The European Physical Journal B

, Volume 38, Issue 2, pp 177–182 | Cite as

Potts model on complex networks

  • S. N. Dorogovtsev
  • A. V. Goltsev
  • J. F. F. Mendes
Article

Abstract.

We consider the general p-state Potts model on random networks with a given degree distribution (random Bethe lattices). We find the effect of the suppression of a first order phase transition in this model when the degree distribution of the network is fat-tailed, that is, in more precise terms, when the second moment of the distribution diverges. In this situation the transition is continuous and of infinite order, and size effect is anomalously strong. In particular, in the case of p = 1, we arrive at the exact solution, which coincides with the known solution of the percolation problem on these networks.

Keywords

Phase Transition Exact Solution Complex Network Degree Distribution Random Network 

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References

  1. 1.
    A.-L. Barabási, R. Albert, Science 286, 509 (1999)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    S.H. Strogatz, Nature 401, 268 (2001)ADSCrossRefGoogle Scholar
  3. 3.
    R. Albert, A.-L. Barabási, Rev. Mod. Phys. 74, 47 (2002)ADSCrossRefGoogle Scholar
  4. 4.
    S.N. Dorogovtsev, J.F.F. Mendes, Adv. Phys. 51, 1079 (2002)ADSCrossRefGoogle Scholar
  5. 5.
    M.E.J. Newman, SIAM Review 45, 167 (2003)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    D.J. Watts, Small Worlds: The Dynamics of Networks between Order and Randomness (Princeton University Press, Princeton, NJ, 1999)Google Scholar
  7. 7.
    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
  8. 8.
    R. Albert, H. Jeong, A.-L. Barabási, Nature 406, 378 (2000)ADSCrossRefGoogle Scholar
  9. 9.
    L.A.N. Amaral, A. Scala, M. Barthelemy, H.E. Stanley, Proc. Nat. Acad. Sci. USA 97, 11149 (2000)ADSCrossRefGoogle Scholar
  10. 10.
    A. Aleksiejuk, J.A. Holyst, D. Stauffer, Physica A 310, 260 (2002)ADSCrossRefGoogle Scholar
  11. 11.
    S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, Phys. Rev. E 66, 016104 (2002)ADSCrossRefGoogle Scholar
  12. 12.
    M. Leone, A. Vázquez, A. Vespignani, R. Zecchina, Eur. Phys. J. B 28, 191 (2002)ADSCrossRefGoogle Scholar
  13. 13.
    M. Gitterman, J. Phys. A 33, 8373 (2000)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    A. Barrat, M. Weigt, Eur. Phys. J. B 13, 547 (2000)ADSCrossRefGoogle Scholar
  15. 15.
    B.J. Kim, H. Hong, P. Holme, G.S. Jeon, P. Minnhagen, M.Y. Choi, Phys. Rev. E 64, 056135 (2001)ADSCrossRefGoogle Scholar
  16. 16.
    H. Hong, B.J. Kim, M.Y. Choi, Phys. Rev. E 66, 018101 (2002)ADSCrossRefGoogle Scholar
  17. 17.
    G. Bianconi, Phys. Lett. A 303, 166 (2002)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    R. Cohen, K. Erez, D. ben-Avraham, S. Havlin, Phys. Rev. Lett. 85, 4625 (2000); Phys. Rev. Lett. 86, 3682 (2001); Phys. Rev. Lett. 87, 219802 (2001)ADSGoogle Scholar
  19. 19.
    D.S. Callaway, M.E.J. Newman, S.H. Strogatz, D.J. Watts, Phys. Rev. Lett. 85, 5468 (2000)ADSCrossRefGoogle Scholar
  20. 20.
    R. Cohen, D. ben-Avraham, S. Havlin, Phys. Rev. E 66, 036113 (2002)ADSCrossRefGoogle Scholar
  21. 21.
    R. Pastor-Satorras, A. Vespignani, Phys. Rev. Lett. 86, 3200 (2001); Phys. Rev. E 63, 066117 (2001)ADSCrossRefGoogle Scholar
  22. 22.
    A.V. Goltsev, S.N. Dorogovtsev, J.F.F. Mendes, Phys. Rev. E 67, 026123 (2003)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    F. Iglói, L. Turban, Phys. Rev. E 66, 036140 (2002)ADSCrossRefGoogle Scholar
  24. 24.
    V.A. Kazakov, Phys. Lett. A 119, 140 (1986); D.V. Boulatov, V.A. Kazakov, Phys. Lett. B 186, 140 (1987); W. Janke, D.A. Johnston, M. Stathakopoulos, J. Phys. A 35, 7575 (2002)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    F.Y. Wu, Rev. Mod. Phys. 54, 235 (1982)ADSCrossRefGoogle Scholar
  26. 26.
    R.J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic Press, London, 1982)Google Scholar
  27. 27.
    P.W. Kosteleyn, C.M. Fortuin, J. Phys. Soc. Jpn Suppl. 26, 11 (1969); C.M. Fortuin, P.W. Kosteleyn, Physica 57, 536 (1972)ADSGoogle Scholar
  28. 28.
    A. Bekessy, P. Bekessy, J. Komlos, Stud. Sci. Math. Hungar. 7, 343 (1972); E.A. Bender, E.R. Canfield, J. Combinatorial Theory A 24, 296 (1978); B. Bollobás, Eur. J. Comb. 1, 311 (1980); N.C. Wormald, J. Combinatorial Theory B 31, 156,168 (1981)MathSciNetGoogle Scholar
  29. 29.
    M. Molloy, B. Reed, Random Structures and Algorithms 6, 161 (1995); Combinatorics, Probability and Computing 7, 295 (1998)MathSciNetCrossRefGoogle Scholar
  30. 30.
    M.E.J. Newman, S.H. Strogatz, D.J. Watts, Phys. Rev. E 64, 026118 (2001); M.E.J. Newman, cond-mat/0202208ADSCrossRefGoogle Scholar
  31. 31.
    S.N. Dorogovtsev, J.F.F. Mendes, Phys. Rev. Lett. 87 , 219801 (2001)ADSCrossRefGoogle Scholar
  32. 32.
    R. Pastor-Satorras, A. Vespignani, Phys. Rev. E 65, 035108 (2002)ADSCrossRefGoogle Scholar
  33. 33.
    D.S. Callaway, J.E. Hopcroft, J.M. Kleinberg, M.E.J. Newman, S.H. Strogatz, Phys. Rev. E 64, 041902 (2001)ADSCrossRefGoogle Scholar
  34. 34.
    S.N. Dorogovtsev, J.F.F. Mendes, A.N. Samukhin, Phys. Rev. E 64, 066110 (2001)ADSCrossRefGoogle Scholar
  35. 35.
    J. Kim, P.L. Krapivsky, B. Kahng, S. Redner, Phys. Rev. E 66, 055101 (2002)ADSCrossRefGoogle Scholar
  36. 36.
    M. Bauer, D. Bernard, J. Stat. Phys. 111, 703 (2003)CrossRefGoogle Scholar
  37. 37.
    A. Vazquez, M. Boguñá, Y. Moreno, R. Pastor-Satorras, A. Vespignani, Phys. Rev. E 67, 046111 (2003)ADSCrossRefGoogle Scholar
  38. 38.
    T. Kihara, Y. Midzuno, J. Shizume, J. Phys. Soc. Jpn 9, 681 (1954)ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin/Heidelberg 2004

Authors and Affiliations

  • S. N. Dorogovtsev
    • 1
    • 3
  • A. V. Goltsev
    • 2
    • 3
  • J. F. F. Mendes
    • 2
  1. 1.Departamento de Física and Centro de Física do PortoFaculdade de Ciências, Universidade do PortoPortoPortugal
  2. 2.Departamento de FísicaUniversidade de AveiroAveiroPortugal
  3. 3.A.F. Ioffe Physico-Technical InstituteSt. PetersburgRussia

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