Deijfen, A. Martin-Löf, J. Stat. Phys. 124, 1377 (2006)MathSciNetADSMATHCrossRefGoogle Scholar
  24. 24.
    P. Erdös, T. Gallai, Matematikai Lapok 11, 264 (1960)Google Scholar
  25. 25.
    J. Billen, M. Wilson, A. Rabinovich, A.R.C. Baljon, Europhys. Lett. 87, 68003 (2009)ADSCrossRefGoogle Scholar
  26. 26.
    S. Perumal, A.A. Minai, Proceedings of the IJCNN 2009 (Atlanta, GA, 2009)Google Scholar
  27. 27.
    J.M. Carlson, J. Doyle, Phys. Rev. Lett. 84, 2529 (2000)ADSCrossRefGoogle Scholar
  28. 28.
    R.R. Vallabhajosyula, D. Chakravarti, S. Lutfeali, A. Ray, A. Raval, PLoS ONE 4, e5344 (2009)ADSCrossRefGoogle Scholar
  29. 29.
    T. Watanabe, e-print arXiv:1108.0742v1 [nlin.CD] (2011)Google Scholar
  30. 30.
    A. Kohn, Punished by rewards (Houghton Miffin Company, New York, 1999)Google Scholar

Copyright information

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

Authors and Affiliations

  • Abhijeet R. Sonawane
    • 1
  • A. Bhattacharyay
    • 1
  • M. S. Santhanam
    • 1
  • G. Ambika
    • 1
  1. 1.Indian Institute of Science Education and ResearchPuneIndia

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