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

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.