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Opinion Dynamics with Set-Based Confidence: Convergence Criteria and Periodic Solutions

Iryna Zabarianska, Anton V. Proskurnikov

TL;DR

The paper extends the Hegselmann-Krause model by replacing distance-based proximity with a general set-based confidence region $\mathcal{O}$, defining trusted neighbors via the Minkowski sum and updating opinions by simple averaging. It proves that when $\mathcal{O}$ is symmetric and contains $0$ in its interior, SCOD converges in a finite number of steps to a clustered equilibrium, mirroring HK behavior, and this convergence persists with identical stubborn agents; however, asymmetry or multiple stubborn agents can induce periodic or non-equilibrium convergence. Through explicit examples, the authors demonstrate non-clustered equilibria, periodic trajectories, and convergent dynamics absent in HK, underscoring the rich dynamics introduced by geometry of $\mathcal{O}$. They also provide a rigorous technical framework based on recurrent averaging inequalities to establish convergence results and discuss open problems, such as convergence rates and heterogeneous extensions, with potential implications for modeling complex opinion dynamics where trust is not distance-based.

Abstract

This paper introduces a new multidimensional extension of the Hegselmann-Krause (HK) opinion dynamics model, where opinion proximity is not determined by a norm or metric. Instead, each agent trusts opinions within the Minkowski sum $ξ+\mathcal{O}$, where $ξ$ is the agent's current opinion and $\mathcal{O}$ is the confidence set defining acceptable deviations. During each iteration, agents update their opinions by simultaneously averaging the trusted opinions. Unlike traditional HK systems, where $\mathcal{O}$ is a ball in some norm, our model allows the confidence set to be non-convex and even unbounded. We demonstrate that the new model, referred to as SCOD (Set-based Confidence Opinion Dynamics), can exhibit properties absent in the conventional HK model. Some solutions may converge to non-equilibrium points in the state space, while others oscillate periodically. These ``pathologies'' disappear if the set $\mathcal{O}$ is symmetric and contains zero in its interior: similar to the usual HK model, SCOD then converges in a finite number of iterations to one of the equilibrium points. The latter property is also preserved if one agent is "stubborn" and resists changing their opinion, yet still influences the others; however, two stubborn agents can lead to oscillations.

Opinion Dynamics with Set-Based Confidence: Convergence Criteria and Periodic Solutions

TL;DR

The paper extends the Hegselmann-Krause model by replacing distance-based proximity with a general set-based confidence region , defining trusted neighbors via the Minkowski sum and updating opinions by simple averaging. It proves that when is symmetric and contains in its interior, SCOD converges in a finite number of steps to a clustered equilibrium, mirroring HK behavior, and this convergence persists with identical stubborn agents; however, asymmetry or multiple stubborn agents can induce periodic or non-equilibrium convergence. Through explicit examples, the authors demonstrate non-clustered equilibria, periodic trajectories, and convergent dynamics absent in HK, underscoring the rich dynamics introduced by geometry of . They also provide a rigorous technical framework based on recurrent averaging inequalities to establish convergence results and discuss open problems, such as convergence rates and heterogeneous extensions, with potential implications for modeling complex opinion dynamics where trust is not distance-based.

Abstract

This paper introduces a new multidimensional extension of the Hegselmann-Krause (HK) opinion dynamics model, where opinion proximity is not determined by a norm or metric. Instead, each agent trusts opinions within the Minkowski sum , where is the agent's current opinion and is the confidence set defining acceptable deviations. During each iteration, agents update their opinions by simultaneously averaging the trusted opinions. Unlike traditional HK systems, where is a ball in some norm, our model allows the confidence set to be non-convex and even unbounded. We demonstrate that the new model, referred to as SCOD (Set-based Confidence Opinion Dynamics), can exhibit properties absent in the conventional HK model. Some solutions may converge to non-equilibrium points in the state space, while others oscillate periodically. These ``pathologies'' disappear if the set is symmetric and contains zero in its interior: similar to the usual HK model, SCOD then converges in a finite number of iterations to one of the equilibrium points. The latter property is also preserved if one agent is "stubborn" and resists changing their opinion, yet still influences the others; however, two stubborn agents can lead to oscillations.
Paper Structure (16 sections, 2 theorems, 16 equations, 8 figures)

This paper contains 16 sections, 2 theorems, 16 equations, 8 figures.

Table of Contents

  1. Introduction
  2. The Model Definition and Examples
  3. Opinions
  4. Confidence graph
  5. The SCOD (Opinion Update Rule)
  6. Extension: Stubborn Agents
  7. The SCOD vs. Previously Known Models
  8. Gallery of Untypical Behaviors
  9. Non-clustered Equilibria
  10. Periodic Solutions
  11. Convergent Solutions Absent in HK models
  12. The SCOD with a Symmetric Confidence Set
  13. Numerical Example
  14. Discussion
  15. Technical Proofs
  16. ...and 1 more sections

Key Result

Theorem 1

Assume that $\mathcal{O}$ obeys Assumptions asm.0, assum.Sym, and $\Xi(0)$ obeys Assumption assum.Stub. The following statements are true: (A) $\Xi(0)$ is an equilibrium if and only if it is clustered. (B) All opinions have finite limits $\xi^i(\infty)=\lim_{t\to\infty}\xi^i(t)$, and $\xi^i(\infty)=

Figures (8)

  • Figure 1: Examples of confidence sets.
  • Figure 2: A union of disconnected cliques
  • Figure 3: Non-clustered equilibrium of the SCOD
  • Figure 4: Example \ref{['ex.period_1d_n3']}. $\xi^2$ oscillates with period 3.
  • Figure 5: Example \ref{['ex.period_2d']}. (a) the confidence set; (b) opinions.
  • ...and 3 more figures

Theorems & Definitions (13)

  • Definition 1
  • Example 1
  • Example 2
  • Remark 1
  • Definition 2
  • Example 3
  • Remark 2
  • Example 4
  • Remark 3
  • Example 5
  • ...and 3 more