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Commitment Versus Persuasion in the Three-Party Constrained Voter Model

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Abstract

In the framework of the three-party constrained voter model, where voters of two radical parties (A and B) interact with “centrists” (C and C ζ ), we study the competition between a persuasive majority and a committed minority. In this model, A’s and B’s are incompatible voters that can convince centrists or be swayed by them. Here, radical voters are more persuasive than centrists, whose sub-population comprises susceptible agents C and a fraction ζ of centrist zealots C ζ . Whereas C’s may adopt the opinions A and B with respective rates 1+δ A and 1+δ B (with δ A ≥δ B >0), C ζ ’s are committed individuals that always remain centrists. Furthermore, A and B voters can become (susceptible) centrists C with a rate 1. The resulting competition between commitment and persuasion is studied in the mean field limit and for a finite population on a complete graph. At mean field level, there is a continuous transition from a coexistence phase when ζ<Δ c =δ A /(1+δ A ) to a phase where centrism prevails when ζ≥Δ c . In a finite population of size N, demographic fluctuations lead to centrism consensus and the dynamics is characterized by the mean consensus time τ. Because of the competition between commitment and persuasion, here consensus is reached much slower (ζ<Δ c ) or faster (ζ≥Δ c ) than in the absence of zealots (when τ∼N). In fact, when ζ<Δ c and there is an initial minority of centrists, the mean consensus time grows as τ∼N −1/2 e Nγ, with N≫1 and . The dynamics is thus characterized by a metastable state where the most persuasive voters and centrists coexist when δ A >δ B , whereas all species coexist when δ A =δ B . When ζ≥Δ c and the initial density of centrists is low, one finds τ∼lnN (when N≫1). Our analytical findings are corroborated by stochastic simulations.

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Notes

  1. These models are equivalent on one-dimensional lattices, but not in higher dimensions where the Ising-Glauber model follows a majority rule, whereas the VM evolves according to a a proportional rule. As a consequence, while there is a surface tension in the two-dimensional Ising-Glauber model this is not the case for the VM, see [10, 41] and references therein.

  2. The voter and Moran models are equivalent on regular graphs but lead to markedly different dynamics on degree-heterogeneous networks, see e.g. [43–47].

  3. It has also been shown that a class of models with voter-like dynamics and describing cooperation dilemmas are characterized by anomalous metastability on scale-free networks, with the mean consensus time and exit probability exhibiting a stretched exponential dependence on the population size and an exponent depending non-trivially on the degree distribution [53].

  4. One slight difference with the generic result given in Ref. [68] lies in the fact that here the absorbing boundary is at z=1−ζ. Furthermore, in the final result (26) we have chosen the same timescale as in Sect. 6.1.1 and divided (22) by N. This allows a direct comparison with the results of the Fokker-Planck treatment and differs by a factor N −1 from that of [66–71].

References

  1. Axelrod, R.: J. Confl. Resolut. 41, 203 (1997)

    Article  Google Scholar 

  2. Axelrod, R.: The Complexity of Cooperation. Princeton University Press, Princeton (1997)

    Google Scholar 

  3. Castellano, C., Marsili, M., Vespignani, A.: Phys. Rev. Lett. 85, 3536 (2000)

    Article  ADS  Google Scholar 

  4. Klemm, K., Eguiluz, V.M., Toral, R., San Miguel, M.: Phys. Rev. E 67, 045101 (2003)

    Article  ADS  Google Scholar 

  5. Deffuant, G., Neau, D., Amblard, F., Weisbuch, G.: Adv. Complex Syst. 3, 87 (2000)

    Article  Google Scholar 

  6. Hegselmann, R., Krause, U.: JASSS 5(3) (2002). http://jasss.soc.surrey.ac.uk/5/3/2/2.pdf

  7. Weisbuch, G., Deffuant, G., Amblard, F., Nadal, J.-P.: Complexity 7, 55 (2002)

    Article  Google Scholar 

  8. Ben-Naim, E., Krapivsky, P.L., Redner, S.: Physica D 183, 190 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  9. Slanina, F.: Eur. Phys. J. B 79, 99 (2011)

    Article  ADS  Google Scholar 

  10. Castellano, C., Fortunato, S., Loreto, V.: Rev. Mod. Phys. 81, 591 (2009)

    Article  ADS  Google Scholar 

  11. Crow, J.F., Kimura, M.: An Introduction to Population Genetics Theory. The Blackburn Press, New Jersey (1970)

    MATH  Google Scholar 

  12. Ewens, W.J.: Mathematical Population Genetics. Springer, USA (2004)

    Book  MATH  Google Scholar 

  13. Nowak, M.A.: Evolutionary Dynamics. Belknap Press, Cambridge, (2006)

    MATH  Google Scholar 

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

    Article  Google Scholar 

  15. Szabó, G., Fáth, G.: Phys. Rep. 446, 97 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  16. Galam, S.: J. Stat. Phys. 61, 943 (1990)

    Article  MathSciNet  ADS  Google Scholar 

  17. Galam, S.: Physica A 274, 132 (1999)

    Article  ADS  Google Scholar 

  18. Galam, S.: Eur. Phys. J. B 25, 403 (2002)

    ADS  Google Scholar 

  19. Galam, S.: Eur. Phys. J. B 45, 569 (2005)

    Article  ADS  Google Scholar 

  20. Sznajd-Weron, K., Sznajd, J.: Int. J. Mod. Phys. C 11, 1157 (2000)

    Article  ADS  Google Scholar 

  21. Mobilia, M., Redner, S.: Phys. Rev. E 68, 046106 (2003)

    Article  ADS  Google Scholar 

  22. Lambiotte, R., Redner, S.: Europhys. Lett. 82, 18007 (2008)

    Article  ADS  Google Scholar 

  23. Slanina, F., Sznajd-Weron, K., Przybyla, D.: Europhys. Lett. 82, 18006 (2008)

    Article  ADS  Google Scholar 

  24. Benczik, I.J., Benczik, S.Z., Schmittmann, B., Zia, R.K.P.: Europhys. Lett. 82, 48006 (2008).

    Article  ADS  Google Scholar 

  25. Volovik, D., Mobilia, M., Redner, S.: Europhys. Lett. 85, 48003 (2009).

    Article  ADS  Google Scholar 

  26. Liggett, T.M.: Interacting Particle Systems. Springer, USA (1985)

    Book  MATH  Google Scholar 

  27. Schelling, T.C.: J. Math. Sociol. 1, 143 (1971)

    Article  Google Scholar 

  28. Schelling, T.C.: Micromotives Macrobehavior. Norton, New York (1978)

    Google Scholar 

  29. Granovetter, M.: Am. J. Sociol. 78, 1360 (1973)

    Google Scholar 

  30. Granovetter, M.: Am. J. Sociol. 83, 1420 (1978).

    Article  Google Scholar 

  31. Vazquez, F., Krapivsky, P.L., Redner, S.: J. Phys. A, Math. Gen. 36, L61 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  32. Vazquez, F., Redner, S.: J. Phys. A, Math. Gen. 37, 8479 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  33. Mobilia, M.: Europhys. Lett. 95, 50002 (2011)

    Article  ADS  Google Scholar 

  34. Lanchier, N.: Ann. Appl. Probab. 22, 860 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  35. Mobilia, M.: Phys. Rev. Lett. 91, 028701 (2003)

    Article  ADS  Google Scholar 

  36. Mobilia, M., Georgiev, I.T.: Phys. Rev. E 71, 046102 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  37. Mobilia, M., Petersen, A., Redner, S.: J. Stat. Mech. 2007, P08029 (2007)

    Article  MathSciNet  Google Scholar 

  38. Galam, S., Jacobs, F.: Physica A 381, 366 (2007)

    Article  ADS  Google Scholar 

  39. Xie, J., Sreenivasan, S., Korniss, G., Zhang, W., Lim, C., Szymanski, B.K.: Phys. Rev. E 84, 011130 (2011)

    Article  ADS  Google Scholar 

  40. Glauber, R.J.: J. Math. Phys. 4, 294 (1963)

    Article  MathSciNet  ADS  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  42. Moran, P.A.P.: The Statistical Processes of Evolutionary Theory. Clarendon University Press, Oxford (1962)

    MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

  48. Gardiner, C.W.: Handbook of Stochastic Methods. Springer, USA (2002)

    Google Scholar 

  49. van Kampen, N.G.: Stochastic Processes in Physics and Chemistry. North-Holland, Amsterdam (1992)

    Google Scholar 

  50. Redner, S.: A Guide to First-Passage Processes. Cambridge University Press, USA (2001)

    Book  MATH  Google Scholar 

  51. Risken, H.: The Fokker-Planck Equation. Springer, New York (1989)

    Book  MATH  Google Scholar 

  52. Karlin, S., Taylor, H.M.: A Second Course in Stochastic Processes. Academic Press, New York (1981)

    MATH  Google Scholar 

  53. Assaf, M., Mobilia, M.: Phys. Rev. Lett. 109, 188701 (2012)

    Article  ADS  Google Scholar 

  54. Krapivsky, P.L.: Phys. Rev. A 45, 1067 (1992)

    Article  ADS  Google Scholar 

  55. Ben-Naim, E., Frachebourg, L., Krapivsky, P.L.: Phys. Rev. E 53, 3078 (1996)

    Article  ADS  Google Scholar 

  56. Frachebourg, L., Krapivsky, P.L.: Phys. Rev. E 53, R3009 (1996)

    Article  ADS  Google Scholar 

  57. Volovik, D., Redner, S.: J. Stat. Mech. 2012, P04003 (2012)

    Article  Google Scholar 

  58. Masuda, N., Gibert, N., Redner, S.: Phys. Rev. E 82, 010103 (2010)

    Article  ADS  Google Scholar 

  59. Mobilia, M.: Phys. Rev. E 86, 011134 (2012)

    Article  ADS  Google Scholar 

  60. Masuda, N.: Sci. Rep. 2, 646 (2012)

    Google Scholar 

  61. Steels, L.: Artif. Life 2, 319 (2005).

    Article  Google Scholar 

  62. Baronchelli, A., Felici, M., Caglioti, E., Loreto, V., Steels, L.: J. Stat. Mech. 2006, P06014 (2006)

    Article  Google Scholar 

  63. Crow, J.F., Kimura, M.: An Introduction to Population Genetics Theory. Blackburn Press, New Jersey (2009)

    Google Scholar 

  64. Ewens, W.J.: Mathematical Population Genetics. Springer, New York (2004)

    Book  MATH  Google Scholar 

  65. Gillespie, D.T.: J. Comput. Phys. 22, 403 (1976)

    Article  MathSciNet  ADS  Google Scholar 

  66. Assaf, M., Meerson, B.: Phys. Rev. Lett. 97, 200602 (2006)

    Article  ADS  Google Scholar 

  67. Assaf, M., Meerson, B.: Phys. Rev. E 75, 031122 (2007)

    Article  ADS  Google Scholar 

  68. Assaf, M., Meerson, B.: Phys. Rev. E 81, 021116 (2010)

    Article  ADS  Google Scholar 

  69. Mobilia, M., Assaf, M.: Europhys. Lett. 91, 10002 (2010)

    Article  ADS  Google Scholar 

  70. Assaf, M., Mobilia, M.: J. Stat. Mech. 2010, P09009 (2010)

    Article  MathSciNet  Google Scholar 

  71. Assaf, M., Mobilia, M.: J. Theor. Biol. 275, 93 (2011)

    Article  MathSciNet  Google Scholar 

  72. Landau, L.D., Lifshitz, E.M.: Quantum Mechanics: Non-Relativistic Theory. Pergamon, London (1977)

    Google Scholar 

  73. Kubo, R., Matsuo, K., Kitahara, K.: J. Stat. Phys. 9, 51 (1973)

    Article  ADS  Google Scholar 

  74. Dykman, M.I., Mori, E., Ross, J., Hunt, P.M.: J. Chem. Phys. 100, 5735 (1973)

    Article  ADS  Google Scholar 

  75. Escudero, C., Kamenev, A.: Phys. Rev. E 79, 041149 (2009)

    Article  ADS  Google Scholar 

  76. Antal, T., Scheuring, I.: Bull. Math. Biol. 68, 1923 (2006)

    Article  MathSciNet  Google Scholar 

  77. Cremer, J., Reichenbach, T., Frey, E.: New J. Phys. 11, 093029 (2009)

    Article  ADS  Google Scholar 

  78. Mobilia, M.: J. Theor. Biol. 264, 1 (2010)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The author is grateful to Sid Redner for useful discussions and insightful comments on a preliminary version of the manuscript.

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  1. Department of Applied Mathematics, School of Mathematics, University of Leeds, Leeds, LS2 9JT, UK

    Mauro Mobilia

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Mobilia, M. Commitment Versus Persuasion in the Three-Party Constrained Voter Model. J Stat Phys 151, 69–91 (2013). https://doi.org/10.1007/s10955-012-0656-x

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