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

TL;DR

This paper introduces a continuous-time weighted-median opinion dynamics model incorporating inertia, providing theoretical analysis and conditions for consensus, disagreement, and external influence, enhancing understanding of opinion evolution in social networks.

Contribution

It presents a novel continuous-time extension of the weighted-median model with inertia, along with comprehensive stability, convergence, and control conditions.

Findings

The equilibria are Lyapunov stable.

Global convergence to consensus is established.

Conditions for persistent disagreement and external control are derived.

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.