Advertisement

The European Physical Journal B

, Volume 84, Issue 2, pp 331–338 | Cite as

Role of fractal dimension in random walks on scale-free networks

  • Zhongzhi ZhangEmail author
  • Yihang Yang
  • Shuyang Gao
Regular Article Interdisciplinary Physics

Abstract

Fractal dimension is central to understanding dynamical processes occurring on networks; however, the relation between fractal dimension and random walks on fractal scale-free networks has been rarely addressed, despite the fact that such networks are ubiquitous in real-life world. In this paper, we study the trapping problem on two families of networks. The first is deterministic, often called (x,y)-flowers; the other is random, which is a combination of (1,3)-flower and (2,4)-flower and thus called hybrid networks. The two network families display rich behavior as observed in various real systems, as well as some unique topological properties not shared by other networks. We derive analytically the average trapping time for random walks on both the (x,y)-flowers and the hybrid networks with an immobile trap positioned at an initial node, i.e., a hub node with the highest degree in the networks. Based on these analytical formulae, we show how the average trapping time scales with the network size. Comparing the obtained results, we further uncover that fractal dimension plays a decisive role in the behavior of average trapping time on fractal scale-free networks, i.e., the average trapping time decreases with an increasing fractal dimension.

Keywords

Fractal Dimension Network Size Degree Distribution Initial Node Network Family 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    R. Albert, A.-L. Barabási, Rev. Mod. Phys. 74, 47 (2002)CrossRefzbMATHADSGoogle Scholar
  2. 2.
    M.E.J. Newman, SIAM Rev. 45, 167 (2003)CrossRefzbMATHADSMathSciNetGoogle Scholar
  3. 3.
    S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, Rev. Mod. Phys. 80, 1275 (2008)CrossRefADSGoogle Scholar
  4. 4.
    S. Havlin, D. Ben-Avraham, Adv. Phys. 36, 695 (1987)CrossRefADSGoogle Scholar
  5. 5.
    R. Metzler, J. Klafter, Phys. Rep. 339, 1 (2000)CrossRefzbMATHADSMathSciNetGoogle Scholar
  6. 6.
    R. Metzler, J. Klafter, J. Phys. A 37, R161 (2004)CrossRefzbMATHADSMathSciNetGoogle Scholar
  7. 7.
    R Burioni, D. Cassi, J. Phys. A 38, R45 (2005)CrossRefzbMATHADSMathSciNetGoogle Scholar
  8. 8.
    O. Bénichou, C. Loverdo, M. Moreau, R. Voituriez, Rev. Mod. Phys. 83, 81 (2011)CrossRefADSGoogle Scholar
  9. 9.
    F. Spitzer, Principles of Random Walk (Van Nostrand, Princeton, New Jersey, 1964)Google Scholar
  10. 10.
    G.H. Weiss, Aspects and Applications of the Random Walk (North Holland, Amsterdam, 1994)Google Scholar
  11. 11.
    B.D. Hughes, Random Walks and Random Environments: Random Walks (Clarendon Press, Oxford, 1996), Vol. 1Google Scholar
  12. 12.
    S. Redner, A Guide to First-Passage Processes (Cambridge University Press, Cambridge, 2001)Google Scholar
  13. 13.
    J.D. Noh, H. Rieger, Phys. Rev. Lett. 92, 118701 (2004)CrossRefADSGoogle Scholar
  14. 14.
    E.M. Bollt, D. Ben-Avraham, New J. Phys. 7, 26 (2005)CrossRefGoogle Scholar
  15. 15.
    S. Condamin, O. Bénichou, V. Tejedor, R. Voituriez, J. Klafter, Nature (London) 450, 77 (2007)CrossRefADSGoogle Scholar
  16. 16.
    O. Bénichou, B. Meyer, V. Tejedor, R. Voituriez, Phys. Rev. Lett. 101, 130601 (2008)CrossRefGoogle Scholar
  17. 17.
    E.W. Montroll, J. Math. Phys. 10, 753 (1969)CrossRefADSGoogle Scholar
  18. 18.
    J.J. Kozak, V. Balakrishnan, Phys. Rev. E 65, 021105 (2002)CrossRefADSMathSciNetGoogle Scholar
  19. 19.
    J.J. Kozak, V. Balakrishnan, Int. J. Bifurc. Chaos Appl. Sci. Eng. 12, 2379 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    E. Agliari, Phys. Rev. E 77, 011128 (2008)CrossRefADSGoogle Scholar
  21. 21.
    A. Garcia Cantú, E. Abad, Phys. Rev. E 77, 031121 (2008)CrossRefADSGoogle Scholar
  22. 22.
    C.P. Haynes, A.P. Roberts, Phys. Rev. E 78, 041111 (2008)CrossRefADSMathSciNetGoogle Scholar
  23. 23.
    J.L. Bentz, J.W. Turner, J.J. Kozak, Phys. Rev. E 82, 011137 (2010)CrossRefADSMathSciNetGoogle Scholar
  24. 24.
    V. Tejedor, O. Bénichou, R. Voituriez, Phys. Rev. E 83, 066102 (2011)CrossRefADSGoogle Scholar
  25. 25.
    A.-L. Barabási, R. Albert, Science 286, 509 (1999)CrossRefMathSciNetGoogle Scholar
  26. 26.
    A. Kittas, S. Carmi, S. Havlin, P. Argyrakis, Europhys. Lett. 84, 40008 (2008)CrossRefADSGoogle Scholar
  27. 27.
    Z.Z. Zhang, Y. Qi, S.G. Zhou, W.L. Xie, J.H. Guan, Phys. Rev. E 79, 021127 (2009)CrossRefADSGoogle Scholar
  28. 28.
    Z.Z. Zhang, J.H. Guan, W.L. Xie, Y. Qi, S.G. Zhou, Europhys. Lett. 86, 10006 (2009)CrossRefADSGoogle Scholar
  29. 29.
    Z.Z. Zhang, S.G. Zhou, W.L. Xie, L.C. Chen, Y. Lin, J.H. Guan, Phys. Rev. E 79, 061113 (2009)CrossRefADSMathSciNetGoogle Scholar
  30. 30.
    E. Agliari, R. Burioni, Phys. Rev. E 80, 031125 (2009)CrossRefADSGoogle Scholar
  31. 31.
    V. Tejedor, O. Bénichou, R. Voituriez, Phys. Rev. E 80, 065104(R) (2009)CrossRefADSGoogle Scholar
  32. 32.
    Z.Z. Zhang, S.Y. Gao, W.L. Xie, Chaos 20, 043112 (2010)CrossRefADSGoogle Scholar
  33. 33.
    E. Agliari, R. Burioni, A. Manzotti, Phys. Rev. E 82, 011118 (2010)CrossRefADSGoogle Scholar
  34. 34.
    C. Song, S. Havlin, H.A. Makse, Nature 433, 392 (2005)CrossRefADSGoogle Scholar
  35. 35.
    C. Song, S. Havlin, H.A. Makse, Nature Physics 2, 275 (2006)CrossRefADSGoogle Scholar
  36. 36.
    Z.Z. Zhang, W.L. Xie, S.G. Zhou, S.Y. Gao, J.H. Guan, Europhys. Lett. 88, 10001 (2009)CrossRefADSGoogle Scholar
  37. 37.
    Z.Z. Zhang, Y. Lin, Y.J. Ma, J. Phys. A 44, 075102 (2011)CrossRefADSMathSciNetGoogle Scholar
  38. 38.
    H.D. Rozenfeld, S. Havlin, D. Ben-Avraham, New J. Phys. 9, 175 (2007)CrossRefADSGoogle Scholar
  39. 39.
    H.D. Rozenfeld, D. Ben-Avraham, Phys. Rev. E 75, 061102 (2007)CrossRefADSGoogle Scholar
  40. 40.
    S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, Phys. Rev. E 65, 066122 (2002)CrossRefADSGoogle Scholar
  41. 41.
    Z.Z. Zhang, S.G. Zhou, L.C. Chen, Eur. Phys. J. B 58, 337 (2007)CrossRefADSGoogle Scholar
  42. 42.
    A.N. Berker, S. Ostlund, J. Phys. C 12, 4961 (1979)CrossRefADSGoogle Scholar
  43. 43.
    M. Kaufman, R.B. Griffiths, Phys. Rev. B 24, 496 (1981)CrossRefADSMathSciNetGoogle Scholar
  44. 44.
    R.B. Griffiths, M. Kaufman, Phys. Rev. B 26, 5022 (1982)CrossRefADSMathSciNetGoogle Scholar
  45. 45.
    M. Hinczewski, A.N. Berker, Phys. Rev. E 73, 066126 (2006)CrossRefADSMathSciNetGoogle Scholar
  46. 46.
    Z.Z. Zhang, S.G. Zhou, T. Zou, L.C. Chen, J.H. Guan, Phys. Rev. E 79, 031110 (2009)CrossRefADSGoogle Scholar
  47. 47.
    D.J. Watts, H. Strogatz, Nature (London) 393, 440 (1998)CrossRefADSGoogle Scholar
  48. 48.
    Z.Z. Zhang, W.L. Xie, S.G. Zhou, M. Li, J.H. Guan, Phys. Rev. E 80, 061111 (2009)CrossRefADSGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.School of Computer ScienceFudan UniversityShanghaiP.R. China
  2. 2.Shanghai Key Lab of Intelligent Information ProcessingFudan UniversityShanghaiP.R. China

Personalised recommendations