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Journal of Statistical Physics

, Volume 176, Issue 1, pp 40–68 | Cite as

Deterministic Versus Stochastic Consensus Dynamics on Graphs

  • Dylan WeberEmail author
  • Ryan Theisen
  • Sebastien Motsch
Article

Abstract

We study two agent based models of opinion formation—one stochastic in nature and one deterministic. Both models are defined in terms of an underlying graph; we study how the structure of the graph affects the long time behavior of the models in all possible cases of graph topology. We are especially interested in the emergence of a consensus among the agents and provide a condition on the graph that is necessary and sufficient for convergence to a consensus in both models. This investigation reveals several contrasts between the models—notably the convergence rates—which are explored through analytical arguments and several numerical experiments.

Keywords

Dynamical systems Agent based modeling Stochastic modeling Deterministic modeling Network dynamics Consensus dynamics 

Notes

Acknowledgements

This work has been supported by the NSF Grants DMS-1515592 and RNMS11-07444 (KI-Net).

References

  1. 1.
    Aldous, D.: Interacting particle systems as stochastic social dynamics. Bernoulli 19(4), 1122–1149 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aldous, D., Fill, J.: Reversible Markov Chains and Random Walks on Graphs. University of California, Berkeley (2002)Google Scholar
  3. 3.
    Aydoğdu, A., Caponigro, M., McQuade, S., Piccoli, B., Duteil, N.P., Rossi, F., Trélat, E.: Interaction Network, State Space, and Control in Social Dynamics. In: Bellomo, N., Degond, P., Tadmor, E. (eds.) Advances in Theory, Models, and Applications, Modeling and Simulation in Science, Engineering and Technology, pp. 99–140. Springer International Publishing, Cham (2017)Google Scholar
  4. 4.
    Ballerini, M., Cabibbo, N., Candelier, R., Cavagna, A., Cisbani, E., Giardina, I., Orlandi, A., Parisi, G., Procaccini, A., Viale, M., Zdravkovic, V.: Empirical investigation of starling flocks: a benchmark study in collective animal behaviour. Anim. Behav. 76(1), 201–215 (2008)CrossRefGoogle Scholar
  5. 5.
    Ben-Naim, E., Krapivsky, P.L., Vazquez, F., Redner, S.: Unity and discord in opinion dynamics. Phys. A 330(1), 99–106 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Blondel, V.D., Hendrickx, J.M., Tsitsiklis, J.N.: On Krause’s multi-agent consensus model with state-dependent connectivity. IEEE Trans. Autom. Control 54(11), 2586–2597 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Buhl, J., Sumpter, D.J.T., Couzin, I.D., Hale, J.J., Despland, E., Miller, E.R., Simpson, S.J.: From disorder to order in marching locusts. Science 312(5778), 1402–1406 (2006)ADSCrossRefGoogle Scholar
  8. 8.
    Carlen, E., Chatelin, R., Degond, P., Wennberg, B.: Kinetic hierarchy and propagation of chaos in biological swarm models. Phys. D 260, 90–111 (2013)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Carlen, E., Degond, P., Wennberg, B.: Kinetic limits for pair-interaction driven master equations and biological swarm models. Math. Models Methods Appl. Sci. 23(07), 1339–1376 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Carro, A., Toral, R., San Miguel, M.: The role of noise and initial conditions in the asymptotic solution of a bounded confidence, continuous-opinion model. J. Stat. Phys. 151, 131–149 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Castellano, C., Vilone, D., Vespignani, A.: Incomplete ordering of the voter model on small-world networks. Europhys. Lett. 63(1), 153 (2003)ADSCrossRefGoogle Scholar
  12. 12.
    Castellano, C., Fortunato, S., Loreto, V.: Statistical physics of social dynamics. Rev. Mod. Phys. 81(2), 591–646 (2009)ADSCrossRefGoogle Scholar
  13. 13.
    Deffuant, G.: Comparing extremism propagation patterns in continuous opinion models. J. Artif. Soc. Simul. 9, 1–8 (2006)Google Scholar
  14. 14.
    Deffuant, G., Neau, D., Amblard, F., Weisbuch, G.: Mixing beliefs among interacting agents. Adv. Complex Syst. 03(01n04), 87–98 (2000)CrossRefGoogle Scholar
  15. 15.
    Dornic, I., Chaté, H., Chave, J., Hinrichsen, H.: Critical coarsening without surface tension: the universality class of the voter model. Phys. Rev. Lett. 87(4), 045701 (2001)ADSCrossRefGoogle Scholar
  16. 16.
    Estrada, E., Vargas-Estrada, E., Ando, H.: Communicability angles reveal critical edges for network consensus dynamics. Phys. Rev. E 92(5), 052809 (2015)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    Fiedler, M.: Algebraic connectivity of graphs. Czechoslov. Math. J. 23(2), 298–305 (1973)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Grinstead, C.M., Snell, J.L.: Introduction to Probability. Chance Project. American Mathematical Society, Providence (2006)Google Scholar
  19. 19.
    Hegselmann, R.: Opinion dynamics and bounded confidence: models, analysis and simulation. J. Artif. Soc. Soc. Simul. 5(3), 1–4 (2002)Google Scholar
  20. 20.
    Herty, M., Ringhofer, C.: Large time behavior of averaged kinetic models on networks. Math. Models Methods Appl. Sci. 25(05), 875–904 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Jabin, P.-E., Motsch, S.: Clustering and asymptotic behavior in opinion formation. J. Diff. Equ. 257(11), 4165–4187 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Kempe, D., Kleinberg, J., Tardos, É.: Maximizing the spread of influence through a social network. In: Proceedings of the 9th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 137–146. ACM (2003)Google Scholar
  23. 23.
    Kolokolnikov, T.: Maximizing algebraic connectivity for certain families of graphs. Linear Algebr. Appl. 471, 122–140 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kolokolnikov, T., Osting, B., Von Brecht, J.: Algebraic connectivity of Erdös-Rényi graphs near the connectivity threshold. Manuscript in preparation (2014)Google Scholar
  25. 25.
    Lanchier, N.: Stochastic Modeling. Universitext. Springer International Publishing, New York (2017)CrossRefzbMATHGoogle Scholar
  26. 26.
    Lanchier, N., Neufer, J.: Stochastic dynamics on hypergraphs and the spatial majority rule model. J. Stat. Phys. 151(1–2), 21–45 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Liggett, T.M.: Interacting Particle Systems. Springer Science & Business Media, New York (2012)zbMATHGoogle Scholar
  28. 28.
    Lorenz, J.: Continuous opinion dynamics under bounded confidence: a survey. Int. J. Mod. Phys. C 18(12), 1819–1838 (2007)ADSCrossRefzbMATHGoogle Scholar
  29. 29.
    Motsch, S., Tadmor, E.: Heterophilious dynamics enhances consensus. SIAM Rev. 56(4), 577–621 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Olfati-Saber, R., Fax, J.A., Murray, R.M.: Consensus and cooperation in networked multi-agent systems. Proc. IEEE 95(1), 215–233 (2007)CrossRefzbMATHGoogle Scholar
  31. 31.
    Pineda, M., Toral, R., Hernández-García, E.: The noisy Hegselmann–Krause model for opinion dynamics. Eur. Phys. J. B 86(12), 490 (2013)ADSMathSciNetCrossRefGoogle Scholar
  32. 32.
    Privault, N.: Stochastic Finance: An Introduction with Market Examples. CRC Press, Boca Raton (2013)CrossRefzbMATHGoogle Scholar
  33. 33.
    Ren, W., Beard, R.W.: Consensus seeking in multiagent systems under dynamically changing interaction topologies. IEEE Trans. Autom. Control 50(5), 655–661 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Saber, RO, Richard M.: Consensus protocols for networks of dynamic agents. In: Proceedings of the 2003 American Control Conference, vol. 2, pp. 951–956. IEEE, Piscatway, NJ (2003)Google Scholar
  35. 35.
    Sood, V., Redner, S.: Voter model on heterogeneous graphs. Phys. Rev. Lett. 94(17), 178701 (2005)ADSCrossRefGoogle Scholar
  36. 36.
    Spanos, D.P., Olfati-Saber, R., Murray, Richard, M.: Dynamic consensus on mobile networks. In: IFAC world congress, pp. 1–6. Citeseer (2005)Google Scholar
  37. 37.
    von Brecht, J., Kolokolnikov, T., Bertozzi, A.L., Sun, H.: Swarming on random graphs. J. Stat. Phys. 151, 150–173 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Xia, H., Wang, H., Xuan, Z.: Opinion dynamics: a multidisciplinary review and perspective on future research. Int. J. Knowl. Syst. Sci. 2(4), 72–91 (2011)CrossRefGoogle Scholar
  39. 39.
    Yu, W., Chen, G., Cao, M., Kurths, J.: Second-order consensus for multiagent systems with directed topologies and nonlinear dynamics. IEEE Trans. Syst. Man Cybern. B 40(3), 881–891 (2010)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.School of Mathematics and Statistical SciencesArizona State UniversityTempeUSA
  2. 2.Department of StatisticsUniversity of California, BerkeleyBerkeleyUSA

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