Stable Boundaries of Opinion Dynamics in Heterogeneous Spatial Complex Networks

TL;DR

This paper studies opinion dynamics on spatial complex networks, revealing that large opinion domains can stabilize instead of merging, which explains persistent opinion diversity through a new mean-field interface model.

Contribution

It introduces a mean-field model for opinion interfaces on GIRGs and proves the existence of stable, non-trivial boundary distributions, explaining persistent opinion diversity.

Findings

Large opinion domains can stabilize and coexist.

The mean-field model predicts stationary opinion boundaries.

Mathematically proven existence of stable interface distributions.

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.