A density description of a bounded-confidence model of opinion dynamics on hypergraphs

Abstract: Social interactions often occur between three or more agents simultaneously. Examining opinion dynamics on hypergraphs allows one to study the effect of such polyadic interactions on the opinions of agents. In this paper, we consider a bounded-confidence model (BCM), in which opinions take continuous values and interacting agents comprise their opinions if they are close enough to each other. We study a density description of a Deffuant--Weisbuch BCM on hypergraphs. We derive a rate equation for the mean-field opinion density as the number of agents becomes infinite, and we prove that this rate equation yields a probability density that converges to noninteracting opinion clusters. Using numerical simulations, we examine bifurcations of the density-based BCM's steady-state opinion clusters and demonstrate that the agent-based BCM converges to the density description of the BCM as the number of agents becomes infinite.

• The paper introduces a density-based bounded-confidence model that extends the Deffuant–Weisbuch framework to account for polyadic interactions on hypergraphs.
• The paper derives mean-field rate equations showing that any initial opinion density converges to isolated clusters represented by Dirac delta functions.
• The paper validates the model through numerical simulations and Monte Carlo methods, demonstrating its computational efficiency and agreement with agent-based results.

A Density Description of a Bounded-Confidence Model of Opinion Dynamics on Hypergraphs

Introduction

The study of opinion dynamics on social networks has advanced significantly with the exploration of models that consider interactions beyond pairwise relations, such as those occurring in hypergraphs. This paper investigates a bounded-confidence model (BCM) of opinion dynamics on hypergraphs, specifically using a density-based approach for a Deffuant–Weisbuch model. By leveraging the hypergraph structure, which accommodates interactions among three or more nodes, the research aims to offer new insights into the group dynamics influencing opinion formation.

Model Formulation

This study formulates a BCM on hypergraphs where opinions are treated as continuous values. It adapts the Deffuant–Weisbuch model, incorporating polyadic interactions by representing them using hyperedges that connect multiple agents. This approach necessitates a density description, achieved by deriving a rate equation for the mean-field opinion density as the number of agents grows to infinity. This enables an analysis where opinions converge to form distinct clusters.

Theoretical Analysis

The authors derive the key mathematical properties of the opinion density dynamics. In the mean-field limit, the rate equation describes how clusters of opinions evolve, stabilize, and potentially bifurcate. The paper includes a rigorous proof showing these mean-field solutions converge to non-interacting opinion clusters over time. Key results demonstrate that any initial density evolves into Dirac delta functions representing isolated opinion clusters, thus reinforcing the stability and long-term fragmentation of opinions on hypergraphs.

Numerical Simulations

The research includes numerical simulations to complement theoretical predictions. These simulations observe bifurcations in cluster formation as initial opinion distributions vary in their breadth (variance), with clusters becoming more numerous under larger initial variances. Such bifurcations indicate transitions between consensus, polarization, and fragmentation states. By employing different discordance functions and hypergraph structures, the study scrutinizes how the confidence bounds affect these transitions.

Comparison with Agent-Based Models

To validate the density-based BCM against agent-based simulations, the study conducts extensive Monte Carlo simulations. As the number of agents increases, the agent-based results approximate the density-based predictions, showcasing convergence in the mean-field limit. This demonstrates the model's applicability for large systems and highlights its computational efficiency over traditional simulation methods.

Conclusion

The density-based BCM on hypergraphs reveals critical insights into opinion formation and polarization. It highlights the role hypergraphs play in shaping complex opinion dynamics driven by polyadic interactions. Future exploration of heterogeneous hypergraph structures and more complex interaction rules could further enhance the understanding of collective decision-making processes across various social settings. This paper sets a foundation for bridging the gap between discrete agent-based models and continuous density descriptions in opinion dynamics research.