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Consensus in the Hegselmann–Krause Model

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Abstract

This paper is concerned with the probability of consensus in a multivariate, socially structured version of the Hegselmann–Krause model for the dynamics of opinions. Individuals are located on the vertices of a finite connected graph representing a social network, and are characterized by their opinion, with the set of opinions \(\Delta \) being a general bounded convex subset of a finite dimensional normed vector space. Having a confidence threshold \(\tau \), two individuals are said to be compatible if the distance (induced by the norm) between their opinions does not exceed the threshold \(\tau \). Each vertex x updates its opinion at rate the number of its compatible neighbors on the social network, which results in the opinion at x to be replaced by a convex combination of the opinion at x and the nearby opinions: \(\alpha \) times the opinion at x plus \((1 - \alpha )\) times the average opinion of its compatible neighbors. The main objective is to derive a lower bound for the probability of consensus when the opinions are initially independent and identically distributed with values in the opinion set \(\Delta \).

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References

  1. Axelrod, R.: The dissemination of culture: a model with local convergence and global polarization. J. Conflict Resolut. 41(2), 203–226 (1997)

    Article  Google Scholar 

  2. Bramson, M., Griffeath, D.: Renormalizing the \(3\)-dimensional voter model. Ann. Probab. 7(3), 418–432 (1979)

    Article  MathSciNet  Google Scholar 

  3. Bramson, M., Griffeath, D.: Clustering and dispersion rates for some interacting particle systems on \({ Z}\). Ann. Probab. 8(2), 183–213 (1980)

    Article  MathSciNet  Google Scholar 

  4. Castellano, C., Fortunato, S., Loreto, V.: Statistical physics of social dynamics. Rev. Mod. Phys. 81(2), 591 (2009)

    Article  ADS  Google Scholar 

  5. Clifford, P., Sudbury, A.: A model for spatial conflict. Biometrika 60, 581–588 (1973)

    Article  MathSciNet  Google Scholar 

  6. Cox, J.T.: Coalescing random walks and voter model consensus times on the torus in \({ Z}^d\). Ann. Probab. 17(4), 1333–1366 (1989)

    Article  MathSciNet  Google Scholar 

  7. Cox, J.T., Griffeath, D.: Diffusive clustering in the two-dimensional voter model. Ann. Probab. 14(2), 347–370 (1986)

    Article  MathSciNet  Google Scholar 

  8. Deffuant, G., Neau, D., Amblard, F., Weisbuch, G.: Mixing beliefs among interacting agents. Adv. Complex Syst. 3(01n04), 87–98 (2000)

    Article  Google Scholar 

  9. Durrett, R., Steif, J.E.: Fixation results for threshold voter systems. Ann. Probab. 21(1), 232–247 (1993)

    MathSciNet  MATH  Google Scholar 

  10. Galam, S.: Minority opinion spreading in random geometry. Eur. Phys. J. B 25(4), 403–406 (2002)

    ADS  Google Scholar 

  11. Häggström, O.: A pairwise averaging procedure with application to consensus formation in the Deffuant model. Acta Appl. Math. 119, 185–201 (2012)

    Article  MathSciNet  Google Scholar 

  12. Hardin, M., Lanchier, N.: Probability of consensus in spatial opinion models with confidence threshold. ArXiv:1912.06746

  13. Hegselmann, R., Krause, U.: Opinion dynamics and bounded confidence models, analysis, and simulation. J. Artif. Soc. Soc. Simul. 5(3), 3–33 (2002)

    Google Scholar 

  14. Hirscher, T.: The Deffuant model on \(\mathbb{Z}\) with higher-dimensional opinion spaces. ALEA Lat. Am. J. Probab. Math. Stat. 11(1), 409–444 (2014)

    MathSciNet  MATH  Google Scholar 

  15. Holley, R.A., Liggett, T.M.: Ergodic theorems for weakly interacting infinite systems and the voter model. Ann. Probab. 3(4), 643–663 (1975)

    Article  MathSciNet  Google Scholar 

  16. Lanchier, N.: The Axelrod model for the dissemination of culture revisited. Ann. Appl. Probab. 22(2), 860–880 (2012)

    Article  MathSciNet  Google Scholar 

  17. Lanchier, N.: The critical value of the Deffuant model equals one half. ALEA Lat. Am. J. Probab. Math. Stat. 9(2), 383–402 (2012)

    MathSciNet  MATH  Google Scholar 

  18. Lanchier, N., Li, H.-L.: Probability of consensus in the multivariate Deffuant model on finite connected graphs. Electron. Commun. Probab. 25, 12 (2020)

    Article  MathSciNet  Google Scholar 

  19. Lanchier, N., Neufer, J.: Stochastic dynamics on hypergraphs and the spatial majority rule model. J. Stat. Phys. 151(1–2), 21–45 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  20. Lanchier, N., Scarlatos, S.: Fixation in the one-dimensional Axelrod model. Ann. Appl. Probab. 23(6), 2538–2559 (2013)

    Article  MathSciNet  Google Scholar 

  21. Lanchier, N., Scarlatos, S.: Clustering and coexistence in the one-dimensional vectorial Deffuant model. ALEA Lat. Am. J. Probab. Math. Stat. 11(1), 541–564 (2014)

    MathSciNet  MATH  Google Scholar 

  22. Lanchier, N., Scarlatos, S.: Limiting behavior for a general class of voter models with confidence threshold. ALEA Lat. Am. J. Probab. Math. Stat. 14(1), 63–92 (2017)

    Article  MathSciNet  Google Scholar 

  23. Lanchier, N., Schweinsberg, J.: Consensus in the two-state Axelrod model. Stoch. Process. Appl. 122(11), 3701–3717 (2012)

    Article  MathSciNet  Google Scholar 

  24. Li, H.-L.: Mixed Hegselmann-Krause dynamics. ArXiv:2010.03050

  25. Li, H.-L.: Mixed Hegselmann-Krause dynamics – nondeterministic case. ArXiv:2105.00577

  26. Li, J.: Axelrod’s model in two dimensions. ProQuest LLC, Ann Arbor, MI. Thesis (Ph.D.)–Duke University (2014)

  27. Liggett, T.M.: Interacting Particle Systems. Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 276. Springer-Verlag, New York (1985)

    Google Scholar 

  28. Liggett, T.M.: Coexistence in threshold voter models. Ann. Probab. 22(2), 764–802 (1994)

    Article  MathSciNet  Google Scholar 

  29. Liggett, T.M.: Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Grundlehren der Mathematischen Wissenschaften, vol. 324. Springer-Verlag, Berlin (1999)

    Book  Google Scholar 

  30. Vazquez, F., Krapivsky, P.L., Redner, S.: Constrained opinion dynamics: freezing and slow evolution. J. Phys. A 36(3), L61–L68 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  31. Zähle, I.: Renormalization of the voter model in equilibrium. Ann. Probab. 29(3), 1262–1302 (2001)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank three anonymous referees for their comments that help improve the presentation and the clarity of this work.

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  1. School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ, 85287, USA

    Nicolas Lanchier & Hsin-Lun Li

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  1. Nicolas Lanchier
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  2. Hsin-Lun Li
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Correspondence to Nicolas Lanchier.

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Communicated by Pierpaolo Vivo.

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Nicolas Lanchier was partially supported by NSF Grant CNS-2000792.

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Lanchier, N., Li, HL. Consensus in the Hegselmann–Krause Model. J Stat Phys 187, 20 (2022). https://doi.org/10.1007/s10955-022-02920-8

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