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Stable Boundaries of Opinion Dynamics in Heterogeneous Spatial Complex Networks

Mats Bierwirth, Johannes Lengler

TL;DR

This work develops and analyzes a tractable mean-field model of the interface between two opinion domains, and rigorously establishes the existence of a stable, non-trivial limiting distribution for the interface profile in a mean-field analysis, demonstrating that the boundary between opinions is stationary.

Abstract

We investigate majority-vote opinion dynamics on Geometric Inhomogeneous Random Graphs (GIRGs), a powerful model for spatial complex networks. In contrast to classic coarsening dynamics where a single opinion typically achieves global consensus, our simulations reveal that sufficiently large, localized opinion domains do not disappear. Instead, they stabilize, leading to a persistent coexistence of competing opinions. To understand the mechanism behind this arrested coarsening, we develop and analyze a tractable mean-field model of the interface between two opinion domains. Our main theoretical result rigorously establishes the existence of a stable, non-trivial limiting distribution for the interface profile in a mean-field analysis. This demonstrates that the boundary between opinions is stationary, providing a mathematical explanation for how complex network geometry can support robust opinion diversity in social systems.

Stable Boundaries of Opinion Dynamics in Heterogeneous Spatial Complex Networks

TL;DR

This work develops and analyzes a tractable mean-field model of the interface between two opinion domains, and rigorously establishes the existence of a stable, non-trivial limiting distribution for the interface profile in a mean-field analysis, demonstrating that the boundary between opinions is stationary.

Abstract

We investigate majority-vote opinion dynamics on Geometric Inhomogeneous Random Graphs (GIRGs), a powerful model for spatial complex networks. In contrast to classic coarsening dynamics where a single opinion typically achieves global consensus, our simulations reveal that sufficiently large, localized opinion domains do not disappear. Instead, they stabilize, leading to a persistent coexistence of competing opinions. To understand the mechanism behind this arrested coarsening, we develop and analyze a tractable mean-field model of the interface between two opinion domains. Our main theoretical result rigorously establishes the existence of a stable, non-trivial limiting distribution for the interface profile in a mean-field analysis. This demonstrates that the boundary between opinions is stationary, providing a mathematical explanation for how complex network geometry can support robust opinion diversity in social systems.
Paper Structure (15 sections, 16 theorems, 93 equations, 3 figures)

This paper contains 15 sections, 16 theorems, 93 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Network Model and Opinion Spreading
  3. Results
  4. Experimental Observations
  5. The Mean-Field Approximation
  6. Framework
  7. The Survival Theorem for Half-spaces
  8. Geometric Advantage Partitioning
  9. A Valid Subsolution
  10. From Half-Spaces to Balls
  11. Localization
  12. Non-Stability of Balls in the Mean-Field
  13. Bounding Local Curvature
  14. Discussion
  15. Future Research Directions

Key Result

Theorem 2.3

Let $\mathrm{G} = (\mathrm{V},\mathrm{E})$ be a GIRG on $n$ vertices and $v \in \mathrm{V}$. Condition on the event that $v$ has weight $w_v$. Then where the first term is contributed by the near-neighbours and the second from the far-neighbours.

Figures (3)

  • Figure 1: Opinion spreading with $\tau=2.15$. Panels 1,2: small square initial configuration ($c_0$ left, final configuration $c^*$ right), only red survives with minor exceptions. Panels 3,4: large square initial configuration ($c_0$ left, $c^*$ right), both opinions survive.
  • Figure 2: Survival probability of a square initial configuration as a function of its side length $s$, for various values of $\tau$. Dots show simulation results (100 runs per data point), and solid curves show logistic fits. The critical size decreases with $\tau$.
  • Figure 3: The two dimensional space $\mathcal{X}$ partitioned into the two regions $\mathcal{R}$ and $\mathcal{B}$.

Theorems & Definitions (42)

  • Definition 2.1: GIRG bringmann2019geometric
  • Definition 2.2: Ball of Influence
  • Theorem 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Definition 2.6: Opinion Spreading
  • Definition 3.1: Mean-Field Assumption
  • Definition 3.2: Mean-Field Update Operator
  • ...and 32 more