Thermal properties of structurally balanced systems on classical random graphs

TL;DR

This paper investigates how social noise influences the transition between balanced and imbalanced states in social networks modeled as classical random graphs, revealing phase transition behaviors dependent on graph density and size.

Contribution

It introduces a detailed analysis of phase transitions in social balance models on random graphs, highlighting the effects of graph density and system size on transition types and critical temperatures.

Findings

First-order phase transition observed at certain densities.

Critical temperature increases with graph density.

Minimal density for phase transition decreases with system size.

Abstract

The dynamics of social relations and the possibility of reaching the state of structural balance (Heider balance) under the influence of the temperature modeling the social noise level are discussed for interacting actors occupying nodes of classical random graphs. Depending on the graph density $D$, either a smooth cross-over or a first-order phase transition from a balanced to an imbalanced state of the system is observed with an increase of the thermal noise level. The minimal graph density $D_\text{min}$ for which the first-order phase transition can be observed decreases with system size $N$ as $D_\text{min}\propto N^{-0.58(1)}$. For graph densities $D>D_\text{min}$ the reduced critical temperature $T_c^\star=T_c/T_c(D=1)$ increases with the graph density as $T_c^\star\propto D^{1.719(6)}$ independently of the system size $N$.