# Feedback Loops in Opinion Dynamics of Agent-Based Models with Multiplicative Noise

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model Description

- $\tilde{U}:{({R}^{d})}^{N}\times {\mathbb{R}}^{N}\to {({\mathbb{R}}^{d})}^{N}$ is a spatial interaction map that models how the positions and opinions of the agents influence the spatial movement of the agents,
- $\tilde{V}:{({R}^{d})}^{N}\times {\mathbb{R}}^{N}\to {({\mathbb{R}}^{d})}^{N}$ is an opinion interaction map that models how the positions and opinions of the agents influence the opinion states of the agents,
- ${B}^{sp}$ and ${B}^{op}$ are independent Brownian motions starting in 0,
- ${\sigma}_{sp}({X}_{t},{\Theta}_{t}),{\sigma}_{op}({X}_{t},{\Theta}_{t})$ 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

**Assumption**

**1.**

#### 3.1. Motivation for the Limiting Equations

**Remark**

**1.**

#### 3.2. Well-Posedness Result of the Coupled Mean-Field SDE

**Remark**

**2.**

**Lemma**

**1.**

**Lemma**

**2.**

**Theorem**

**1.**

#### 3.3. Convergence of the Microscopic Model to the Mean-Field Equation

**Theorem**

**2.**

## 4. Characterization of the Empirical Measure and Its Limit

#### 4.1. Derivation of the PDE for the Law of the Coupled Mean-Field SDEs

**Remark**

**3.**

#### 4.2. SPDE Description for the Empirical Measure ${\mu}^{N}$

**Remark**

**4.**

#### 4.3. Numerical Experiment

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1

**Proof of Lemma 1.**

**Proof of Lemma 2.**

**Proof of Theorem 1.**

#### Appendix A.2

**Proof of Theorem 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.; GonzÃ¡lez-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]

**Figure 1.**Snapshots from numerical simulations at final time $T=2.5$ for different influences of additive noise: (

**a**) $\sigma =0.01$, (

**b**) $\sigma =0.05$, and (

**c**) $\sigma =0.15$. Positions of agents indicate their positions in a social space. Colour of agents denotes their opinions according to the colour-bar. Other parameters are fixed to $R=0.015$ and $\alpha =\beta =20$.

**Figure 2.**Opinion trajectories of agents over time-period $[0,2.5]$ for different influences of additive noise: (

**a**) $\sigma =0.01$, (

**b**) $\sigma =0.05$, and (

**c**) $\sigma =0.15$. Other parameters are fixed to $R=0.015$ and $\alpha =\beta =20$.

**Figure 3.**Distribution of agents’ opinions at final time $T=2.5$ for different influences of additive noise: (

**a**) $\sigma =0.01$, (

**b**) $\sigma =0.05$, and (

**c**) $\sigma =0.15$. Other parameters are fixed to $R=0.15$ and $\alpha =\beta =20$.

**Figure 4.**Snapshots from numerical simulations at final time $T=2.5$ for different influences of opinion and spatial strength: (

**a**) $\alpha =\beta =50$, (

**b**) $\alpha =\beta =5$, (

**c**) $\alpha =50,\beta =5$. Position of agents indicate their position in a social space. Colour of agents denotes their opinions according to the colour-bar. Other parameters are fixed to $R=0.15$ and $\sigma =0.05$.

**Figure 5.**Distribution of agents’ opinions at final time $T=2.5$ for different influences of opinion and spatial strength: (

**a**) $\alpha =\beta =50$, (

**b**) $\alpha =\beta =5$, (

**c**) $\alpha =50,\beta =5$. Positions of agents indicate their positions in a social space. Colours of agents denote their opinions according to the colour-bar. Other parameters are fixed to $R=0.15$ and $\sigma =0.05$.

**Figure 6.**Results of one simulation of the ABM with multiplicative noise, for $\alpha =\beta =20$ and $R=0.15$. (

**Left**): Snapshot of the dynamics at $T=2.5$. (

**Middle**): Opinion trajectories during $[0,2.5]$. (

**Right**): Distribution of agents’ opinions at $T=2.5$. Positions of agents indicate their positions in a social space. Colours of agents denote their opinions according to the colour-bar. Other parameters are fixed to $R=0.015$ and $\sigma =0.05$.

**Figure 7.**Empirical density of agents in mean-field limit given by Equation (20) at initial time $t=0$, intermediate time $t=0.5$, and final time $t=1$.

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

**MDPI and ACS Style**

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

**AMA Style**

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 Style**

Djurdjevac 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