Equilibria and consensus in discrete-time opinion dynamics with mixed weighted-averaging and weighted-median mechanisms
Abstract
Opinion dynamics models relate interpersonal influence to consensus or persistent disagreement. We study a discrete-time model combining weighted-average and weighted-median updates on a fixed directed network. Individuals update synchronously with possibly different adjustment coefficients. We characterize equilibria through local balance conditions and opinion level sets, and establish convergence to a Lyapunov-stable equilibrium using nonexpansiveness and Krasnosel'skii--Mann theory. We define type-dependent cohesive sets: each weighted-average member assigns all its weight to the set, whereas each weighted-median member assigns at least one half. Consensus is reached from every initial condition if and only if no two such sets are disjoint. Otherwise, an open set of initial conditions leads to persistent disagreement. Replacing weighted-median individuals by weighted-average individuals preserves consensus whenever it is already guaranteed for all initial conditions. Four numerical studies examine the number and locations of weighted-average individuals. The frequency of reaching a prescribed consensus tolerance is nondecreasing with their fraction in the sampled ensembles, but individual realizations need not follow this trend.
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BibTeX
@misc{cheng2026equilibria,
title = {{Equilibria and consensus in discrete-time opinion dynamics with mixed weighted-averaging and weighted-median mechanisms}},
author = {Junkai Cheng and Wenjun Mei},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16406},
eprint = {2610.16406},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16406},
}
RIS
TY - UNPB
TI - Equilibria and consensus in discrete-time opinion dynamics with mixed weighted-averaging and weighted-median mechanisms
AU - Junkai Cheng
AU - Wenjun Mei
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16406
UR - https://arxiv.org/abs/2610.16406
AB - Opinion dynamics models relate interpersonal influence to consensus or persistent disagreement. We study a discrete-time model combining weighted-average and weighted-median updates on a fixed directed network. Individuals update synchronously with possibly different adjustment coefficients. We characterize equilibria through local balance conditions and opinion level sets, and establish convergence to a Lyapunov-stable equilibrium using nonexpansiveness and Krasnosel'skii--Mann theory. We define type-dependent cohesive sets: each weighted-average member assigns all its weight to the set, whereas each weighted-median member assigns at least one half. Consensus is reached from every initial condition if and only if no two such sets are disjoint. Otherwise, an open set of initial conditions leads to persistent disagreement. Replacing weighted-median individuals by weighted-average individuals preserves consensus whenever it is already guaranteed for all initial conditions. Four numerical studies examine the number and locations of weighted-average individuals. The frequency of reaching a prescribed consensus tolerance is nondecreasing with their fraction in the sampled ensembles, but individual realizations need not follow this trend.
ER -