Convergence, Consensus and Dissensus in the Weighted-Median Opinion Dynamics

Mechanistic and tractable mathematical models play a key role in understanding how social influence shapes public opinions. Recently, a weighted-median mechanism has been proposed as a new micro-foundation of opinion dynamics and validated via experimental data. Numerical studies indicate that this new mechanism recreates some non-trivial real-world features of opinion evolution. In this paper, we conduct a thorough analysis of the weighted-median opinion dynamics. We fully characterize the equilibria set, and establish the almost-sure convergence for any initial condition. Moreover, we prove a necessary and sufficient condition for the almost-sure convergence to consensus, as well as a sufficient condition for almost-sure dissensus. We related the rich dynamical bevaior of the weighted-median opinion dynamics to two delicate network structures: the cohesive sets and the decisive links. To complement our sufficient conditions for almost-sure dissensus, we further prove that, given the influence network, determining whether the system almost surely achieves persistent dissensus is NP-hard, which reflects the complexity the network topology contributes to opinion evolution.