Advertisement

Fractality and degree correlations in scale-free networks

  • Yuka Fujiki
  • Shogo Mizutaka
  • Kousuke Yakubo
Regular Article

Abstract

Fractal scale-free networks are empirically known to exhibit disassortative degree mixing. It is, however, not obvious whether a negative degree correlation between nearest neighbor nodes makes a scale-free network fractal. Here we examine the possibility that disassortativity in complex networks is the origin of fractality. To this end, maximally disassortative (MD) networks are prepared by rewiring edges while keeping the degree sequence of an initial uncorrelated scale-free network. We show that there are many MD networks with different topologies if the degree sequence is the same with that of the (u,v)-flower but most of them are not fractal. These results demonstrate that disassortativity does not cause the fractal property of networks. In addition, we suggest that fractality of scale-free networks requires a long-range repulsive correlation, in the sense of the shortest path distance, in similar degrees.

Keywords

Statistical and Nonlinear Physics 

References

  1. 1.
    R. Albert, A.-L. Barabási, Rev. Mod. Phys. 74, 47 (2002)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, Rev. Mod. Phys. 80, 1275 (2008)ADSCrossRefGoogle Scholar
  3. 3.
    R. Pastor-Satorras, C. Castellano, P. Van Mieghem, A. Vespignani, Rev. Mod. Phys. 87, 925 (2015)ADSCrossRefGoogle Scholar
  4. 4.
    A.-L. Barabási, R. Albert, Science 286, 509 (1999)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    M.E.J. Newman, Phys. Rev. Lett. 89, 208701 (2002)ADSCrossRefGoogle Scholar
  6. 6.
    M.E.J. Newman, Phys. Rev. E 67, 026126 (2003)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    F. Kawasaki, K. Yakubo, Phys. Rev. E 82, 036113 (2010)ADSCrossRefGoogle Scholar
  8. 8.
    D.J. Watts, S.H. Strogatz, Nature 393, 440 (1998)ADSCrossRefGoogle Scholar
  9. 9.
    C. Song, S. Havlin, H.A. Makse, Nature 433, 392 (2005)ADSCrossRefGoogle Scholar
  10. 10.
    L.K. Gallos, C. Song, H.A. Makse, Phys. Rev. Lett. 100, 248701 (2008)ADSCrossRefGoogle Scholar
  11. 11.
    L.K. Gallos, H.A. Makse, M. Sigman, Proc. Natl. Acad. Sci. USA 109, 2825 (2012)ADSCrossRefGoogle Scholar
  12. 12.
    A. Watanabe, S. Mizutaka, K. Yakubo, J. Phys. Soc. Jpn 84, 114003 (2015)ADSCrossRefGoogle Scholar
  13. 13.
    S.-H. Yook, F. Radicchi, H. Meyer-Ortmanns, Phys. Rev. E 72, 045105 (2005)ADSCrossRefGoogle Scholar
  14. 14.
    C. Song, S. Havlin, H.A. Makse, Nat. Phys. 2, 275 (2006)CrossRefGoogle Scholar
  15. 15.
    H.D. Rozenfeld, S. Havlin, D. ben-Avraham, New J. Phys. 9, 175 (2007)ADSCrossRefGoogle Scholar
  16. 16.
    J.S. Kim, B. Kahng, D. Kim, Phys. Rev. E 79, 067103 (2009)ADSCrossRefGoogle Scholar
  17. 17.
    J. Menche, A. Valleriani, R. Lipowsky, Phys. Rev. E 81, 046103 (2010)ADSCrossRefGoogle Scholar
  18. 18.
    C.M. Schneider, A.A. Moreira, J.S. Andrade, S. Havlin, H.J. Herrmann, Proc. Natl. Acad. Sci. USA 108, 3838 (2011)ADSCrossRefGoogle Scholar
  19. 19.
    S. Johnson, J.J. Torres, J. Marro, M.A. Muñoz, Phys. Rev. Lett. 104, 108702 (2010)ADSCrossRefGoogle Scholar
  20. 20.
    S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, Phys. Rev. E 65, 066122 (2002)ADSCrossRefGoogle Scholar
  21. 21.
    S. Maslov, K. Sneppen, Science 296, 910 (2002)ADSCrossRefGoogle Scholar
  22. 22.
    A. Bekessy, P. Bekessy, J. Komlos, Stud. Sci. Math. Hungar. 7, 343 (1972)Google Scholar
  23. 23.
    M. Molloy, B. Reed, Random Struct. Algorithms 6, 161 (1995)CrossRefGoogle Scholar
  24. 24.
    M.E.J. Newman, S.H. Strogatz, D.J. Watts, Phys. Rev. E 64, 026118 (2001)ADSCrossRefGoogle Scholar
  25. 25.
    S.N. Dorogovtsev, A.L. Ferreira, A.V. Goltsev, J.F.F. Mendes, Phys. Rev. E 81, 031135 (2010)ADSCrossRefGoogle Scholar
  26. 26.
    N. Litvak, R. van der Hofstad, Phys. Rev. E 87, 022801 (2013)ADSCrossRefGoogle Scholar
  27. 27.
    M. Raschke, M. Schläpfer, R. Nibali, Phys. Rev. E 82, 037102 (2010)ADSCrossRefGoogle Scholar
  28. 28.
    W.-Y. Zhang, Z.-W. Wei, B.-H. Wang, X.-P. Han, Physica A 451, 440 (2016)ADSCrossRefGoogle Scholar
  29. 29.
    L. Donetti, P.I. Hurtado, M.A. Muñoz, Phys. Rev. Lett. 95, 188701 (2005)ADSCrossRefGoogle Scholar
  30. 30.
    T. Nishikawa, A.E. Motter, Y.-C. Lai, F.C. Hoppensteadt, Phys. Rev. E 66, 046139 (2002)ADSCrossRefGoogle Scholar
  31. 31.
    S.K. Dwivedi, S. Jalan, Phys. Rev. E 92, 022802 (2015)ADSCrossRefGoogle Scholar
  32. 32.
    R.C. Wilson, P. Zhu, Patt. Recog. 41, 2833 (2008)CrossRefGoogle Scholar
  33. 33.
    J. Gu, B. Hua, S. Liu, Discrete Appl. Math. 190-191, 56 (2015)CrossRefGoogle Scholar
  34. 34.
    A. Barvinok, J.A. Hartigan, Random Struct. Algor. 42, 301 (2013)CrossRefGoogle Scholar
  35. 35.
    C. Song, L.K. Gallos, S. Havlin, H.A. Makse, J. Stat. Mech.: Theory Exp., P03006 (2007)Google Scholar
  36. 36.

Copyright information

© EDP Sciences, SIF, Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of Applied PhysicsHokkaido UniversitySapporoJapan
  2. 2.School of Statistical Thinking, The Institute of Statistical MathematicsTachikawaJapan

Personalised recommendations