Opinion evolution in the presence of constant propaganda: homogeneous and localized cases


The opinion evolution of a group of agents arranged in a square lattice in the presence of external constant propaganda is studied. The contagion between agents is modeled according to the voter model, but the effect of external propaganda and a social temperature is also considered. At a first stage, the influence of the contagion probability, the temperature, and the value of the propaganda are analyzed for a homogeneous application of this last parameter. An analytical expression was found for the stationary state in some cases. Also, the effect of the spatial location of the propaganda is analyzed, where it is applied only to a subset of agents. The random distribution of agents affected by the propaganda was found to be the most effective, while for the case of distribution in patches and stripes, it depends on their size.

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Data Availability Statement

This manuscript has no associated data or the data will not be deposited. [Authors’ comment: The article in its current form has all the information needed to reproduce the presented results. There is no extra information or data that has been omitted.]


  1. 1.

    W. Weidlich, Sociodynamics: A Systematic Approach to Mathematical Modelling in the Social Sciences (Harwood Academic Publishers, Amsterdam, 2000)

    Google Scholar 

  2. 2.

    D. Stauffer, Introduction to statistical physics outside physics. Phys. A 336, 1–5 (2004)

    Google Scholar 

  3. 3.

    S. Galam, Application of statistical physics to politics. Phys. A 274, 132–139 (1999)

    Google Scholar 

  4. 4.

    S. Galam, Sociophysics: a personal testimony. Phys. A 336, 49–55 (2004a)

    Google Scholar 

  5. 5.

    S. Galam, Sociophysics: a review of Galam models. Int. J. Mod. Phys. C 19, 409–440 (2008)

    ADS  MATH  Google Scholar 

  6. 6.

    S. Galam, Contrarian deterministic effects on opinion dynamics: “the hung elections scenario”. Phys. A 333, 453 (2004)

    MathSciNet  Google Scholar 

  7. 7.

    R. Axelrod, The Complexity of Cooperation (Princeton University Press, Princeton, New Jersey, 1997)

    Google Scholar 

  8. 8.

    R. Axelrod, R. Hamilton, The evolution of cooperation. Science 211, 1390–1396 (1981)

    ADS  MathSciNet  MATH  Google Scholar 

  9. 9.

    N. Crokidakis, Effects of mass media on opinion spreading in the sznajd sociophysics model. Phys. A 391, 1729–1734 (2012)

    Google Scholar 

  10. 10.

    S.E. Parsegov, A.V. Proskurnikov, R. Tempo, N.E. Friedkin, Novel multidimensional models of opinion dynamics in social networks. IEEE Trans. Autom. Control 62(5), 2270–2285 (2017)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    F. Schweitzer, Sociophysics. Phys. Today 71(2), 40 (2018)

    Google Scholar 

  12. 12.

    M. Perc, The social physics collective. Sci. Rep. 9, 16549 (2019)

    ADS  Google Scholar 

  13. 13.

    M.G. Zimmermann, V.M. Eguíluz, M.S. Miguel, Cooperation, adaption and the emergence of leadership, in Economics and Heterogeneous Interacting Agents, ed. by J.B. Zimmermann, A. Kirman (Springer, Berlin, Heidelberg, 2001), pp. 73–86

  14. 14.

    H. Ebel, S. Bornholdt, Evolutionary games and the emergence of complex networks. arXiv:cond-mat/0211666, (2002)

  15. 15.

    S.H. Strogatz, Exploring complex networks. Nature 410, 268–276 (2001)

    ADS  MATH  Google Scholar 

  16. 16.

    R. Albert, A.-L. Barabasi, Statistical mechanics of complex networks. Rev. Mod. Phys. 74, 47–97 (2002)

    ADS  MathSciNet  MATH  Google Scholar 

  17. 17.

    A. Matjaž Perc, Szolnoki, coevolutionary games—a mini review. BioSystems 99, 109–125 (2010)

    Google Scholar 

  18. 18.

    M. Perc, J. Gómez-Gardeñes, A. Szolnoki, L.M. Floría, Y. Moreno, Interface Evolutionary dynamics of group interactions on structured populations: a review. J. R. Soc. 10, 20120997 (2013)

    Google Scholar 

  19. 19.

    F. Ding, Y. Liu, B. Shen, X. Si, An evolutionary game theory model of binary opinion formation. Phys. A 389, 1745–1752 (2010)

    Google Scholar 

  20. 20.

    M. Perc, J.J. Jordan, D.G. Rand, Z. Wang, S. Boccaletti, A. Szolnoki, Statistical physics of human cooperation. Phys. Rep. 687, 1–51 (2017)

    ADS  MathSciNet  MATH  Google Scholar 

  21. 21.

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

    Google Scholar 

  22. 22.

    C. Castellano, S. Fortunato, V. Loreto, Statistical physics of social dynamics. Rev. Mod. Phys. 81, 591 (2009)

    ADS  Google Scholar 

  23. 23.

    F. Vazquez, S. Redner, Ultimate fate of constrained voters. J. Phys. A 37, 8479 (2004)

    ADS  MathSciNet  MATH  Google Scholar 

  24. 24.

    F. Vazquez, S. Kaprivsky, S. Redner, Constrained opinion dynamics: Freezing and slow evolution. J. Phys. A 36, L61 (2003)

    ADS  MathSciNet  MATH  Google Scholar 

  25. 25.

    P. Balenzuela, J.P. Pinasco, V. Semeshenko, The undecided have the key: Interaction-driven opinion dynamics in a three state model. PLos One (2015). https://doi.org/10.1371/journal.pone.0139572

    Article  MATH  Google Scholar 

  26. 26.

    M.C. Gimenez, A.P. Paz García, M.A. Burgos Paci, L. Reinaudi, Range of interaction in an opinion evolution model of ideological self-positioning: contagion, hesitance and polarization. Phys. A 447, 320–330 (2016)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    R. Hegselmann, U. Krause, Opinion dynamics and bounded confidence models, analysis and simulation. J. Artif. Soc. Soc. Simul. 5(3), 2 (2002)

  28. 28.

    G. Deffuant, D. Neau, F. Amblard, G. Weisbuch, Mixing beliefs among interacting agents. Adv. Complex Syst. 3, 87–98 (2000)

    Google Scholar 

  29. 29.

    G. Deffuant, F. Amblard, G. Weisbuch, T. Faure, How can extremism prevail? a study based on the relative agreement interaction model. J. Artif. Soc. Soc. Simul. 5(4), 1 (2002)

  30. 30.

    J.C. Dittmer, Consensus formation under bounded confidence. Nonlinear Anal. 47, 4615–4621 (2001)

    MathSciNet  MATH  Google Scholar 

  31. 31.

    M.F. Laguna, G. Abramson, D.H. Zanette, Minorities in a model for opinion formation. Complexity 9(4), 31 (2004)

    MathSciNet  Google Scholar 

  32. 32.

    J. Zhang, Y. Hong, Opinion evolution analysis for short-range and long-range deffuant-weisbuch models. Phys. A 392, 5289–5297 (2013)

    MathSciNet  MATH  Google Scholar 

  33. 33.

    P.L. Krapivsky, S. Redner, Dynamics of majority rule in two-state interacting spin systems. Phys. Rev. Lett. 90(23), 238701 (2003)

    ADS  Google Scholar 

  34. 34.

    M. Mobilia, S. Redner, Majority versus minority dynamics: phase transition in an interacting two-state spin system. Phys. Rev. E 68, 046106 (2003)

    ADS  Google Scholar 

  35. 35.

    M. Kuperman, D. Zanette, Stochastic resonance in a model of opinion formation on small-world networks. Eur. Phys. J. B 26, 387–391 (2002)

    ADS  Google Scholar 

  36. 36.

    K. Sznajd-Weron, J. Sznajd, Opinion evolution in closed community. Int. J. Mod. Phys. C 11, 1157–1165 (2000)

    ADS  MATH  Google Scholar 

  37. 37.

    J.R. Sánchez, A modified one-dimensional sznajd model. arXiv:cond-mat/0408518, (2004)

  38. 38.

    D. Stauffer, Monte Carlo simulations of sznajd models. J. Artif. Soc. Soc. Simul. 5, 477 (2002)

    Google Scholar 

  39. 39.

    M.S. de la Lama, J.M. López, H.S. Wio, Spontaneous emergence of contrarian-like behaviour in an opinion spreading model. Europhys. Lett. 72, 851 (2005)

    ADS  Google Scholar 

  40. 40.

    H.S. Wio, M.S. de la Lama, J.M. López, Contrarian-like behaviour and system size stochastic resonance in an opinion spreading model. Phys. A 371, 108–111 (2006)

    Google Scholar 

  41. 41.

    M.C. Gimenez, J.A. Revelli, M.S. de la Lama, J.M. López, H.S. Wio, Interplay between social debate and propaganda in an opinion formation model. Phys. A 392, 278–286 (2013)

    Google Scholar 

  42. 42.

    M.C. Gimenez, J.A. Revelli, H.S. Wio, Non local effects in the sznajd model: Stochastic resonance aspects. ICST Trans. Complex Syst. e3, 10–12 (2012)

    Google Scholar 

  43. 43.

    P. Clifford, A. Sudbury, A model for spatial conflict. Biometrika 60(3), 581 (1973)

    MathSciNet  MATH  Google Scholar 

  44. 44.

    R. Holley, T.M. Ligget, Ergodic theorem for weakly interacting infinite systems and the voter model. Ann. Probab. 3(4), 643–663 (1975)

    MathSciNet  Google Scholar 

  45. 45.

    K. Suchecki, V.M. Eguíluz, M. San Miguel, Voter model dynamics in complex networks: role of dimensionality, disorder and degree distribution. Phys. Rev. E 72, 036132 (2005)

    ADS  Google Scholar 

  46. 46.

    F. Vazquez, V. Eguíluz, Analytical solution of the voter model on uncorrelated networks. New J. Phys. 10, 063011 (2008)

    ADS  Google Scholar 

  47. 47.

    E. Ben-waim, L. Frachebourg, P.L. Krapivsky, Coarsening and persistence in the voter model. Phys. Rev. E 53, 1996 (1996)

    Google Scholar 

  48. 48.

    F.M. Bass, A new product growth for model consumer durables. Manag. Sci. 15, 215–227 (1969)

    MATH  Google Scholar 

  49. 49.

    S. Gonçalves, M.F. Laguna, J.R. Iglesias, Why, when, and how fast innovations are adopted. Eur. Phys. J. B 85, 192 (2012)

    ADS  Google Scholar 

  50. 50.

    A.H. Rodríguez, Y. Moreno, Effects of mass media action on the axelrod model with social influence. Phys. Rev. E 82, 016111 (2010)

    ADS  Google Scholar 

  51. 51.

    S. Galam, S. Moscovici, Towards a theory of collective phenomena: consensus and attitude changes in groups. Eur. J. Soc. Psychol. 21, 49–74 (1991)

    Google Scholar 

  52. 52.

    Adrián Tarín Sanz, Communication, ideology and power: notes for the debate between the intentional propaganda theory and the spontaneous reproduction of propaganda theory. Comun. Soc. 32, 191–209 (2018)

    Google Scholar 

  53. 53.

    F.J.Z. Borgesius, J. Möller, S. Kruikemeier, R. Fathaigh, K. Irion, T. Dobber, B. Bodo, C. de Vreese, Online political microtargeting: Promises and threats for democracy. Utrecht Law Rev. 14(1), 82–96 (2018)

    Google Scholar 

  54. 54.

    D. Spohr, Fake news and ideological polarization: filter bubbles and selective exposure on social media. Bus. Inf. Rev. 34(3), 150–160 (2017)

    Google Scholar 

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The authors are grateful acknowledge to CONICET, Argentina, Foncyt, and Secyt, for financial support. Dr. Horacio Wio is also acknowledged for inspiring ideas. Computer clusters BACO, from UNSL and Verseo, from the Department of Theoretical and Computational Chemistry, UNC, were employed. PMC and AJR-P are grateful to CONICET (Argentina) under project number PIP 112-201101-00615, and Universidad Nacional de San Luis (Argentina) under project No. 03-0816.

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MCG performed the simulations. All of the authors have contributed equally to the discussion and analysis of the model, interpretation of the results, and writing and editing of the manuscript.

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Correspondence to Luis Reinaudi.

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Gimenez, M.C., Reinaudi, L., Paz-García, A.P. et al. Opinion evolution in the presence of constant propaganda: homogeneous and localized cases. Eur. Phys. J. B 94, 35 (2021). https://doi.org/10.1140/epjb/s10051-021-00047-5

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