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K-Averaging Agent-Based Model: Propagation of Chaos and Convergence to Equilibrium

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Abstract

The paper treats an agent-based model with averaging dynamics to which we refer as the K-averaging model. Broadly speaking, our model can be added to the growing list of dynamics exhibiting self-organization such as the well-known Vicsek-type models (Aldana et al. in: Phys Rev Lett 98(9):095702, 2007; Aldana and Huepe in: J Stat Phys 112(1–2):135–153, 2003; Pimentel in: Phys. Rev. E 77(6):061138, 2008). In the K-averaging model, each of the N particles updates their position by averaging over K randomly selected particles with additional noise. To make the K-averaging dynamics more tractable, we first establish a propagation of chaos type result in the limit of infinite particle number (i.e. \(N \rightarrow \infty \)) using a martingale technique. Then, we prove the convergence of the limit equation toward a suitable Gaussian distribution in the sense of Wasserstein distance as well as relative entropy. We provide additional numerical simulations to illustrate both results.

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References

  1. Aldana, M., Dossetti, V., Huepe, C., Kenkre, V.M., Larralde, H.: Phase transitions in systems of self-propelled agents and related network models. Phys. Rev. Lett. 98(9), 095702 (2007)

  2. Aldana, M., Huepe, C.: Phase transitions in self-driven many-particle systems and related non-equilibrium models: a network approach. J. Stat. Phys. 112(1–2), 135–153 (2003)

    Article  Google Scholar 

  3. Artstein, S., Ball, K., Barthe, F., Naor, A.: Solution of Shannon’s problem on the monotonicity of entropy. J. Am. Math. Soc. 17(4), 975–982 (2004)

    Article  MathSciNet  Google Scholar 

  4. Bakry, D., Gentil, I., Ledoux, M.: Analysis and Geometry of Markov Diffusion Operators, vol. 348. Springer, New York (2013)

    MATH  Google Scholar 

  5. Barbaro, A.B.T., Degond, P.: Phase transition and diffusion among socially interacting self-propelled agents. arXiv:1207.1926 (2012)

  6. Baumann, F., Lorenz-Spreen, P., Sokolov, I.M., Starnini, M.: Modeling echo chambers and polarization dynamics in social networks. Phys. Rev. Lett. 124(4), 048301 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  7. Belmonte, J.M., Thomas, G.L., Brunnet, L.G., de Almeida, R.M.C., Chaté, H.: Self-propelled particle model for cell-sorting phenomena. Phys. Rev. Lett. 100(24), 248702 (2008)

    Article  ADS  Google Scholar 

  8. Berti, P., Pratelli, L., Rigo, P.: Almost sure weak convergence of random probability measures. Stoch. Stoch. Rep 78(2), 91–97 (2006)

    Article  MathSciNet  Google Scholar 

  9. Bertin, E., Droz, M., Grégoire, G.: Boltzmann and hydrodynamic description for self-propelled particles. Phys. Rev. E 74(2), 022101 (2006)

    Article  ADS  Google Scholar 

  10. Bertin, E., Droz, M., Grégoire, G.: Hydrodynamic equations for self-propelled particles: microscopic derivation and stability analysis. J. Phys. A 42(44), 445001 (2009)

    Article  Google Scholar 

  11. Billingsley, P.: Convergence of Probability Measures. Wiley, New York (2013)

    MATH  Google Scholar 

  12. Boissard, E., Degond, P., Motsch, S.: Trail formation based on directed pheromone deposition. J. Math. Biol. 66(6), 1267–1301 (2013)

    Article  MathSciNet  Google Scholar 

  13. Carlen, E., Chatelin, R., Degond, P., Wennberg, B.: Kinetic hierarchy and propagation of chaos in biological swarm models. Physica D 260, 90–111 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  14. Chaté, H., Ginelli, F., Grégoire, G., Peruani, F., Raynaud, F.: Modeling collective motion: variations on the Vicsek model. Eur. Phys. J. B 64(3), 451–456 (2008)

    Article  ADS  Google Scholar 

  15. Chuang, Y.-L., Dorsogna, M.R., Marthaler, D., Bertozzi, A.L., Chayes, L.S.: State transitions and the continuum limit for a 2D interacting, self-propelled particle system. Physica D 232(1), 33–47 (2007)

  16. Cover, T.M.: Elements of Information Theory. Wiley, New York (1999)

    Google Scholar 

  17. Dai Pra, P.: Stochastic mean-field dynamics and applications to life sciences. In: International Workshop on Stochastic Dynamics Out of Equilibrium, pp. 3–27. Springer, New York (2017)

  18. Degond, P., Frouvelle, A., Raoul, G.: Local stability of perfect alignment for a spatially homogeneous kinetic model. J. Stat. Phys. 157(1), 84–112 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  19. Hardin, M., Lanchier, N.: Probability of consensus in spatial opinion models with confidence threshold. arXiv:1912.06746 (2019)

  20. Hauray, M., Jabin, P.-E.: N-particles approximation of the Vlasov equations with singular potential. Arch. Ratl. Mech. Anal. 183(3), 489–524 (2007)

    Article  MathSciNet  Google Scholar 

  21. Jabin, P.-E., Wang, Z.: Quantitative estimates of propagation of chaos for stochastic systems with \(W^{-1,\infty }\) kernels. Invent. Math. 214(1), 523–591 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  22. Lanchier, N., Li, H.-L.: Probability of consensus in the multivariate Deffuant model on finite connected graphs. Electr. Commun. Probab. 25, 1–12 (2020)

    MathSciNet  Google Scholar 

  23. Liggett, T.M.: Interacting Particle Systems, vol. 276. Springer, New York (2012)

    MATH  Google Scholar 

  24. Madiman, M., Barron, A.: Generalized entropy power inequalities and monotonicity properties of information. IEEE Trans. Inform. Theory 53(7), 2317–2329 (2007)

    Article  MathSciNet  Google Scholar 

  25. Merle, M., Salez, J.: Cutoff for the mean-field zero-range process. Ann. Probab. 47(5), 3170–3201 (2019)

    Article  MathSciNet  Google Scholar 

  26. Méléard, S., Roelly-Coppoletta, S.: A propagation of chaos result for a system of particles with moderate interaction. Stoch. Process. Appl. 26, 317–332 (1987)

    Article  MathSciNet  Google Scholar 

  27. Motsch, S., Tadmor, E.: Heterophilious dynamics enhances consensus. SIAM Rev. 56(4), 577–621 (2014)

    Article  MathSciNet  Google Scholar 

  28. Naldi, G., Pareschi, L., Toscani, G.: Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences. Springer, New York (2010)

    Book  Google Scholar 

  29. Oelschlager, K.: A martingale approach to the law of large numbers for weakly interacting stochastic processes. Ann. Probab. 12, 458–479 (1984)

    Article  MathSciNet  Google Scholar 

  30. Pimentel, J.A., Aldana, M., Huepe, C., Larralde, H.: Intrinsic and extrinsic noise effects on phase transitions of network models with applications to swarming systems. Phys. Rev. E 77(6), 061138 (2008)

    Article  ADS  Google Scholar 

  31. Popoviciu, T.: Sur les équations algébriques ayant toutes leurs racines réelles. Mathematica 9, 129–145 (1935)

    MATH  Google Scholar 

  32. Porfiri, M.: Linear analysis of the vectorial network model. IEEE Trans. Circ. Syst. II 61(1), 44–48 (2013)

    Google Scholar 

  33. Porfiri, M., Ariel, G.: On effective temperature in network models of collective behavior. Chaos 26(4), 043109 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  34. Rezakhanlou, F., Villani, C., Golse, F.: Entropy methods for the Boltzmann equation: lectures from a special semester at the Centre Émile Borel, Institut H. Poincaré, Paris, 2001, No. 1916. Springer, New York (2008)

  35. Sznitman, A.-S.: Topics in propagation of chaos. In: Ecole d’été de Probabilités de Saint-Flour XIX–1989, pp. 165–251. Springer, Berlin (1991)

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Acknowledgements

It is a pleasure to thank my Ph.D. advisor Sébastien Motsch for his tremendous help on various portions of this manuscript, and I thank the anonymous reviewer for providing helpful comments on earlier drafts of the manuscript.

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  1. Arizona State University - School of Mathematical and Statistical Sciences, 900 S Palm Walk, Tempe, AZ, 85287-1804, USA

    Fei Cao

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  1. Fei Cao
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Correspondence to Fei Cao.

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Communicated by Irene Giardina.

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Cao, F. K-Averaging Agent-Based Model: Propagation of Chaos and Convergence to Equilibrium. J Stat Phys 184, 18 (2021). https://doi.org/10.1007/s10955-021-02807-0

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