Feedback Loops in Opinion Dynamics of Agent-Based Models with Multiplicative Noise
Abstract
:1. Introduction
2. Model Description
- is a spatial interaction map that models how the positions and opinions of the agents influence the spatial movement of the agents,
- is an opinion interaction map that models how the positions and opinions of the agents influence the opinion states of the agents,
- and are independent Brownian motions starting in 0,
- are diffusion coefficients for spatial and opinion dynamics, respectively.
2.1. Pairwise Interactions
2.2. Multi-Body Interactions
2.3. Stochastic Influence: Multiplicative Noise
2.4. Numerical Simulations of the ABM
3. Theoretical Analysis: Coupled Mean-Field Limit
3.1. Motivation for the Limiting Equations
3.2. Well-Posedness Result of the Coupled Mean-Field SDE
3.3. Convergence of the Microscopic Model to the Mean-Field Equation
4. Characterization of the Empirical Measure and Its Limit
4.1. Derivation of the PDE for the Law of the Coupled Mean-Field SDEs
4.2. SPDE Description for the Empirical Measure
4.3. Numerical Experiment
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
Appendix A.1
Appendix A.2
References
- Lewandowsky, S.; Smillie, L.; Garcia, D.; Hertwig, R.; Weatherall, J.; Egidy, S.; Robertson, R.E.; O’Connor, C.; Kozyreva, A.; Lorenz-Spreen, P.; et al. Technology and Democracy: Understanding the Influence of Online Technologies on Political Behaviour and Decision-Making; Technical Report; Publications Office of the European Union: Luxembourg, 2020. [Google Scholar]
- Porten-Cheé, P.; Eilders, C. The effects of likes on public opinion perception and personal opinion. Communications 2020, 45, 223–239. [Google Scholar] [CrossRef]
- Peralta, A.F.; Kertész, J.; Iñiguez, G. Opinion dynamics in social networks: From models to data. arXiv 2022, arXiv:2201.01322. [Google Scholar]
- Sîrbu, A.; Loreto, V.; Servedio, V.D.P.; Tria, F. Opinion Dynamics: Models, Extensions and External Effects. In Participatory Sensing, Opinions and Collective Awareness; Loreto, V., Haklay, M., Hotho, A., Servedio, V.D., Stumme, G., Theunis, J., Tria, F., Eds.; Springer International Publishing: Cham, Switzerland, 2017; pp. 363–401. [Google Scholar] [CrossRef]
- Holley, R.A.; Liggett, T.M. Ergodic theorems for weakly interacting infinite systems and the voter model. Ann. Probab. 1975, 3, 643–663. [Google Scholar] [CrossRef]
- Schweitzer, F.; Hołyst, J.A. Modelling collective opinion formation by means of active Brownian particles. Eur. Phys. J. B Condens. Matter Complex Syst. 2000, 15, 723–732. [Google Scholar] [CrossRef]
- Starnini, M.; Frasca, M.; Baronchelli, A. Emergence of metapopulations and echo chambers in mobile agents. Sci. Rep. 2016, 6, 31834. [Google Scholar] [CrossRef]
- Kan, U.; Feng, M.; Porter, M.A. An Adaptive Bounded-Confidence Model of Opinion Dynamics on Networks. arXiv 2021, arXiv:2112.05856. [Google Scholar]
- Stauffer, D. Opinion Dynamics and Sociophysics. In Encyclopedia of Complexity and Systems Science; Meyers, R.A., Ed.; Springer: New York, NY, USA, 2009; pp. 6380–6388. [Google Scholar] [CrossRef]
- Clifford, P.; Sudbury, A. A model for spatial conflict. Biometrika 1973, 60, 581–588. [Google Scholar] [CrossRef]
- Degroot, M.H. Reaching a Consensus. J. Am. Stat. Assoc. 1974, 69, 118–121. [Google Scholar] [CrossRef]
- Hegselmann, R.; Krause, U. Opinion dynamics and bounded confidence: Models, analysis and simulation. J. Artif. Soc. Soc. Simul. 2002, 5, 1–33. [Google Scholar]
- Schweitzer, F.; Farmer, J.D. Brownian Agents and Active Particles: Collective Dynamics in the Natural and Social Sciences; Springer: Berlin/Heidelberg, Germany, 2003; Volume 1. [Google Scholar]
- Pineda, M.; Toral, R.; Hernández-García, E. The noisy Hegselmann-Krause model for opinion dynamics. Eur. Phys. J. B 2013, 86, 1–10. [Google Scholar] [CrossRef]
- Goddard, B.D.; Gooding, B.; Short, H.; Pavliotis, G. Noisy bounded confidence models for opinion dynamics: The effect of boundary conditions on phase transitions. IMA J. Appl. Math. 2022, 87, 80–110. [Google Scholar] [CrossRef]
- Wang, C.; Li, Q.; Weinan, E.; Chazelle, B. Noisy Hegselmann-Krause systems: Phase transition and the 2R-conjecture. J. Stat. Phys. 2017, 166, 1209–1225. [Google Scholar] [CrossRef]
- Gomes, S.N.; Pavliotis, G.A.; Vaes, U. Mean field limits for interacting diffusions with colored noise: Phase transitions and spectral numerical methods. Multiscale Model. Simul. 2020, 18, 1343–1370. [Google Scholar] [CrossRef]
- Crokidakis, N.; Anteneodo, C. Role of conviction in nonequilibrium models of opinion formation. Phys. Rev. E 2012, 86, 061127. [Google Scholar] [CrossRef]
- Mavrodiev, P.; Schweitzer, F. The ambigous role of social influence on the wisdom of crowds: An analytic approach. Phys. A Stat. Mech. Its Appl. 2021, 567, 125624. [Google Scholar] [CrossRef]
- Milli, L. Opinion Dynamic Modeling of News Perception. Appl. Netw. Sci. 2021, 6, 1–19. [Google Scholar] [CrossRef]
- Hegselmann, R.; Krause, U. Opinion dynamics under the influence of radical groups, charismatic leaders, and other constant signals: A simple unifying model. Netw. Heterog. Media 2015, 10, 477. [Google Scholar] [CrossRef]
- Yu, Y.; Xiao, G.; Li, G.; Tay, W.P.; Teoh, H.F. Opinion diversity and community formation in adaptive networks. Chaos Interdiscip. J. Nonlinear Sci. 2017, 27, 103115. [Google Scholar] [CrossRef]
- Buscarino, A.; Fortuna, L.; Frasca, M.; Rizzo, A. Local and global epidemic outbreaks in populations moving in inhomogeneous environments. Phys. Rev. E 2014, 90, 042813. [Google Scholar] [CrossRef]
- Centola, D.; González-Avella, J.C.; EguÃluz, V.M.; Miguel, M.S. Homophily, Cultural Drift, and the Co-Evolution of Cultural Groups. J. Confl. Resolut. 2007, 51, 905–929. [Google Scholar] [CrossRef]
- Vazquez, F.; González-Avella, J.C.; Eguíluz, V.M.; San Miguel, M. Time-scale competition leading to fragmentation and recombination transitions in the coevolution of network and states. Phys. Rev. E 2007, 76, 046120. [Google Scholar] [CrossRef] [Green Version]
- Levis, D.; Diaz-Guilera, A.; Pagonabarraga, I.; Starnini, M. Flocking-enhanced social contagion. Phys. Rev. Res. 2020, 2, 032056. [Google Scholar] [CrossRef]
- Sznitman, A.S. Topics in propagation of chaos. In Proceedings of the Ecole d’Eté de Probabilités de Saint-Flour XIX—1989; Burkholder, D.L., Pardoux, E., Sznitman, A.S., Hennequin, P.L., Eds.; Springer: Berlin/Heidelberg, Germany, 1991; pp. 165–251. [Google Scholar]
- Gärtner, J. On the McKean-Vlasov limit for interacting diffusions. Math. Nachrichten 1988, 137, 197–248. [Google Scholar] [CrossRef]
- Krylov, N.V.; Röckner, M. Strong solutions of stochastic equations with singular time dependent drift. Probab. Theory Relat. Fields 2005, 131, 154–196. [Google Scholar] [CrossRef]
- Dean, D.S. Langevin equation for the density of a system of interacting Langevin processes. J. Phys. A Math. Gen. 1996, 29, L613–L617. [Google Scholar] [CrossRef]
- Helfmann, L.; Conrad, N.D.; Djurdjevac, A.; Winkelmann, S.; Schütte, C. From interacting agents to density-based modeling with stochastic PDEs. Commun. Appl. Math. Comput. Sci. 2021, 16, 1–32. [Google Scholar] [CrossRef]
- Weisbuch, G.; Deffuant, G.; Amblard, F.; Nadal, J.P. Interacting Agents and Continuous Opinions Dynamics. In Heterogenous Agents, Interactions and Economic Performance; Lecture Notes in Economics and Mathematical Systems; Beckmann, M., Künzi, H.P., Fandel, G., Trockel, W., Aliprantis, C.D., Basile, A., Drexl, A., Feichtinger, G., Güth, W., Inderfurth, K., et al., Eds.; Springer: Berlin/Heidelberg, Germany, 2003; Volume 521, pp. 225–242. [Google Scholar] [CrossRef]
- Friedkin, N.; Johnsen, E. Social Influence Networks and Opinion Change. Adv. Group Process. 1999, 16, 1–29. [Google Scholar]
- Battiston, F.; Cencetti, G.; Iacopini, I.; Latora, V.; Lucas, M.; Patania, A.; Young, J.G.; Petri, G. Networks beyond pairwise interactions: Structure and dynamics. Phys. Rep. 2020, 874, 1–92. [Google Scholar] [CrossRef]
- Battiston, F.; Amico, E.; Barrat, A.; Bianconi, G.; Ferraz de Arruda, G.; Franceschiello, B.; Iacopini, I.; Kéfi, S.; Latora, V.; Moreno, Y.; et al. The physics of higher-order interactions in complex systems. Nat. Phys. 2021, 17, 1093–1098. [Google Scholar] [CrossRef]
- Neuhäuser, L.; Schaub, M.T.; Mellor, A.; Lambiotte, R. Opinion Dynamics with Multi-body Interactions. In Network Games, Control and Optimization. NETGCOOP 2021; Lasaulce, S., Mertikopoulos, P., Orda, A., Eds.; Springer International Publishing: Cham, Switzerland, 2021; pp. 261–271. [Google Scholar]
- Pineda, M.; Toral, R.; Hernández-García, E. Noisy continuous-opinion dynamics. J. Stat. Mech. Theory Exp. 2009, 2009, P08001. [Google Scholar] [CrossRef]
- Mäs, M.; Flache, A.; Helbing, D. Individualization as Driving Force of Clustering Phenomena in Humans. PLoS Comput. Biol. 2010, 6, e1000959. [Google Scholar] [CrossRef] [Green Version]
- Preisler, H.K.; Ager, A.A.; Johnson, B.K.; Kie, J.G. Modeling animal movements using stochastic differential equations. Environmetrics 2004, 15, 643–657. [Google Scholar] [CrossRef]
- Sun, Y.; Lin, W. A positive role of multiplicative noise on the emergence of flocking in a stochastic Cucker-Smale system. Chaos Interdiscip. J. Nonlinear Sci. 2015, 25, 083118. [Google Scholar] [CrossRef]
- Kloeden, P.E.; Platen, E. Higher-order implicit strong numerical schemes for stochastic differential equations. J. Stat. Phys. 1992, 66, 283–314. [Google Scholar] [CrossRef]
- Higham, D.J. An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev. 2001, 43, 525–546. [Google Scholar] [CrossRef]
- Stern, S.; Livan, G. The impact of noise and topology on opinion dynamics in social networks. R. Soc. Open Sci. 2021, 8, 201943. [Google Scholar] [CrossRef]
- Zhang, X. Stochastic differential equations with Sobolev diffusion and singular drift and applications. Ann. Appl. Probab. 2016, 26, 2697–2732. [Google Scholar] [CrossRef]
- Hao, Z.; Röckner, M.; Zhang, X. Strong convergence of propagation of chaos for McKean-Vlasov SDEs with singular interactions. arXiv 2022, arXiv:2204.07952. [Google Scholar]
- Jabin, P.E.; Wang, Z. Mean field limit and propagation of chaos for Vlasov systems with bounded forces. J. Funct. Anal. 2016, 271, 3588–3627. [Google Scholar] [CrossRef]
- Dos Reis, G.; Engelhardt, S.; Smith, G. Simulation of McKean–Vlasov SDEs with super-linear growth. IMA J. Numer. Anal. 2022, 42, 874–922. [Google Scholar] [CrossRef]
- Hammersley, W.R.; Šiška, D.; Szpruch, Ł. McKean–Vlasov SDEs under measure dependent Lyapunov conditions. In Annales de l’Institut Henri Poincaré, Probabilités et Statistiques; Institut Henri Poincaré: Paris, France, 2021; Volume 57, pp. 1032–1057. [Google Scholar]
- Bresch, D.; Jabin, P.E.; Wang, Z. Mean-field limit and quantitative estimates with singular attractive kernels. arXiv 2020, arXiv:2011.08022. [Google Scholar]
- Lacker, D. On a strong form of propagation of chaos for McKean-Vlasov equations. Electron. Commun. Probab. 2018, 23, 1–11. [Google Scholar] [CrossRef]
- Dudley, R.M. Real Analysis and Probability, 2nd ed.; Cambridge University Press: Cambridge, UK, 2002. [Google Scholar] [CrossRef]
- Kawasaki, K. Stochastic model of slow dynamics in supercooled liquids and dense colloidal suspensions. Phys. A Stat. Mech. Its Appl. 1994, 208, 35–64. [Google Scholar] [CrossRef]
- Villani, C. Optimal Transport; Vol. 338, Grundlehren der Mathematischen Wissenschaften; Springer: Berlin/Heidelberg, Germany, 2009. [Google Scholar] [CrossRef] [Green Version]
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |
© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Djurdjevac Conrad, N.; Köppl, J.; Djurdjevac, A. Feedback Loops in Opinion Dynamics of Agent-Based Models with Multiplicative Noise. Entropy 2022, 24, 1352. https://doi.org/10.3390/e24101352
Djurdjevac Conrad N, Köppl J, Djurdjevac A. Feedback Loops in Opinion Dynamics of Agent-Based Models with Multiplicative Noise. Entropy. 2022; 24(10):1352. https://doi.org/10.3390/e24101352
Chicago/Turabian StyleDjurdjevac Conrad, Nataša, Jonas Köppl, and Ana Djurdjevac. 2022. "Feedback Loops in Opinion Dynamics of Agent-Based Models with Multiplicative Noise" Entropy 24, no. 10: 1352. https://doi.org/10.3390/e24101352