Probability of consensus in the multivariate Deffuant model on finite connected graphs

Nicolas Lanchier,Hsin-Lun Li

doi:10.1214/20-ecp359

Abstract

The Deffuant model is a spatial stochastic model for the dynamics of opinions in which individuals are located on a connected graph representing a social network and characterized by a number in the unit interval representing their opinion. The system evolves according to the following averaging procedure: at each time step, two neighbors are randomly chosen and interact if and only if the distance between their opinions does not exceed a certain confidence threshold, with each interaction resulting in the neighbors’ opinions getting closer to each other. Most of the analytical results established so far about this model assume that the individuals are located on the integers. In contrast, we study the more realistic case where the social network can be any finite connected graph. In addition, we extend the opinion space to any bounded convex subset of a normed vector space where the norm is used to measure the level of disagreement or distance between the opinions. Our main result gives a lower bound for the probability of consensus. Our proof leads to a universal lower bound that depends on the confidence threshold, the opinion space (convex subset and norm) and the initial distribution, but not on the size or the topology of the social network.

Highlights

This paper is concerned with opinion dynamics on connected graphs
The Deffuant model is a spatial stochastic model for the dynamics of opinions in which individuals are located on a connected graph representing a social network and characterized by a number in the unit interval representing their opinion
Multivariate Deffuant model on finite connected graphs proved that the process on the infinite square lattice clusters in one and two dimensions whereas opinions coexist at equilibrium in higher dimensions [15]

Summary

Introduction

This paper is concerned with opinion dynamics on connected graphs. The first and most popular stochastic model in this topic is the voter model, introduced in [5, 15]. Other spatial stochastic models of opinion dynamics include homophily in the form of a confidence threshold: individuals interact with their neighbors on the graph if and only if the level of disagreement between the two individuals before the interaction does not exceed a certain threshold The simplest such model is the constrained voter model [27], the voter model with three opinions (leftist, centrist and rightist) where leftists and rightists do not interact. Because [8] is purely based on numerical simulations, the authors only considered specific graphs: the complete graph and the two-dimensional torus Their simulations on large graphs suggest the following conjecture for the infinite system obtained by assuming that pairs of neighbors are chosen in continuous time at rate one: the process exhibits a phase transition at the critical threshold one-half in that a consensus is reached when τ > 1/2 whereas disagreements persist in the long run when τ < 1/2. While our bound depends on the choice of the opinion space, it is uniform in all possible choices of the social network

Model description and main results

Limiting configurations

Stopping time and consensus event