The European Physical Journal B

, Volume 77, Issue 2, pp 213–217 | Cite as

Percolation of arbitrary uncorrelated nested subgraphs

Statistical and Nonlinear Physics

Abstract.

The study of percolation in so-called nested subgraphs implies a generalization of the concept of percolation since the results are not linked to specific graph process. Here the behavior of such graphs at criticallity is studied for the case where the nesting operation is performed in an uncorrelated way. Specifically, I provide an analyitic derivation for the percolation inequality showing that the cluster size distribution under a generalized process of uncorrelated nesting at criticality follows a power law with universal exponent γ = 3/2. The relevance of the result comes from the wide variety of processes responsible for the emergence of the giant component that fall within the category of nesting operations, whose outcome is a family of nested subgraphs.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    R. Albert, H. Jeong, A.-L. Barabási, Nature 406, 378 (2000) CrossRefADSGoogle Scholar
  2. 2.
    R. Cohen, K. Erez, D. ben-Avraham, S. Havlin, Phys. Rev. Lett. 85, 4626 (2000) CrossRefADSGoogle Scholar
  3. 3.
    S.N. Dorogovtsev, J.F.F. Mendes, Evolution of Networks: From Biological Nets to the Internet and WWW (Oxford University Press, Oxford, 2003) Google Scholar
  4. 4.
    D.S. Callaway, M.E.J. Newman, S.E. Strogatz, D.J. Watts, Phys. Rev. Lett. 85, 5468 (2000) CrossRefADSGoogle Scholar
  5. 5.
    M.E.J. Newman, S.H. Strogatz, D.J. Watts, Phys. Rev. E 64, 026118 (2001) CrossRefADSGoogle Scholar
  6. 6.
    R. Cohen, D. ben-Avraham, S. Havlin, Phys. Rev. E 66, 036113 (2002) CrossRefADSGoogle Scholar
  7. 7.
    R.V. Solé, J.M. Montoya, Proc. R. Soc. Lond. B 268, 2039 (2001) CrossRefGoogle Scholar
  8. 8.
    R. Pastor-Satorras, A. Vespignani, Phys. Rev. Lett. 86, 3200 (2001) CrossRefADSGoogle Scholar
  9. 9.
    S.N. Dorogovtsev, A.V. Goltsev, J.F.F Mendes, Phys. Rev. Lett. 96, 040601 (2006) CrossRefADSGoogle Scholar
  10. 10.
    A.V. Goltsev, S.N. Dorogovtsev, J.F.F. Mendes, Phys. Rev. E 78, 051105 (2008) CrossRefMathSciNetADSGoogle Scholar
  11. 11.
    A. Bekessy, P. Bekessy, J. Komlos, Stud. Sci. Math, Hungar. 7, 343 (1972) MathSciNetGoogle Scholar
  12. 12.
    E.A. Bender, E.R. Canfield, J. Combinatorial Theory A 24, 296 (1978) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    B. Bollobás, Eur. J. Comb. 1, 311 (1980) MATHGoogle Scholar
  14. 14.
    N.C. Wormald, J. Comb. Theor. B 31, 156 (1981) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    B. Bollobás, Random Graphs, 2nd edn. (Cambridge University Press, Cambridge, 2001) Google Scholar
  16. 16.
    J. Park, M.E.J. Newman, Phys. Rev. E 70, 066117 (2004) CrossRefMathSciNetADSGoogle Scholar
  17. 17.
    P. Erdös, A. Rényi, Publicationes Mathematicae 6, 290 (1959) MATHGoogle Scholar
  18. 18.
    H.E. Stanley, Introduction to Phase Transitions and Critical Phenomena (Oxford Univ. Press, Oxford, 1971) Google Scholar
  19. 19.
    B. Corominas-Murtra, J.F.F. Mendes, R.J. Solé, Phys. A: Math. Theor. 41, 385003 (2008) CrossRefADSGoogle Scholar
  20. 20.
    D. Fernholz, V. Ramachandran, Technical Report TR04-13, University of Texas at Austin. G/A 2004 Google Scholar
  21. 21.
    C. Rodriguez-Caso, M.A. Medina, R.V. Solé, FEBS Journal 272, 6423 (2005) CrossRefGoogle Scholar
  22. 22.
    B. Corominas-Murtra, S. Valverde, C. Rodríguez-Caso, R.V. Solé, Europhys. Lett. 7, 18004 (2007) CrossRefGoogle Scholar
  23. 23.
    H.S. Wilf, Generatingfunctionology, 2nd edn. (Academic Press, Boston, London, 1994) Google Scholar
  24. 24.
    S. Janson, D.E. Knuth, T. Łuczak, B. Pittel, Rand. Struct. Alg. 4 , 231 (1993) Google Scholar
  25. 25.
    C. Moore, M.E.J. Newman, Phys. Rev. E 62, 7059 (2000) CrossRefADSGoogle Scholar
  26. 26.
    A.V. Goltsev, S.N. Dorogovtsev, J.F.F. Mendes, Phys. Rev. E 73, 056101 1 (2006) Google Scholar
  27. 27.
    M. Molloy, B. Reed, Rand Struct. Alg. 6, 161 (1995) MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    B. Bollobás, Modern graph theory Graduate Texts in Mathematics (Springer, New York, 1998), Vol. 184 Google Scholar
  29. 29.
    Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, edited by M. Abramowitz, I.A. Stegun (New York, Dover, 1972) Google Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.ICREA-Complex Systems Lab, Parc de Recerca Biomèdica-Universitat Pompeu FabraBarcelonaSpain

Personalised recommendations