- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review E

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Modularity, asymmetry, and polarization shape consensus speed in the voter model

Madi Yerlanov*, Zachary P. Kilpatrick†, and Nancy Rodriguez‡

  • *Contact author: madi.yerlanov@colorado.edu
  • †Contact author: zpkilpat@colorado.edu
  • ‡Contact author: rodrign@colorado.edu
PDF

Phys. Rev. E 114, 044313 – Published 13 October, 2026

DOI: https://doi.org/10.1103/br97-fw3p

Abstract

In populations with community structure, the formation of consensus requires both alignment within and diffusion of beliefs across groups, processes that evolve on distinct timescales. How do modularity, asymmetry, and polarization shape this process? We study a variant of the voter model in which a population is divided into two cliques. At each time step, a pair of nodes is selected; if their binary opinions differ, then each agent adopts the opinion of the other with a certain probability. The type of pair selected depends on a network structure parameter—modularity that is central in this paper. We analyze how this coupling strength, population imbalance, and initial polarization jointly determine the time to consensus. Formation of consensus generally starts with interclique interactions rapidly synchronizing the two cliques' opinion fractions, after which consensus is reached through a slower diffusion along the synchronized curve. This slow stage is largely insensitive to modularity except when the cliques are nearly disconnected. To analyze these dynamics, we derive time-continuous approximations in the large-population limit and assess their accuracy against the discrete model. While modularity primarily affects the synchronization stage, initially polarized and asymmetric populations exhibit nontrivial effects, including regimes in which an intermediate level coupling minimizes consensus time. A small-clique scaling analysis reveals that this optimum arises from a competition between fast alignment drift and noise amplification in the smaller group.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (44)

  1. M. Girvan and M. E. Newman, Community structure in social and biological networks, Proc. Natl. Acad. Sci. USA 99, 7821 (2002).
  2. S. Fortunato, Community detection in graphs, Phys. Rep. 486, 75 (2010).
  3. D. Centola, How Behavior Spreads: The Science of Complex Contagions (Princeton University Press, Princeton, NJ, 2018), Vol. 3.
  4. C. Castellano, S. Fortunato, and V. Loreto, Statistical physics of social dynamics, Rev. Mod. Phys. 81, 591 (2009).
  5. D. Centola, The spread of behavior in an online social network experiment, Science 329, 1194 (2010).
  6. T. M. Liggett, Stochastic Interacting Systems: Contact, Voter and Exclusion Processes (Springer Science & Business Media, Berlin, 2013), Vol. 324.
  7. P. Clifford and A. Sudbury, A model for spatial conflict, Biometrika 60, 581 (1973).
  8. R. A. Holley and T. M. Liggett, Ergodic theorems for weakly interacting infinite systems and the voter model, Ann. Probab. 3, 643 (1975).
  9. V. Sood and S. Redner, Voter model on heterogeneous graphs, Phys. Rev. Lett. 94, 178701 (2005).
  10. G. J. Baxter, R. A. Blythe, and A. J. McKane, Fixation and consensus times on a network: A unified approach, Phys. Rev. Lett. 101, 258701 (2008).
  11. T. Antal, S. Redner, and V. Sood, Evolutionary dynamics on degree-heterogeneous graphs, Phys. Rev. Lett. 96, 188104 (2006).
  12. N. Masuda, Voter model on the two-clique graph, Phys. Rev. E 90, 012802 (2014).
  13. M. T. Gastner and K. Ishida, Voter model on networks partitioned into two cliques of arbitrary sizes, J. Phys. A: Math. Theor. 52, 505701 (2019).
  14. G. Deffuant, D. Neau, F. Amblard, and G. Weisbuch, Mixing beliefs among interacting agents, Adv. Complex Syst. 03, 87 (2000).
  15. R. Hegselmann and U. Krause, Opinion dynamics and bounded confidence models, analysis and simulation, J. Artif. Soc. Social Simul. 5, 1, (2002).
  16. G. Toscani, Kinetic models of opinion formation, Commun. Math. Sci. 4, 481 (2006).
  17. G. Albi, L. Pareschi, G. Toscani, and M. Zanella, Recent advances in opinion modeling: Control and social influence, in Active Particles, Volume 1, edited by N. Bellomo, P. Degond, and E. Tadmor, Modeling and Simulation in Science, Engineering and Technology (Birkhäuser, Cham, 2017), pp. 49–98.
  18. B. Düring, J. Franceschi, M.-T. Wolfram, and M. Zanella, Breaking consensus in kinetic opinion formation models on graphons, J. Nonlinear Sci. 34, 79 (2024).
  19. J. Grilli, T. Rogers, and S. Allesina, Modularity and stability in ecological communities, Nat. Commun. 7, 12031 (2016).
  20. É. Danchin, L.-A. Giraldeau, T. J. Valone, and R. H. Wagner, Public information: From nosy neighbors to cultural evolution, Science 305, 487 (2004).
  21. T. C. Bergstrom, Evolution of social behavior: Individual and group selection, J. Econ. Perspect. 16, 67 (2002).
  22. B. Allen, G. Lippner, Y.-T. Chen, B. Fotouhi, N. Momeni, S.-T. Yau, and M. A. Nowak, Evolutionary dynamics on any population structure, Nature (London) 544, 227 (2017).
  23. D. Hathcock and S. H. Strogatz, Fitness dependence of the fixation-time distribution for evolutionary dynamics on graphs, Phys. Rev. E 100, 012408 (2019).
  24. G. W. A. Constable, T. Rogers, A. J. McKane, and C. E. Tarnita, Demographic noise can reverse the direction of deterministic selection, Proc. Natl. Acad. Sci. USA 113, E4745 (2016).
  25. G. W. A. Constable and A. J. McKane, Exploiting fast-variables to understand population dynamics and evolution, J. Stat. Phys. 172, 3 (2018).
  26. S. Ghosh, Distributed Systems: An Algorithmic Approach (Chapman and Hall/CRC, Boca Raton, FL, 2006).
  27. R. Olfati-Saber and R. M. Murray, Consensus problems in networks of agents with switching topology and time-delays, IEEE Trans. Autom. Control 49, 1520 (2004).
  28. A. Olshevsky and J. N. Tsitsiklis, Convergence speed in distributed consensus and averaging, SIAM J. Control Optim. 48, 33 (2009).
  29. S. Redner, Reality-inspired voter models: A mini-review, C. R. Phys. 20, 275 (2019).
  30. D. Bhat and S. Redner, Polarization and consensus by opposing external sources, J. Stat. Mech. (2020) 013402.
  31. L. Becchetti, V. Bonifaci, E. Cruciani, and F. Pasquale, in Proceedings of the 32nd International Joint Conference on Artificial Intelligence (IJCAI), edited by E. Elkind (International Joint Conferences on Artificial Intelligence, 2023), pp. 38–45.
  32. C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry, and the Natural Sciences, 2nd ed., Springer Series in Synergetics, Vol. 13 (Springer, Berlin, 1985).
  33. S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence (John Wiley & Sons, New York, NY, 1986).
  34. N. G. van Kampen, Stochastic Processes in Physics and Chemistry, 3rd ed. (Elsevier, Amsterdam, 2007).
  35. H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications (Springer, Berlin, 1989).
  36. A. J. McGann, The tyranny of the supermajority, J. Theor. Politics 16, 53 (2004).
  37. S. Chapman and T. G. Cowling, The Mathematical Theory of Non-uniform Gases, 3rd ed. (Cambridge University Press, Cambridge, UK, 1970).
  38. G. A. Pavliotis, Stochastic Processes and Applications (Springer, Berlin, 2014), Vol. 60.
  39. M. Yerlanov, Voter model code, 2025, https://github.com/MaYeatCo/Voter_Model.
  40. P. Trubowitz and N. Mellow, “Going bipartisan”: Politics by other means, Political Sci. Q. 120, 433 (2005).
  41. C. A. Bail, L. P. Argyle, T. W. Brown, J. P. Bumpus, H. Chen, M. F. Hunzaker, J. Lee, M. Mann, F. Merhout, and A. Volfovsky, Exposure to opposing views on social media can increase political polarization, Proc. Natl. Acad. Sci. USA 115, 9216 (2018).
  42. P. Dandekar, A. Goel, and D. T. Lee, Biased assimilation, homophily, and the dynamics of polarization, Proc. Natl. Acad. Sci. USA 110, 5791 (2013).
  43. W. J. Ewens, Mathematical Population Genetics: Theoretical Introduction (Springer, Berlin, 2004), Vol. 27.
  44. M. Mobilia, Does a single zealot affect an infinite group of voters? Phys. Rev. Lett. 91, 028701 (2003).

Outline

Information

Phys. Rev. E 114, 044313– Published 13 October, 2026
  • Received 30 March 2026
  • Revised 25 June 2026
  • Accepted 29 September 2026
Article part of CHORUS
Crossmark - Check for updates button

©2026 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation