Multidimensional Opinion Dynamics with Heterogeneous Bounded Confidences and Random Interactions

TL;DR

This paper proposes a randomized multidimensional bounded confidence model for opinion dynamics, demonstrating finite-time convergence to fixed opinions and conditions for consensus, addressing limitations of traditional all-to-all interaction models.

Contribution

It introduces a heterogeneous, randomized BC model that converges in finite time and provides conditions for consensus, improving realism over standard models.

Findings

Agents' opinions converge to fixed points almost surely.

Convergence rate follows a negative exponential distribution.

Finite-time consensus is achievable under certain conditions.

Abstract

This paper introduces a heterogeneous multidimensional bounded confidence (BC) opinion dynamics with random pairwise interactions, whereby each pair of agents accesses each other's opinions with a specific probability. This revised model is motivated by the observation that the standard Hegselmann-Krause (HK) dynamics requires unrealistic all-to-all interactions at certain configurations. For this randomized BC opinion dynamics, regardless of initial opinions and positive confidence bounds, we show that the agents' states converge to fixed final opinions in finite time almost surely and that the convergence rate follows a negative exponential distribution in mean square. Furthermore, we establish sufficient conditions for the heterogeneous BC opinion dynamics with random interactions to achieve consensus in finite time.