# Mean-field solution of structural balance dynamics in nonzero temperature

## TL;DR

This paper models the dynamics of signed networks at nonzero temperatures using a mean-field approach, revealing a first-order phase transition from imbalance to balance influenced by network tension tolerance.

## Contribution

It introduces a generalized model incorporating temperature to study structural balance, revealing a phase transition and hysteresis in network dynamics.

## Key findings

- Identifies a critical temperature $T_{c}$ for phase transition.
- Shows the existence of hysteresis in the transition.
- Demonstrates the impact of tension tolerance on network balance.

## Abstract

In signed networks with simultaneous friendly and hostile interactions, there is a general tendency to a global structural balance, based on the dynamical model of links status. Although the structural balance represents a state of the network with a lack of contentious situations, there are always tensions in real networks. To study such networks, we generalize the balance dynamics in nonzero temperatures. The presented model uses elements from Boltzmann-Gibbs statistical physics to assign an energy to each type of triad, and it introduces the temperature as a measure of tension tolerance of the network. Based on the mean-field solution of the model, we find out that the model undergoes a first-order phase transition from an imbalanced random state to structural balance with a critical temperature $T_{c}$, where in the case of $T > T_{c}$ there is no chance to reach the balanced state. A main feature of the first-order phase transition is the occurrence of a hysteresis loop crossing the balanced and imbalanced regimes.