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Multidimensional Opinion Dynamics with Confirmation Bias: A Multi-Layer Framework

M. Hossein Abedinzadeh, Emrah Akyol

Abstract

We study multidimensional opinion dynamics under confirmation bias in social networks. Each agent holds a vector of correlated opinions across multiple topic layers. Peer interaction is modeled through a static, informationally symmetric social channel, while external information enters through a dynamic, informationally asymmetric source channel. Source influence is described by nonnegative state-dependent functions of agent--source opinion mismatch, which captures confirmation bias without hard thresholds. For general Lipschitz source-influence functions, we give sufficient conditions under which the dynamics are contractive and converge to a unique steady state independent of the initial condition. For affine confirmation-bias functions, we show that the steady state can be computed through a finite sign-consistency search and identify a regime in which it admits a closed form. For broader classes of bounded nonlinear source-influence functions, we derive explicit lower and upper bounds on the fixed point. Numerical examples and a study on a real-world adolescent lifestyle network illustrate the role of multidimensional coupling and show that source-design conclusions can change qualitatively when confirmation bias is ignored.

Multidimensional Opinion Dynamics with Confirmation Bias: A Multi-Layer Framework

Abstract

We study multidimensional opinion dynamics under confirmation bias in social networks. Each agent holds a vector of correlated opinions across multiple topic layers. Peer interaction is modeled through a static, informationally symmetric social channel, while external information enters through a dynamic, informationally asymmetric source channel. Source influence is described by nonnegative state-dependent functions of agent--source opinion mismatch, which captures confirmation bias without hard thresholds. For general Lipschitz source-influence functions, we give sufficient conditions under which the dynamics are contractive and converge to a unique steady state independent of the initial condition. For affine confirmation-bias functions, we show that the steady state can be computed through a finite sign-consistency search and identify a regime in which it admits a closed form. For broader classes of bounded nonlinear source-influence functions, we derive explicit lower and upper bounds on the fixed point. Numerical examples and a study on a real-world adolescent lifestyle network illustrate the role of multidimensional coupling and show that source-design conclusions can change qualitatively when confirmation bias is ignored.
Paper Structure (14 sections, 5 theorems, 92 equations, 9 figures, 2 tables, 1 algorithm)

This paper contains 14 sections, 5 theorems, 92 equations, 9 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Opinion Dynamics Model
  4. Convergence and Fixed-Point Analysis
  5. Convergence Analysis
  6. Fixed-Point Analysis
  7. Fixed Point Under the Affine Rule
  8. Bounded Source-Influence Functions
  9. Special Case with a Closed-Form Solution
  10. Network Simulation
  11. Simulation
  12. Conclusion
  13. Proof of Theorem \ref{['thm:th1']}
  14. Proof of Theorem \ref{['thm:th2']}

Key Result

Theorem 1

Under Assumptions asm:Positivity, asm:Wrow, asm: ConvexCondition2, asm: LC, and asm: CC, the map $f$ in eq:CBUR is a contraction on $\mathbb{I}^{nq}$ under $\|\cdot\|_\infty$. Hence eq:CBUR admits a unique fixed point $\mathbf{x}^*\in\mathbb{I}^{nq}$, and every trajectory converges to it. The fixed

Figures (9)

  • Figure 1: Network used in Example \ref{['ex: Example 1']} with two agents $\mathrm{v}_1$ and $\mathrm{v}_2$, and one information source $\mathrm{u}_1$.
  • Figure 2: Example \ref{['ex: Example 1']}: true two-dimensional dynamics versus decoupled scalar approximation.
  • Figure 3: Network used in Example \ref{['ex: Example 2']} with three agents $\mathrm{v}_1$, $\mathrm{v}_2$, and $\mathrm{v}_3$, and one information source $\mathrm{u}_1$.
  • Figure 4: Example \ref{['ex: Example 2']}: bounded-confidence dynamics for several thresholds.
  • Figure 5: Example \ref{['ex: Example 2']}: affine confirmation-bias dynamics for several values of $\gamma$ and $\varepsilon$ combinations.
  • ...and 4 more figures

Theorems & Definitions (20)

  • Remark 1
  • Remark 2
  • Remark 3
  • Example 1
  • Remark 4
  • Example 2
  • Theorem 1
  • Remark 5
  • Remark 6
  • Example 3
  • ...and 10 more