The Continuous-Time Weighted-Median Opinion Dynamics
<|reference_start|>The Continuous-Time Weighted-Median Opinion Dynamics: Opinion dynamics models are important in understanding and predicting opinion formation processes within social groups. Although the weighted-averaging opinion-update mechanism is widely adopted as the micro-foundation of opinion dynamics, it bears a non-negligibly unrealistic implication: opinion attractiveness increases with opinion distance. Recently, the weighted-median mechanism has been proposed as a new microscopic mechanism of opinion exchange. Numerous advancements have been achieved regarding this new micro-foundation, from theoretical analysis to empirical validation, in a discrete-time asynchronous setup. However, the original discrete-time weighted-median model does not allow for "compromise behavior" in opinion exchanges, i.e., no intermediate opinions are created between disagreeing agents. To resolve this problem, this paper propose a novel continuous-time weighted-median opinion dynamics model, in which agents' opinions move towards the weighted-medians of their out-neighbors' opinions. It turns out that the proof methods for the original discrete-time asynchronous model are no longer applicable to the analysis of the continuous-time model. In this paper, we first establish the existence and uniqueness of the solution to the continuous-time weighted-median opinion dynamics by showing that the weighted-median mapping is contractive on any graph. We also characterize the set of all the equilibria. Then, by leveraging a new LaSalle invariance principle argument, we prove the convergence of the continuous-time weighted-median model for any initial condition and derive a necessary and sufficient condition for the convergence to consensus.<|reference_end|>
@article{han2024the,
    title={The Continuous-Time Weighted-Median Opinion Dynamics},
    author={Yi Han, Ge Chen, Florian D"orfler, Wenjun Mei},
    journal={arXiv preprint arXiv:2404.16318},
    year={2024},
    archivePrefix={arXiv},
    eprint={2404.16318},
    primaryClass={eess.SY cs.SY}
}