Journal of Statistical Physics

, Volume 156, Issue 2, pp 336–367 | Cite as

Stochastic Dynamics of the Multi-State Voter Model Over a Network Based on Interacting Cliques and Zealot Candidates

  • Filippo Palombi
  • Simona Toti


The stochastic dynamics of the multi-state voter model is investigated on a class of complex networks made of non-overlapping cliques, each hosting a political candidate and interacting with the others via Erdős–Rényi links. Numerical simulations of the model are interpreted in terms of an ad-hoc mean field theory, specifically tuned to resolve the inter/intra-clique interactions. Under a proper definition of the thermodynamic limit (with the average degree of the agents kept fixed while increasing the network size), the model is found to display the empirical scaling discovered by Fortunato and Castellano (Phys Rev Lett 99(13):138701, 2007) , while the vote distribution resembles roughly that observed in Brazilian elections.


Social physics Proportional elections Voter model Mean field theory 



We thank R. Filippini for her participation in the early stage of this work. The computing resources used for our numerical study and the related technical support have been provided by the CRESCO/ENEAGRID High Performance Computing infrastructure and its staff [25]. CRESCO (Computational RESearch centre on COmplex systems) is funded by ENEA and by Italian and European research programmes.


  1. 1.
    Fortunato, S., Castellano, C.: Scaling and universality in proportional elections. Phys. Rev. Lett. 99(13), 138701 (2007)ADSCrossRefGoogle Scholar
  2. 2.
    Chatterjee, A., Mitrović, M., Fortunato, S.: Universality in voting behavior: an empirical analysis. Sci. Rep. 3 (2013)Google Scholar
  3. 3.
    Costa Filho, R.N., Almeida, M.P., Andrade, J.S., Moreira, J.E.: Scaling behavior in a proportional voting process. Phys. Rev. E 60(1), 1067–1068 (1999)ADSCrossRefGoogle Scholar
  4. 4.
    Clifford, P., Sudbury, A.: A model for spatial conflict. Biometrika 60(3), 581–588 (1973)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Holley, R., Liggett, T.M.: Ergodic theorems for weakly interacting infinite systems and the voter model. Ann. Probab. 3(4), 643–663 (1975)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Böhme, G.A., Gross, T.: Fragmentation transitions in multistate voter models. Phys. Rev. E 85, 066117 (2012)ADSCrossRefGoogle Scholar
  7. 7.
    Hubbell, S.P.: The Unified Neutral Theory of Biodiversity and Biogeography (MPB-32) (Monographs in Population Biology). Princeton University Press, Princeton (2001)Google Scholar
  8. 8.
    McKane, A.J., Alonso, D., Solé, R.V.: Analytic solution of hubbell’s model of local community dynamics. Theor. Popul. Biol. 65(1), 67–73 (2004)CrossRefzbMATHGoogle Scholar
  9. 9.
    Pigolotti, S., Flammini, A., Marsili, M., Maritan, A.: Species lifetime distribution for simple models of ecologies. Proc. Natl. Acad. USA 102(44), 15747–15751 (2005)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Starnini, M., Baronchelli, A., Pastor-Satorras, R.: Ordering dynamics of the multi-state voter model. J. Stat. Mech. P10027 (2012)Google Scholar
  11. 11.
    Mobilia, M.: Does a single zealot affect an infinite group of voters? Phys. Rev. Lett. 91, 028701 (2003)ADSCrossRefGoogle Scholar
  12. 12.
    Acemoglu, D., Como, G., Fagnani, F., Ozdaglar, A.E.: Opinion fluctuations and disagreement in social networks. Levine’s working paper archive, Levine, D.K. (2010)Google Scholar
  13. 13.
    Yildiz, E., Acemoglu, D., Ozdaglar, A.E., Saberi, A., Scaglione, A.: Discrete opinion dynamics with stubborn agents. LIDS report 2858, to appear in ACM Transactions on Economics and Computation (2012)Google Scholar
  14. 14.
    Wu, Y., Shen, J.: Opinion dynamics with stubborn vertices. Electron. J. Linear Algebr. 23, 790–800 (2012)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Xie, J., Sreenivasan, S., Korniss, G., Zhang, W., Lim, C., Szymanski, B.K.: Social consensus through the influence of committed minorities. Phys. Rev. E 84(1), 011130 (2011)ADSCrossRefGoogle Scholar
  16. 16.
    Xie, J., Emenheiser, J., Kirby, M., Sreenivasan, S., Szymanski, B.K., Korniss, G.: Evolution of opinions on social networks in the presence of competing committed groups. PLoS One 7(3), e33215 (2012)ADSCrossRefGoogle Scholar
  17. 17.
    Singh, P., Sreenivasan, S., Szymanski, B.K., Korniss, G.: Accelerating consensus on coevolving networks: the effect of committed individuals. Phys. Rev. E 85, 046104 (2012)ADSCrossRefGoogle Scholar
  18. 18.
    Mobilia, M.: Commitment versus persuasion in the three-party constrained voter model. J. Stat. Phys. 151, 69–91 (2013)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Watts, D.J., Strogatz, S.H.: Collective dynamics of “small-world” networks. Nature 393(6684), 409–10 (1998)CrossRefGoogle Scholar
  20. 20.
    Gardiner, C.W.: Handbook of Stochastic Methods. Springer, Berlin (1994)Google Scholar
  21. 21.
    Bastian, M., Heymann, S., Jacomy, M.: Gephi: an open source software for exploring and manipulating networks (2009)Google Scholar
  22. 22.
    Mobilia, M., Petersen, A., Redner, S.: On the role of zealotry in the voter model. J. Stat. Mech. 08, P08029 (2007)MathSciNetGoogle Scholar
  23. 23.
    Maruyama, G.: Continuous Markov processes and stochastic equations. Rend. Circ. Mat. Palermo 4(1), 48–90 (1955)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Słomiński, L.: On approximation of solutions of multidimensional sde’s with reflecting boundary conditions. Stoch. Process. Appl. 50(2), 197–219 (1994)CrossRefzbMATHGoogle Scholar
  25. 25. Accessed 1 Jan 2014
  26. 26.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Publications, New York (1964)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.ENEA—Italian Agency for New Technologies, Energy and Sustainable Economic DevelopmentFrascatiItaly
  2. 2.ISTAT—Istituto Nazionale di StatisticaRomeItaly

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