Bounded Confidence Opinion Dynamics in Non-Euclidean Norms: Containment and Convergence with Stubborn Agents

Bounded confidence models of opinion formation are paradigmatic examples of coevolutionary networks, where agents update their opinions by averaging those of peers deemed sufficiently similar. In this paper, we study a multidimensional extension of the Hegselmann–Krause (HK) model with stubborn agents (MHK-S), in which the confidence sets are defined as balls in a general (not necessarily Euclidean) norm and some agents remain stubborn, being resistant to social influence. We establish a behavioral dichotomy: an agent’s opinion either terminates in finite time (if not influenced by stubborn agents) or converges asymptotically to the convex hull of the stubborn agents’ opinions. In the special case where the stubborn agents’ opinions are sufficiently close, the influenced agents converge to their barycenter; we further extend this result to scenarios featuring multiple clusters of stubborn opinions that are sufficiently distant from each other.