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arXiv · 2404.16318

Modeling, Analysis, and Control of Continuous-Time Weighted-Median Opinion Dynamics

Yi Han·Julien M. Hendrickx·Ge Chen·Wenjun Mei
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Abstract

Simple yet predictive mathematical models are essential for mechanistic understanding of opinion evolution in social groups. The weighted-median mechanism has recently been proposed as a well-founded alternative to conventional DeGroot-type opinion dynamics. However, the original weighted-median model excludes compromise behavior, as individuals directly adopt their neighbors' opinions without forming intermediate values. In this paper, we introduce a parsimonious continuous-time extension of the weighted-median model by incorporating individual inertia, allowing opinions to move gradually toward the neighbors' weighted median. Empirical evidence shows that this model outperforms both the original weighted-median and DeGroot models with inertia in predicting opinion shifts. We provide a complete theoretical analysis of the proposed dynamics: the equilibria are characterized and shown to be Lyapunov stable; global convergence is established via the Bony-Brezis method, yielding necessary and sufficient conditions for consensus from arbitrary initial states. In addition, we derive a graph-theoretic condition for persistent disagreement and a necessary and sufficient condition for steering the system to any prescribed consensus value through constant external inputs to a subset of individuals. These results reveal how a social group's resilience to external manipulation fundamentally depends on its internal network structure.

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Yi Han, Julien M. Hendrickx, Ge Chen, Wenjun Mei. 2024-04-25. Modeling, Analysis, and Control of Continuous-Time Weighted-Median Opinion Dynamics. https://arxiv.org/abs/2404.16318

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