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Heterophobic interactions hinder consensus formation in sparse random networks

Alejandro Castro1, Tuan Minh Pham2,3, Ernesto Ortega4, and David Machado1,5,6,*

  • *Contact author: david.machado@uniroma1.it
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Phys. Rev. E 112, 064309 – Published 15 December, 2025

DOI: https://doi.org/10.1103/cxsr-dh25

Abstract

Heterophobic interactions, which drive individuals to be repelled from others with opposite opinions, play a role as important as homophilic ones in shaping many dynamical processes on social networks, such as opinion formation, social balance, or epidemic spreading. In this paper, we use belief propagation and Monte Carlo simulations on treelike signed graphs to predict that a sufficient propensity to heterophobia can impede a consensus that would otherwise emerge via a phase transition. As the strength of heterophobic interactions and the rationality of individuals with respect to social stress decrease, this transition changes from continuous to discontinuous, with a strong dependence on the initial conditions. The size of the parameter region where consensus can be reached from any initial condition decays as a power-law function of the number of discussed topics.

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References (58)

  1. C. Castellano, S. Fortunato, and V. Loreto, Statistical physics of social dynamics, Rev. Mod. Phys. 81, 591 (2009).
  2. M. Jusup, P. Holme, K. Kanazawa, M. Takayasu, I. Romić, Z. Wang, S. Geček, T. Lipić, B. Podobnik, L. Wang, W. Luo, T. Klanjšček, J. Fan, S. Boccaletti, and M. Perc, Social physics, Phys. Rep. 948, 1 (2022).
  3. S. Redner, Reality-inspired voter models: A mini-review, C. R. Phys. 20, 275 (2019).
  4. P. DiMaggio, J. Evans, and B. Bryson, Have American's social attitudes become more polarized? Am. J. Sociol. 102, 690 (1996).
  5. M. P. Fiorina and S. J. Abrams, Political polarization in the American public, Annu. Rev. Polit. Sci. 11, 563 (2008).
  6. M. Granovetter, Threshold models of collective behavior, Am. J. Sociol. 83, 1420 (1978).
  7. J. P Gleeson, Cascades on correlated and modular random networks, Phys. Rev. E 77, 046117 (2008).
  8. E. Bonabeau, G. Theraulaz, and J.-L. Deneubourg, Phase diagram of a model of self-organizing hierarchies, Phys. A: Stat. Mech. Appl. 217, 373 (1995).
  9. D. Mistry, Q. Zhang, N. Perra, and A. Baronchelli, Committed activists and the resilience of crowd dynamics, Phys. Rev. E 92, 042805 (2015).
  10. N. E. Friedkin, Norm formation in social influence networks, Soc. Netw. 23, 167 (2001).
  11. V. Sood and S. Redner, Voter model on heterogeneous graphs, Phys. Rev. Lett. 94, 178701 (2005).
  12. P. Holme and M. E. J. Newman, Nonequilibrium phase transition in the coevolution of networks and opinions, Phys. Rev. E 74, 056108 (2006).
  13. F. Vazquez, V. M. Eguíluz, and M. S. Miguel, Generic absorbing transition in coevolution dynamics, Phys. Rev. Lett. 100, 108702 (2008).
  14. R. Durrett, J. P. Gleeson, A. L. Lloyd, P. J. Mucha, F. Shi, D. Sivakoff, J. E. S. Socolar, and C. Varghese, Graph fission in an evolving voter model, Proc. Natl. Acad. Sci. USA 109, 3682 (2012).
  15. F. Baumann, P. Lorenz-Spreen, I. M. Sokolov, and M. Starnini, Modeling echo chambers and polarization dynamics in social networks, Phys. Rev. Lett. 124, 048301 (2020).
  16. T. Pham, S. Redner, L. Waldorp, J. Armas, and H. L. J. van der Maas, Polarisation in increasingly connected societies, arXiv:2503.24098.
  17. R. Axelrod, The dissemination of culture: A model with local convergence and global polarization, J. Conflict Resolut. 41, 203 (1997).
  18. T. M. Pham, A. C. Alexander, J. Korbel, R. Hanel, and S. Thurner, Balance and fragmentation in societies with homophily and social balance, Sci. Rep. 11, 17188 (2021).
  19. P. J. Górski, K. Bochenina, J. A. Hołyst, and R. M. D' Souza, Homophily based on few attributes can impede structural balance, Phys. Rev. Lett. 125, 078302 (2020).
  20. P. J. Górski, C. Atkisson, and J. A. Hołyst, A general model for how attributes can reduce polarization in social groups, Net. Sci. 11, 536 (2023).
  21. P. J Górski, A. Sulik, G. Andres, G. Vaccario, and J. A Hołyst, Coexistence of balance and hierarchies: An ego perspective to explain empirical networks, PNAS Nexus 4, pgaf130 (2025).
  22. T. M. Pham, J. Korbel, R. Hanel, and S. Thurner, Empirical social triad statistics can be explained with dyadic homophylic interactions, Proc. Natl. Acad. Sci. USA 119, e2121103119 (2022).
  23. J. Korbel, S. D. Lindner, T. M. Pham, R. Hanel, and S. Thurner, Homophily-based social group formation in a spin glass self-assembly framework, Phys. Rev. Lett. 130, 057401 (2023).
  24. A. Flache, M. Mäs, T. Feliciani, E. Chattoe-Brown, G. Deffuant, S. Huet, and J. Lorenz, Models of social influence: Towards the next frontiers, JASSS 20, 2 (2017).
  25. M. W. Macy, J. A. Kitts, A. Flache, and S. Benard, Polarization in dynamic networks: A Hopfield model of emergent structure, in Dynamic Social Network Modeling and Analysis: Workshop Summary and Papers (National Academies Press, Washington, DC, 2003), p. 162.
  26. M. Galesic, H. Olsson, T. Pham, J. Sorger, and S. Thurner, Experimental evidence confirms that triadic social balance can be achieved through dyadic interactions, npj Complex. 2, 1 (2025).
  27. Y. Kabashima and D. Saad, Belief propagation vs TAP for decoding corrupted messages, Europhys. Lett. 44, 668 (1998).
  28. J. S Yedidia, W. T Freeman, and Y. Weiss, Constructing free-energy approximations and generalized belief propagation algorithms, Inf. Theory IEEE Trans. 51, 2282 (2005).
  29. J. Pearl, Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference (Morgan Kaufmann, Burlington, MA, 1988).
  30. F. R. Kschischang, B. J. Frey, and H.-A. Loeliger, Factor graphs and the sum-product algorithm, IEEE Trans. Inf. Theory 47, 498 (2001).
  31. J. S Yedidia, W. Freeman, and Y. Weiss, Generalized belief propagation, in Advances in Neural Information Processing Systems, edited by T. Leen, T. Dietterich, and V. Tresp (MIT Press, Cambridge, MA, 2000), Vol. 13, pp. 689–695.
  32. J. S Yedidia, W. T Freeman, and Y. Weiss, Understanding belief propagation and its generalizations, in Exploring Artificial Intelligence in the New Millennium (Morgan Kaufmann, Burlington, MA, 2003), pp. 239–269.
  33. T. Rizzo, A. Lage-Castellanos, R. Mulet, and F. Ricci-Tersenghi, Replica cluster variational method, J. Stat. Phys. 139, 375 (2010).
  34. F. L. Metz, I. Neri, and D. Bollé, Spectra of sparse regular graphs with loops, Phys. Rev. E 84, 055101(R) (2011).
  35. D. Bollé, F. L. Metz, and I. Neri, On the spectra of large sparse graphs with cycles, in Spectral Analysis, Differential Equations and Mathematical Physics: A Festschrift in Honor of Fritz Gesztesy's 60th Birthday, edited by H. Holden et al. (American Mathematical Society, Providence, RI, 2013), pp. 35–58.
  36. A. Kirkley, G. T. Cantwell, and M. E. J. Newman, Belief propagation for networks with loops, Sci. Adv. 7, eabf1211 (2021).
  37. G. T. Cantwell, A. Kirkley, and F. Radicchi, Heterogeneous message passing for heterogeneous networks, Phys. Rev. E 108, 034310 (2023).
  38. Y. Wang, Y. E. Zhang, F. Pan, and P. Zhang, Tensor network message passing, Phys. Rev. Lett. 132, 117401 (2024).
  39. Grover E. C. Guzman, P. F. Stadler, and A. Fujita, Cavity approach for the approximation of spectral density of graphs with heterogeneous structures, Phys. Rev. E 109, 034303 (2024).
  40. T. M. Pham, T. Peron, and F. L. Metz, Effects of clustering heterogeneity on the spectral density of sparse networks, Phys. Rev. E 110, 054307 (2024).
  41. G. Deffuant, D. Neau, F. Amblard, and G. Weisbuch, Mixing beliefs among interacting agents, Adv. Complex Syst. 03, 87 (2000).
  42. F. Baumann, P. Lorenz-Spreen, I. M. Sokolov, and M. Starnini, Emergence of polarized ideological opinions in multidimensional topic spaces, Phys. Rev. X 11, 011012 (2021).
  43. A. Jedrzejewski and K. Sznajd-Weron, Person-situation debate revisited: Phase transitions with quenched and annealed disorders, Entropy 19, 415 (2017).
  44. N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, Equation of state calculations by fast computing machines, J. Chem. Phys. 21, 1087 (1953).
  45. D. B. Bahr and E. Passerini, Statistical mechanics of opinion formation and collective behavior: Micro-sociology, J. Math. Soc. 23, 1 (1998).
  46. M. Mézard and A. Montanari, Information, Physics, and Computation (Oxford University Press, Oxford, UK, 2009).
  47. A. Bizyaeva, A. Franci, and N. E. Leonard, Multitopic belief formation through bifurcations over signed social networks, IEEE Trans. Autom. Control 70, 5082 (2025).
  48. S. Arrhenius, Über die dissociationswärme und den einfluss der temperatur auf den dissociationsgrad der elektrolyte, Z. Phys. Chem. 4U, 96 (1889).
  49. See https://github.com/d4v1d-cub/Homophily_RandGraphs and https://doi.org/10.5281/zenodo.15789890.
  50. M. Mézard and G. Parisi, The Bethe lattice spin glass revisited, Eur. Phys. J. B 20, 217 (2001).
  51. A. Mozeika and A. C. C. Coolen, Dynamical replica analysis of processes on finitely connected random graphs: II Dynamics in the griffiths phase of the diluted ising ferromagnet, J. Phys. A: Math. Theor. 42, 195006 (2009).
  52. J. Korbel, S. Havlin, and S. Thurner, Microscopic origin of abrupt mixed-order phase transitions, Nat. Commun. 16, 2628 (2025).
  53. M. G. Noudehi, A. Kargaran, N. Azimi-Tafreshi, and G. R. Jafari, Second- to first-order phase transition: Coevolutionary versus structural balance, Phys. Rev. E 106, 044303 (2022).
  54. A. Lage-Castellanos and R. Mulet, Zero temperature solutions of the Edwards-Anderson model in random Husimi lattices, Eur. Phys. J. B 65, 117 (2008).
  55. L. Pedraza, S. Pinto, J. P. Pinasco, and P. Balenzuela, Analytical approach to the axelrod model based on similarity vectors, Phys. Rev. E 103, 012307 (2021).
  56. A. Montanari, F. Ricci-Tersenghi, and G. Semerjian, Clusters of solutions and replica symmetry breaking in random k-satisfiability, J. Stat. Mech. (2008) P04004.
  57. F. R. -Tersenghi and G. Semerjian, On the cavity method for decimated random constraint satisfaction problems and the analysis of belief propagation guided decimation algorithms, J. Stat. Mech. (2009) P09001.
  58. L. Zdeborová and F. Krzakala, Generalization of the cavity method for adiabatic evolution of Gibbs states, Phys. Rev. B 81, 224205 (2010).

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Phys. Rev. E 112, 064309– Published 15 December, 2025
  • Received 2 July 2025
  • Accepted 24 November 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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