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Convergence Analysis of Weighted Median Opinion Dynamics with Higher-Order Effects

The weighted median mechanism provides a robust alternative to weighted averaging in opinion dynamics. Existing models, however, are predominantly formulated on pairwise interaction graphs, which limits their ability to represent higher-order environmental effects. In this work, a generalized weighted median opinion dynamics model is proposed by incorporating high-order interactions through a simplicial complex representation. The resulting dynamics are formulated as a nonlinear discrete-time system with synchronous opinion updates, in which intrinsic agent interactions and external environmental influences are jointly modeled. Sufficient conditions for asymptotic consensus are established for heterogeneous systems composed of opinionated and unopinionated agents. For homogeneous opinionated systems, convergence and convergence rates are rigorously analyzed using the Banach fixed-point theorem. Theoretical results demonstrate the stability of the proposed dynamics under mild conditions, and numerical simulations are provided to corroborate the analysis. This work extends median-based opinion dynamics to high-order interaction settings and provides a system-level framework for stability and consensus analysis.