Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Yi Han, Julien M. Hendrickx, Ge Chen, Wenjun Mei

TL;DR

A parsimonious continuous-time extension of the weighted-median model is introduced by incorporating individual inertia, allowing opinions to move gradually toward the neighbors'weighted median, revealing how a social group's resilience to external manipulation fundamentally depends on its internal network structure.

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.

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

TL;DR

A parsimonious continuous-time extension of the weighted-median model is introduced by incorporating individual inertia, allowing opinions to move gradually toward the neighbors'weighted median, revealing how a social group's resilience to external manipulation fundamentally depends on its internal network structure.

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.
Paper Structure (19 sections, 20 theorems, 67 equations, 3 figures, 1 table)

This paper contains 19 sections, 20 theorems, 67 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background and Motivations
  3. Literature Review
  4. Contribution
  5. Organization
  6. Empirical Evidence and Model Setup
  7. Empirical Support for the Weighted-Median Mechanism with Inertia
  8. Basic Definitions And Model Setup
  9. Dynamical behavior of the continuous-time weighted-median opinion dynamics
  10. Lemmas on Positive Invariance
  11. Main results
  12. Further Discussion and Simulations
  13. Conclusions
  14. Proof of Lemma \ref{['lem:CT-from-eq-set-to-single-eq']}
  15. Proof of Lemma \ref{['lem:refine-omega-limit-set']}
  16. ...and 4 more sections

Key Result

Lemma 1

Given any $n$-tuple of real values $x=(x_1,\dots, x_n)$ and any $n$-tuple of non-negative weights $w=(w_1,\dots, w_n)$ with $\sum_{i=1}^n w_i=1$,

Figures (3)

  • Figure 1: (a) Distributions of prediction errors under the $\text{H}_\text{A}$ and $\text{H}_\text{M}$ models. (b) Distributions of prediction errors under the weighted-median model with and without inertia. The top and bottom 10% outliers are not displayed in panel (a) and (b). In panel (b), A nonlinear axis scale and a white gap are used to accommodate the first bin, which is disproportionaly high. The last bin (0.4$<$) aggregates all error rates greater than 0.4 to compactly represent the tail.
  • Figure 2: How network structure affects the minimal number of externally manipulated nodes required to steer the CTWM opinion dynamics to a prescribed consensus opinion. Panel A illustrates the effect of link density, based on simulations over Erdős–Rényi (ER) random networks. Panel B illustrates the effect of clustering coefficient (negatively correlated with rewiring probability), based on simulations over Watts–Strogatz (WS) small-world networks. In both plots, each data point represents the sample mean of 150 independent simulations under the same parameter setting, and the boxes denote 95% confidence intervals computed from the sample standard deviation.
  • Figure 3: An illustration of the two repulsive classes. For simplicity, the vector $x\in\mathbb{R}^n$ is assumed to satisfy $x_{1}\geq x_{2}\geq \cdots \geq x_{n}$, which are distributed the axis. (1) Upward-repulsive class: suppose $\mathcal{C}=\{3,\dots,k,l\}$ is a cohesive set. The minimum within $\mathcal{C}$ is uniquely attained at $l$, and it holds that $\sum_{h\in \mathcal{C},,h\neq l} w_{lh} > \tfrac{1}{2}$. Under these conditions, $x$ belongs to the upward-repulsive class. (2) Downward-repulsive class: suppose $\mathcal{M}=\{3,4,k,m,l\}$ is a maximal cohesive set. Let $\Theta = \arg\max_{i\in \mathcal{M}} x_{i}=\{3,4\}$. No cohesive set contained in $\Theta \cup \bar{\mathcal{M}}=\{1,2,3,4,\cdots,j,n\}$ includes any element of $\Theta$. Under these conditions, $x$ belongs to the downward-repulsive class.

Theorems & Definitions (25)

  • Definition 1: Weighted median
  • Lemma 1: Properties of weighted median
  • Definition 2
  • Definition 3: Cohesive set
  • Definition 4: Decisive link and decisive graph
  • Lemma 2: WM-JMH-GC-FB-FD:24
  • Lemma 3: Convergence to unique equilibrium
  • Lemma 4: Bony-Brezis theorem
  • Proposition 5
  • Lemma 6
  • ...and 15 more