Paradise-disorder transition in structural balance dynamics on Erd\"os-R\'enyi graphs

TL;DR

This paper explores how structural balance dynamics on Erd"os-Rényi graphs transition from a harmonious state to disorder, providing analytical predictions and numerical validation for different network configurations.

Contribution

It introduces a mean-field analytical framework for the Heider balance model on Erd"os-Rényi graphs, revealing how the transition scales with connection probability and network density.

Findings

First-order transition predicted by mean-field theory.

Critical temperature scales as p^2 for monolayer networks.

Numerical simulations confirm analytical predictions in dense graphs.

Abstract

Structural balance has been posited as one of the factors influencing how friendly and hostile relations of social actors evolve over time. This study investigates the behavior of the Heider balance model in Erd\"os-R\'enyi random graphs in the presence of a noisy environment, particularly the transition from an initially entirely positively polarized paradise state to a disordered phase. We examine both single-layer and bilayer network configurations and provide a mean-field solution for the average link polarization that predicts a first-order transition where the critical temperature scales with the connection probability $p$ as $p^2$ for a monolayer system and in a more complex way for a bilayer. We show that to mimic the dynamics observed in complete graphs, the intralayer Heider interaction strengths should be scaled as $p^{-2}$, while the interlayer interaction strengths should be scaled as $p^{-1}$ for random graphs. Numerical simulations have been performed, and their results confirm our analytical predictions, provided that graphs are dense enough.