Mixed Hegselmann-Krause dynamics

The original Hegselmann-Krause (HK) model consists of a set of~$n$ agents that are characterized by their opinion, number in~$[0, 1]$. Each agent, say agent~$i$, updates its opinion~$x_i$ taking the average opinion all neighbors, whose differs from~$x_i$ at most~$\epsilon$. There two types of~HK models: synchronous~HK and asynchronous~HK model. For synchronous model, update simultaneously each time step, whereas for only one agent chosen uniformly random step. This paper is concerned with variant the~HK dynamics, called mixed~HK where can choose degree stubbornness mix neighbors update. be different and/or vary over time. An not stubborn or absolutely open-minded if new does change particular case where, In contrast, asynchronous corresponds to except who open-minded. We first show some common properties such as finite-time convergence, do hold mixed then investigate conditions under which asymptotic stability holds, consensus achieved