Convergence, stability and robustness of multidimensional opinion dynamics in continuous time

TL;DR

This paper studies the stability and robustness of multidimensional opinion dynamics in continuous time, analyzing how equilibria behave under small perturbations and heterogeneous interactions.

Contribution

It introduces a comprehensive analysis of stability and robustness for multidimensional opinion models with heterogeneous interactions, extending previous one-dimensional results.

Findings

All equilibria in the interior are strongly Lyapunov stable.

Necessary and sufficient conditions for equilibrium robustness are provided for indicator interaction functions.

Robustness conditions generalize known one-dimensional results to multidimensional settings.

Abstract

We analyze a continuous time multidimensional opinion model where agents have heterogeneous but symmetric and compactly supported interaction functions. We consider Filippov solutions of the resulting dynamics and show strong Lyapunov stability of all equilibria in the relative interior of the set of equilibria. We investigate robustness of equilibria when a new agent with arbitrarily small weight is introduced to the system in equilibrium. Assuming the interaction functions to be indicators, we provide a necessary condition and a sufficient condition for robustness of the equilibria. Our necessary condition coincides with the necessary and sufficient condition obtained by Blondel et al. for one dimensional opinions.