Heterogenous mean-field analysis of a generalized voter-like model on networks

Abstract

We propose a generalized framework for the study of voter models in complex networks at the heterogeneous mean-field (HMF) level that (i) yields a unified picture for existing copy/invasion processes and (ii) allows for the introduction of further heterogeneity through degree-selectivity rules. In the context of the HMF approximation, our model is capable of providing straightforward estimates for central quantities such as the exit probability and the consensus/fixation time, based on the statistical properties of the complex network alone. The HMF approach has the advantage of being readily applicable also in those cases in which exact solutions are difficult to work out. Finally, the unified formalism allows one to understand previously proposed voter-like processes as simple limits of the generalized model.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    C. Castellano, S. Fortunato, V. Loreto, Rev. Mod. Phys. 81, (2009)

  2. 2.

    B. Drossel, Adv. Phys. 50, 209 (2001)

    ADS  Article  Google Scholar 

  3. 3.

    P. Clifford, A. Sudbury, Biometrika 60, 581 (1973)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    P. Moran, Proc. Camb. Phil. Soc. 54, 60 (1958)

    ADS  Article  MATH  Google Scholar 

  5. 5.

    T.M. Liggett, Interacting Particle Systems (Springer-Verlag, New York, 1985)

  6. 6.

    P. Krapivsky, S. Redner, E. Ben-Naim, A Kinetic View of Statistical Physics (Cambridge University Press, Cambridge, 2010)

  7. 7.

    R. Albert, A.L. Barabási, Rev. Mod. Phys. 74, 47 (2002)

    ADS  Article  MATH  Google Scholar 

  8. 8.

    S.N. Dorogovtsev, J.F.F. Mendes, Evolution of networks: From biological nets to the Internet and WWW (Oxford University Press, Oxford, 2003)

  9. 9.

    M. Newman, SIAM Rev. 45, 167 (2003)

    MathSciNet  ADS  Article  MATH  Google Scholar 

  10. 10.

    K. Suchecki, V. Eguíluz, M.S. Miguel, Phys. Rev. E 72, 036132 (2005)

    ADS  Article  Google Scholar 

  11. 11.

    V. Sood, S. Redner, Phys. Rev. Lett. 94, 178701 (2005)

    ADS  Article  Google Scholar 

  12. 12.

    T. Antal, S. Redner, V. Sood, Phys. Rev. Lett. 96, 188104 (2006)

    ADS  Article  Google Scholar 

  13. 13.

    V. Sood, T. Antal, S. Redner, Phys. Rev. E 77, 041121 (2008)

    MathSciNet  ADS  Article  Google Scholar 

  14. 14.

    C.M. Schneider-Mizell, L.M. Sander, J. Stat. Phys. 136, 59 (2009)

    MathSciNet  ADS  Article  MATH  Google Scholar 

  15. 15.

    H.X. Yang, Z.X. Wu, C. Zhou, T. Zhou, B.H. Wang, Phys. Rev. E 80, 046108 (2009)

    ADS  Article  Google Scholar 

  16. 16.

    Y. Lin, H. Yang, Z. Rong, B. Wang, Int. J. Mod. Phys. C 21, 1011 (2010)

    ADS  Article  Google Scholar 

  17. 17.

    A. Baronchelli, C. Castellano, R. Pastor-Satorras, Phys. Rev. E 83, 066117 (2011)

    ADS  Article  Google Scholar 

  18. 18.

    M.Á. Serrano, K. Klemm, F. Vazquez, V.M. Eguíluz, M.S. Miguel, J. Stat. Mech. P10024 (2009)

  19. 19.

    A. Barrat, M. Barthélemy, A. Vespignani, Dynamical Processes on Complex Networks (Cambridge University Press, Cambridge, 2008)

  20. 20.

    S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, Rev. Mod. Phys. 80, 1275 (2008)

    ADS  Article  Google Scholar 

  21. 21.

    M. Boguñá, R. Pastor-Satorras, Phys. Rev. E 66, 047104 (2002)

    ADS  Article  Google Scholar 

  22. 22.

    A. Baronchelli, R. Pastor-Satorras, J. Stat. Mech. L11001 (2009)

  23. 23.

    G.J. Baxter, R.A. Blythe, A.J. McKane, Phys. Rev. Lett. 101, 258701 (2008)

    ADS  Article  Google Scholar 

  24. 24.

    R.A. Blythe, J. Phys. A Math. Theor. 43, 385003 (2010)

    MathSciNet  ADS  Article  Google Scholar 

  25. 25.

    R.A. Blythe, A.J. McKane, J. Stat. Mech. P07018 (2007)

  26. 26.

    A. Baronchelli, R. Pastor-Satorras, Phys. Rev. E 82, 011111 (2010)

    ADS  Article  Google Scholar 

  27. 27.

    F.R. Gantmacher, The Theory of Matrices (Chelsea Publishing Company, 1959)

  28. 28.

    C. Castellano, AIP Conf. Proc. 779, 114 (2005)

    ADS  Article  Google Scholar 

  29. 29.

    M.A. Nowak, Evolutionary Dynamics (Berknap, Harvard, Cambridge, 2006)

  30. 30.

    M. McPherson, L.S. Lovin, J.M. Cook, Ann. Rev. Soc. 27, 415 (2001)

    Article  Google Scholar 

  31. 31.

    M. Catanzaro, M. Boguñá, R. Pastor-Satorras, Phys. Rev. E 71, 027103 (2005)

    ADS  Article  Google Scholar 

  32. 32.

    A. Baronchelli, L. Dall’Asta, A. Barrat, V. Loreto, Phys. Rev. E 73, 015102 (2006)

    MathSciNet  ADS  Article  Google Scholar 

  33. 33.

    L. Dall’Asta, C. Castellano, Europhys. Lett. 77, 60005 (2007)

    MathSciNet  ADS  Article  Google Scholar 

  34. 34.

    F. Vazquez, V.M. Eguíluz, New J. Phys. 10, 063011 (2008)

    Article  Google Scholar 

  35. 35.

    J. Gleeson, Phys. Rev. Lett. 107, 68701 (2011)

    ADS  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to P. Moretti.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Moretti, P., Liu, S.Y., Baronchelli, A. et al. Heterogenous mean-field analysis of a generalized voter-like model on networks. Eur. Phys. J. B 85, 88 (2012). https://doi.org/10.1140/epjb/e2012-20501-1

Download citation

Keywords

  • Statistical and Nonlinear Physics