Consensus Stability in the Hegselmann-Krause Model with Coopetition and Cooperosity

Heterogeneous Hegselmann-Krause (HK) models have been used to represent opinion dynamics in social networks. In this framework, the concepts of coopetition and cooperosity have been recently introduced by the authors in order to characterize different connectivity thresholds for the agents. Inspired by this application, in this paper a sufficient condition for the asymptotic stability of the origin in piecewise linear systems is proved. The result is based on continuous Lyapunov functions which are piecewise differentiable in time. By considering a piecewise quadratic Lyapunov function, the stability result is applied for the consensus in heterogeneous HK models. Examples of heterogeneous HK models with different number of agents show the effectiveness of the proposed approach.