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

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.