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arXiv · 2212.08808

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

Wenjun Mei·Julien M. Hendrickx·Ge Chen·Francesco Bullo·Florian Dörfler
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Abstract

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 also indicate that this new mechanism recreates some non-trivial real-world features of opinion evolution. In this paper, we conduct a thorough theoretical analysis of the weighted-median opinion dynamics. We fully characterize the set of all equilibria, and we establish the almost-sure finite-time convergence for any initial condition. Moreover, we prove a necessary and sufficient graph-theoretic condition for the almost-sure convergence to consensus, as well as a sufficient graph-theoretic condition for almost-sure persistent dissensus. It turns out that the weighted-median opinion dynamics, despite its simplicity in form, exhibit rich dynamical behavior that depends on some delicate network structures. 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.

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Wenjun Mei, Julien M. Hendrickx, Ge Chen, Francesco Bullo, Florian Dörfler. 2022-12-17. Convergence, Consensus and Dissensus in the Weighted-Median Opinion Dynamics. https://arxiv.org/abs/2212.08808

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