Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Convergence Analysis of Weighted-Median Opinion Dynamics with Prejudice

Ruichang Zhang, Zhixin Liu, Ge Chen, Wenjun Mei

TL;DR

The paper investigates discrete-time, synchronous weighted-median opinion dynamics with prejudice, addressing limitations of the Friedkin-Johnsen model. It introduces the prejudice-augmented update $x_i(t+1)=\lambda_i u_i+(1-\lambda_i)\mathrm{Med}_i(\boldsymbol{x}(t);W)$ and shows that the associated operator $F(\boldsymbol{x})=\Lambda\boldsymbol{u}+(I_n-\Lambda)\mathrm{Med}(\boldsymbol{x};W)$ is a contraction with factor $1-\lambda_{\min}$, ensuring a unique fixed point $\boldsymbol{x}^*$ and exponential convergence; a closed-form limit $\boldsymbol{x}^*=(I-A)^{-1}\Lambda\boldsymbol{u}$ is provided. For partial prejudice, the paper proves a necessary-and-sufficient consensus condition: asymptotic consensus occurs iff the influence network contains no cohesive unprejudiced subset; otherwise, cohesion can prevent convergence, distinguishing this model from FJ. Simulations corroborate the theory, showing consensus under full prejudice and multi-cluster outcomes under partial prejudice, reflecting more nuanced social dynamics.

Abstract

The Friedkin-Johnsen (FJ) model introduces prejudice into the opinion evolution and has been successfully validated in many practical scenarios; however, due to its weighted average mechanism, only one prejudiced agent can always guide all unprejudiced agents synchronizing to its prejudice under the connected influence network, which may not be in line with some social realities. To fundamentally address the limitation of the weighted average mechanism, a weighted-median opinion dynamics has been recently proposed; however, its theoretical analysis is challenging due to its nonlinear nature. This paper studies the weighted-median opinion dynamics with prejudice, and obtains the convergence and convergence rate when all agents have prejudice, and a necessary and sufficient condition for asymptotic consensus when a portion of agents have prejudice. These results are the first time to analyze the discrete-time and synchronous opinion dynamics with the weighted median mechanism, and address the phenomenon of the FJ model that connectivity leads to consensus when a few agents with the same prejudice join in an unprejudiced group.

Convergence Analysis of Weighted-Median Opinion Dynamics with Prejudice

TL;DR

The paper investigates discrete-time, synchronous weighted-median opinion dynamics with prejudice, addressing limitations of the Friedkin-Johnsen model. It introduces the prejudice-augmented update and shows that the associated operator is a contraction with factor , ensuring a unique fixed point and exponential convergence; a closed-form limit is provided. For partial prejudice, the paper proves a necessary-and-sufficient consensus condition: asymptotic consensus occurs iff the influence network contains no cohesive unprejudiced subset; otherwise, cohesion can prevent convergence, distinguishing this model from FJ. Simulations corroborate the theory, showing consensus under full prejudice and multi-cluster outcomes under partial prejudice, reflecting more nuanced social dynamics.

Abstract

The Friedkin-Johnsen (FJ) model introduces prejudice into the opinion evolution and has been successfully validated in many practical scenarios; however, due to its weighted average mechanism, only one prejudiced agent can always guide all unprejudiced agents synchronizing to its prejudice under the connected influence network, which may not be in line with some social realities. To fundamentally address the limitation of the weighted average mechanism, a weighted-median opinion dynamics has been recently proposed; however, its theoretical analysis is challenging due to its nonlinear nature. This paper studies the weighted-median opinion dynamics with prejudice, and obtains the convergence and convergence rate when all agents have prejudice, and a necessary and sufficient condition for asymptotic consensus when a portion of agents have prejudice. These results are the first time to analyze the discrete-time and synchronous opinion dynamics with the weighted median mechanism, and address the phenomenon of the FJ model that connectivity leads to consensus when a few agents with the same prejudice join in an unprejudiced group.
Paper Structure (9 sections, 13 theorems, 80 equations, 2 figures)

This paper contains 9 sections, 13 theorems, 80 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Basic Definition and Related Work
  3. Weighted-median opinion dynamics
  4. Friedkin-Johnsen model
  5. Weighted-median Prejudice Opinion Dynamics
  6. Weighted-median Opinion Dynamics with Partial Prejudice
  7. Proof of Theorem \ref{['Main_result1']}
  8. Simulations
  9. Concluding Remarks

Key Result

Lemma 2.1

If there exists ${x}^{\ast }\in\cup_{i=1}^n\{x_i\}$ such that then ${x}^{\ast }$ is the unique weighted median of $\boldsymbol{x}$ associated with $\boldsymbol{\vartheta}$; Otherwise there exists $z\in\cup_{i=1}^n\{x_i\}$ such that then the weighted median of $\boldsymbol{x}$ associated with $\boldsymbol{\vartheta}$ is not unique.

Figures (2)

  • Figure 1: The symmetrical influence network $\mathcal{G}(W)$ used in our simulations.
  • Figure 2: Opinion evolutions of our system (\ref{['doc10']}) with prejudiced agents (a) and mixed agents (c), and the FJ model (\ref{['FJ']}) with prejudiced agents (b) and mixed agents (d).

Theorems & Definitions (20)

  • Definition 2.1: Weighted median
  • Lemma 2.1: Appendix A in mei2022micro
  • Definition 3.1
  • Theorem 3.1: Convergence and convergence rate of weighted-median prejudice opinion dynamics
  • Remark 1
  • Lemma 3.1: Banach's Fixed Point Theorem
  • Lemma 3.2: Non-expansion of weighted median mapping, Theorem 8 in han2024continuoustime
  • Corollary 3.1
  • Remark 2
  • Proposition 3.1
  • ...and 10 more