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Degree correlation in scale-free graphs

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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(ℓ|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 ℓ. 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.

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References

  1. A. Broder, R. Kumar, F. Maghoul, P. Raghavan, S. Rajagopalan, R. Stata, A. Tomkins, J. Wiener, Comp. Net. 33, 309 (2000)

    Article  Google Scholar 

  2. L.A. Adamic, B.A. Huberman, Science 287, 2115 (2000)

    Article  ADS  Google Scholar 

  3. H. Jeong, B. Tombor, R. Albert, Z.N. Oltvai, A.L. Barabási, Nature 407, 651 (2000)

    Article  ADS  Google Scholar 

  4. I. Farkas, H. Jeong, T. Vicsek, A. Barabási, Z. Oltvai, Phys. A 318, 601 (2003)

    Article  Google Scholar 

  5. S. Redner, Eur. Phys. J. B 4, 131 (1998)

    Article  ADS  Google Scholar 

  6. A.L. Barabási, H. Jeong, Z. Néda, E. Ravasz, A. Schubert, T. Vicsek, Phys. A 311, 590 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  7. A.L. Barabási, R. Albert, Science 286, 509 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  8. R.R. Albert, A.L. Barabási, Phys. Rev. Lett. 85, 5234 (2000)

    Article  ADS  Google Scholar 

  9. A.L. Barabási, R. Albert, H. Jeong, Physica A 272, 173 (1999)

    Article  ADS  Google Scholar 

  10. R. Xulvi-Brunet, W. Pietsch, I.M. Sokolov, Phys. Rev. E 68, 036119 (2003)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  12. N. Schwartz, R. Cohen, D. Ben-Avraham, A.L. Barabási, S. Havlin, Phys. Rev. E 66, 015104 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  13. K. Deng, H. Zhao, D. Li, Physica A 379, 714 (2007)

    Article  ADS  Google Scholar 

  14. L.K. Gallos, R. Cohen, P. Argyrakis, A. Bunde, S. Havlin, Phys. Rev. Lett. 94, 188701 (2005)

    Article  ADS  Google Scholar 

  15. M. Boguná, R. Pastor-Satorras, A. Vespignani, Stat. Mech. Complex Net., Lec. Notes Phys. 625, 127 (2003)

    Article  ADS  Google Scholar 

  16. Y. Moreno, J.B. Gómez, A.F. Pacheco, Phys. Rev. E 68, 035103 (2003)

    Article  ADS  Google Scholar 

  17. P.L. Krapivsky, S. Redner, Phys Rev. E 63, 066123 (2001)

    Article  ADS  Google Scholar 

  18. P. Krapivsky, S. Redner, F. Leyvraz, Phys. Rev. Lett. 85, 4629 (2000)

    Article  ADS  Google Scholar 

  19. S.N. Dorogovtsev, J.F.F. Mendes, A.N. Samukhin, Phys. Rev. Lett. 85, 4633 (2000)

    Article  ADS  Google Scholar 

  20. P. Krapivsky, G. Rodgers, S. Redner, Phys. Rev. Lett. 86, 5401 (2001)

    Article  ADS  Google Scholar 

  21. P. Krapivsky, S. Redner, J. Phys. A 35, 9517 (2002)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  22. P. Krapivsky, S. Redner, Stat. Mech. Complex Net. 625, 3 (2003)

    Article  ADS  Google Scholar 

  23. B. Bollobás, Random graphs (Cambridge University Press, 2001), Vol. 73

  24. M.E.J. Newman, SIAM Rev. 45, 167 (2003)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  25. V.M. Eguiluz, K. Klemm, Phys. Rev. Lett. 89, 108701 (2002)

    Article  ADS  Google Scholar 

  26. M. Barthélemy, A. Barrat, R. Pastor-Satorras, A. Vespignani, Phys. Rev. Lett. 92, 178701 (2004)

    Article  ADS  Google Scholar 

  27. C. Castellano, R. Pastor-Satorras, Phys. Rev. Lett. 96, 038701 (2006)

    Article  ADS  Google Scholar 

  28. C. Castellano, R. Pastor-Satorras, Phys. Rev. Lett. 100, 148701 (2008)

    Article  ADS  Google Scholar 

  29. A. Baronchelli, M. Catanzaro, R. Pastor-Satorras, Phys. Rev. E 78, 016111 (2008)

    Article  ADS  Google Scholar 

  30. S.C. Ferreira, R.S. Ferreira, R. Pastor-Satorras, Phys. Rev. E 83, 066113 (2011)

    Article  ADS  Google Scholar 

  31. R. Cohen, D. Ben-Avraham, S. Havlin, Phys. Rev. E 66, 036113 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  32. Z. Dezso, A.L. Barabási, Phys. Rev. E 65, 055103 (2002)

    Article  ADS  Google Scholar 

  33. Y. Wang, G. Xiao, J. Hu, T.H. Cheng, L. Wang, Phys. A 388, 2535 (2009)

    Article  Google Scholar 

  34. Y. Moreno, M. Nekovee, A.F. Pacheco, Phys. Rev. E 69, 066130 (2004)

    Article  ADS  Google Scholar 

  35. Y. Moreno, R. Pastor-Satorras, A. Vespignani, Eur. Phys. J. B 26, 521 (2002)

    ADS  Google Scholar 

  36. A. Singh, Y. Singh, in Proccedings of the 2013 Nat. Conf. Communications (NCC, IEEE), Delhi, 2013, p. 1

  37. C.Y. Xia, S.W. Sun, Z.X. Liu, Z.Q. Chen, Z.Z. Yuan, Int. J. Mod. Phys. B 23, 2203 (2009)

    Article  ADS  MATH  Google Scholar 

  38. S. Mizutaka, K. Yakubo, Phys. Rev. E 88, 012803 (2013)

    Article  ADS  Google Scholar 

  39. J. Wang, Safety Science 53, 219 (2013)

    Article  Google Scholar 

  40. M. Boguñá, C. Castellano, R. Pastor-Satorras, Phys. Rev. E 79, 036110 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  41. L.K. Gallos, R. Cohen, P. Argyrakis, A. Bunde, S. Havlin, Phys. Rev. Lett. 94, 188701 (2005)

    Article  ADS  Google Scholar 

  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. 4

  43. R. Cohen, K. Erez, D. Ben-Avraham, S. Havlin, Phys. Rev. Lett. 85, 4626 (2000)

    Article  ADS  Google Scholar 

  44. R. Cohen, K. Erez, D. Ben-Avraham, S. Havlin, Phys. Rev. Lett. 86, 3682 (2001)

    Article  ADS  Google Scholar 

  45. J.G. Liu, Z.T. Wang, Y.Z. Dang, Mod. Phys. Lett. B 20, 815 (2006)

    Article  ADS  MATH  Google Scholar 

  46. F. Caccioli, L. Dallâ-Asta, J. Stat. Mech. 2009, 10004 (2009)

    Article  Google Scholar 

  47. L.K. Gallos, C. Song, H.A. Makse, Phys. Rev. Lett. 100, 248701 (2008)

    Article  ADS  Google Scholar 

  48. Q. Li, L. Braunstein, H. Wang, J. Shao, H.E. Stanley, S. Havlin, J. Stat. Phys. 151, 92 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  49. T. Konno, Physica A 392, 5453 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  50. R. Pastor-Satorras, A. Vespignani, Phys. Rev. Lett. 86, 3200 (2001)

    Article  ADS  Google Scholar 

  51. B. Bollobás, Random Structures and Algorithms 18, 270 (2001)

    Google Scholar 

  52. M.E.J. Newman, Phys. Rev. Lett. 89, 208701 (2002)

    Article  ADS  Google Scholar 

  53. A. Vázquez, R. Pastor-Satorras, A. Vespignani, Phys. Rev. E 65, 066130 (2002)

    Article  ADS  Google Scholar 

  54. S. Maslov, K. Sneppen, Science 296, 910 (2002)

    Article  ADS  Google Scholar 

  55. P.V. Mieghem, X. Ge, P. Schumm, S. Trakanovski, H. Wang, Phys. Rev. E 82, 056113 (2010)

    Article  ADS  Google Scholar 

  56. H. Wang, W. Winterbach, P.V. Mieghem, Eur. Phy. J. B 83, 203 (2011)

    Article  ADS  Google Scholar 

  57. D. Liu, S. Trajanovski, P.V. Mieghem, Phy. Rev. E 87, 012816 (2013)

    Article  ADS  Google Scholar 

  58. D. Rybski, D. Rozenfeld, J.P. Kropp, Europhys. Lett. 90, 28002 (2010)

    Article  ADS  Google Scholar 

  59. B. Sury, Eur. J. Combinatorics 14, 351 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  60. W.Y.G. Wa, J. Math. Res. Exposition 31, 1021 (2011)

    MATH  MathSciNet  Google Scholar 

  61. E.S. Andersen, Math. Scand. 1, 261 (1953)

    MATH  MathSciNet  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Electrical and Computer Engineering, McGill University, H3A 0E9, Montréal, Québec, Canada

    Babak Fotouhi & Michael G. Rabbat

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  1. Babak Fotouhi
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  2. Michael G. Rabbat
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Correspondence to Babak Fotouhi.

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Fotouhi, B., Rabbat, M.G. Degree correlation in scale-free graphs. Eur. Phys. J. B 86, 510 (2013). https://doi.org/10.1140/epjb/e2013-40920-6

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