Degree correlation in scale-free graphs

Regular Article

Abstract

We obtain closed form expressions for the expected conditional degree distribution and the joint degree distribution of the linear preferential attachment model for network growth in the steady state. We consider the multiple-destination preferential attachment growth model, where incoming nodes at each timestep attach to β existing nodes, selected by degree-proportional probabilities. By the conditional degree distribution p(|k), we mean the degree distribution of nodes that are connected to a node of degree k. By the joint degree distribution p(k,ℓ), we mean the proportion of links that connect nodes of degrees k and . In addition to this growth model, we consider the shifted-linear preferential growth model and solve for the same quantities, as well as a closed form expression for its steady-state degree distribution.

Keywords

Statistical and Nonlinear Physics 

References

  1. 1.
    A. Broder, R. Kumar, F. Maghoul, P. Raghavan, S. Rajagopalan, R. Stata, A. Tomkins, J. Wiener, Comp. Net. 33, 309 (2000)CrossRefGoogle Scholar
  2. 2.
    L.A. Adamic, B.A. Huberman, Science 287, 2115 (2000) ADSCrossRefGoogle Scholar
  3. 3.
    H. Jeong, B. Tombor, R. Albert, Z.N. Oltvai, A.L. Barabási, Nature 407, 651 (2000) ADSCrossRefGoogle Scholar
  4. 4.
    I. Farkas, H. Jeong, T. Vicsek, A. Barabási, Z. Oltvai, Phys. A 318, 601 (2003) CrossRefGoogle Scholar
  5. 5.
    S. Redner, Eur. Phys. J. B 4, 131 (1998)ADSCrossRefGoogle Scholar
  6. 6.
    A.L. Barabási, H. Jeong, Z. Néda, E. Ravasz, A. Schubert, T. Vicsek, Phys. A 311, 590 (2002) CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    A.L. Barabási, R. Albert, Science 286, 509 (1999) ADSCrossRefMathSciNetGoogle Scholar
  8. 8.
    R.R. Albert, A.L. Barabási, Phys. Rev. Lett. 85, 5234 (2000) ADSCrossRefGoogle Scholar
  9. 9.
    A.L. Barabási, R. Albert, H. Jeong, Physica A 272, 173 (1999) ADSCrossRefGoogle Scholar
  10. 10.
    R. Xulvi-Brunet, W. Pietsch, I.M. Sokolov, Phys. Rev. E 68, 036119 (2003) ADSCrossRefGoogle Scholar
  11. 11.
    H. Hooyberghs, B.V. Schaeybroeck, A.A. Moreira, J.S. Andrade Jr., H.J. Herrmann, J.O. Indekeu, Phys. Rev. E 81, 011102 (2010) ADSCrossRefGoogle Scholar
  12. 12.
    N. Schwartz, R. Cohen, D. Ben-Avraham, A.L. Barabási, S. Havlin, Phys. Rev. E 66, 015104 (2002) ADSCrossRefMathSciNetGoogle Scholar
  13. 13.
    K. Deng, H. Zhao, D. Li, Physica A 379, 714 (2007) ADSCrossRefGoogle Scholar
  14. 14.
    L.K. Gallos, R. Cohen, P. Argyrakis, A. Bunde, S. Havlin, Phys. Rev. Lett. 94, 188701 (2005) ADSCrossRefGoogle Scholar
  15. 15.
    M. Boguná, R. Pastor-Satorras, A. Vespignani, Stat. Mech. Complex Net., Lec. Notes Phys. 625, 127 (2003)ADSCrossRefGoogle Scholar
  16. 16.
    Y. Moreno, J.B. Gómez, A.F. Pacheco, Phys. Rev. E 68, 035103 (2003) ADSCrossRefGoogle Scholar
  17. 17.
    P.L. Krapivsky, S. Redner, Phys Rev. E 63, 066123 (2001) ADSCrossRefGoogle Scholar
  18. 18.
    P. Krapivsky, S. Redner, F. Leyvraz, Phys. Rev. Lett. 85, 4629 (2000) ADSCrossRefGoogle Scholar
  19. 19.
    S.N. Dorogovtsev, J.F.F. Mendes, A.N. Samukhin, Phys. Rev. Lett. 85, 4633 (2000) ADSCrossRefGoogle Scholar
  20. 20.
    P. Krapivsky, G. Rodgers, S. Redner, Phys. Rev. Lett. 86, 5401 (2001) ADSCrossRefGoogle Scholar
  21. 21.
    P. Krapivsky, S. Redner, J. Phys. A 35, 9517 (2002) ADSCrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    P. Krapivsky, S. Redner, Stat. Mech. Complex Net. 625, 3 (2003)ADSCrossRefGoogle Scholar
  23. 23.
    B. Bollobás, Random graphs (Cambridge University Press, 2001), Vol. 73Google Scholar
  24. 24.
    M.E.J. Newman, SIAM Rev. 45, 167 (2003)ADSCrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    V.M. Eguiluz, K. Klemm, Phys. Rev. Lett. 89, 108701 (2002) ADSCrossRefGoogle Scholar
  26. 26.
    M. Barthélemy, A. Barrat, R. Pastor-Satorras, A. Vespignani, Phys. Rev. Lett. 92, 178701 (2004) ADSCrossRefGoogle Scholar
  27. 27.
    C. Castellano, R. Pastor-Satorras, Phys. Rev. Lett. 96, 038701 (2006) ADSCrossRefGoogle Scholar
  28. 28.
    C. Castellano, R. Pastor-Satorras, Phys. Rev. Lett. 100, 148701 (2008) ADSCrossRefGoogle Scholar
  29. 29.
    A. Baronchelli, M. Catanzaro, R. Pastor-Satorras, Phys. Rev. E 78, 016111 (2008) ADSCrossRefGoogle Scholar
  30. 30.
    S.C. Ferreira, R.S. Ferreira, R. Pastor-Satorras, Phys. Rev. E 83, 066113 (2011) ADSCrossRefGoogle Scholar
  31. 31.
    R. Cohen, D. Ben-Avraham, S. Havlin, Phys. Rev. E 66, 036113 (2002) ADSCrossRefMathSciNetGoogle Scholar
  32. 32.
    Z. Dezso, A.L. Barabási, Phys. Rev. E 65, 055103 (2002) ADSCrossRefGoogle Scholar
  33. 33.
    Y. Wang, G. Xiao, J. Hu, T.H. Cheng, L. Wang, Phys. A 388, 2535 (2009) CrossRefGoogle Scholar
  34. 34.
    Y. Moreno, M. Nekovee, A.F. Pacheco, Phys. Rev. E 69, 066130 (2004) ADSCrossRefGoogle Scholar
  35. 35.
    Y. Moreno, R. Pastor-Satorras, A. Vespignani, Eur. Phys. J. B 26, 521 (2002)ADSGoogle Scholar
  36. 36.
    A. Singh, Y. Singh, in Proccedings of the 2013 Nat. Conf. Communications (NCC, IEEE), Delhi, 2013, p. 1Google Scholar
  37. 37.
    C.Y. Xia, S.W. Sun, Z.X. Liu, Z.Q. Chen, Z.Z. Yuan, Int. J. Mod. Phys. B 23, 2203 (2009) ADSCrossRefMATHGoogle Scholar
  38. 38.
    S. Mizutaka, K. Yakubo, Phys. Rev. E 88, 012803 (2013) ADSCrossRefGoogle Scholar
  39. 39.
    J. Wang, Safety Science 53, 219 (2013)CrossRefGoogle Scholar
  40. 40.
    M. Boguñá, C. Castellano, R. Pastor-Satorras, Phys. Rev. E 79, 036110 (2009) ADSCrossRefMathSciNetGoogle Scholar
  41. 41.
    L.K. Gallos, R. Cohen, P. Argyrakis, A. Bunde, S. Havlin, Phys. Rev. Lett. 94, 188701 (2005) ADSCrossRefGoogle Scholar
  42. 42.
    R. Cohen, S. Havlin, D. Ben-Avraham, Handbook of graphs and networks, edited by S. Bornnoldt and H.G. Schuster (Wiley-VCH, 2002), chap. 4Google Scholar
  43. 43.
    R. Cohen, K. Erez, D. Ben-Avraham, S. Havlin, Phys. Rev. Lett. 85, 4626 (2000) ADSCrossRefGoogle Scholar
  44. 44.
    R. Cohen, K. Erez, D. Ben-Avraham, S. Havlin, Phys. Rev. Lett. 86, 3682 (2001) ADSCrossRefGoogle Scholar
  45. 45.
    J.G. Liu, Z.T. Wang, Y.Z. Dang, Mod. Phys. Lett. B 20, 815 (2006)ADSCrossRefMATHGoogle Scholar
  46. 46.
    F. Caccioli, L. Dallâ-Asta, J. Stat. Mech. 2009, 10004 (2009) CrossRefGoogle Scholar
  47. 47.
    L.K. Gallos, C. Song, H.A. Makse, Phys. Rev. Lett. 100, 248701 (2008) ADSCrossRefGoogle Scholar
  48. 48.
    Q. Li, L. Braunstein, H. Wang, J. Shao, H.E. Stanley, S. Havlin, J. Stat. Phys. 151, 92 (2012)ADSCrossRefMathSciNetGoogle Scholar
  49. 49.
    T. Konno, Physica A 392, 5453 (2013) ADSCrossRefMathSciNetGoogle Scholar
  50. 50.
    R. Pastor-Satorras, A. Vespignani, Phys. Rev. Lett. 86, 3200 (2001) ADSCrossRefGoogle Scholar
  51. 51.
    B. Bollobás, Random Structures and Algorithms 18, 270 (2001)Google Scholar
  52. 52.
    M.E.J. Newman, Phys. Rev. Lett. 89, 208701 (2002) ADSCrossRefGoogle Scholar
  53. 53.
    A. Vázquez, R. Pastor-Satorras, A. Vespignani, Phys. Rev. E 65, 066130 (2002) ADSCrossRefGoogle Scholar
  54. 54.
    S. Maslov, K. Sneppen, Science 296, 910 (2002) ADSCrossRefGoogle Scholar
  55. 55.
    P.V. Mieghem, X. Ge, P. Schumm, S. Trakanovski, H. Wang, Phys. Rev. E 82, 056113 (2010) ADSCrossRefGoogle Scholar
  56. 56.
    H. Wang, W. Winterbach, P.V. Mieghem, Eur. Phy. J. B 83, 203 (2011)ADSCrossRefGoogle Scholar
  57. 57.
    D. Liu, S. Trajanovski, P.V. Mieghem, Phy. Rev. E 87, 012816 (2013) ADSCrossRefGoogle Scholar
  58. 58.
    D. Rybski, D. Rozenfeld, J.P. Kropp, Europhys. Lett. 90, 28002 (2010) ADSCrossRefGoogle Scholar
  59. 59.
    B. Sury, Eur. J. Combinatorics 14, 351 (1993)CrossRefMATHMathSciNetGoogle Scholar
  60. 60.
    W.Y.G. Wa, J. Math. Res. Exposition 31, 1021 (2011) MATHMathSciNetGoogle Scholar
  61. 61.
    E.S. Andersen, Math. Scand. 1, 261 (1953)MATHMathSciNetGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of Electrical and Computer EngineeringMcGill UniversityMontréalCanada

Personalised recommendations