Heider balance of a square lattice in an external field

TL;DR

This paper investigates the effects of an external social field on Heider balance in lattice networks, using multiple methods including exact calculations, simulations, and analytical solutions, revealing insights into social balance dynamics.

Contribution

It introduces the external social field into the Heider model and leverages the Ising model equivalence to derive exact analytical results for system susceptibility.

Findings

External field breaks symmetry between hostile and friendly relationships.

Exact susceptibility formulas were derived for systems in equilibrium.

Comparative analysis of triangular and square networks was conducted.

Abstract

We discuss the Heider model in the presence of an external social field. This field was introduced to break the symmetry between the probabilities of hostile and friendly relationships. We consider the system in the presence of fluctuations generated by thermal noise and present the results of a comparative study of two-dimensional triangular and square networks with periodic boundary conditions. The results were obtained using three different methods: exact calculations for small systems, Monte Carlo simulations of medium-sized systems, and exact calculations in the thermodynamic limit (corresponding to infinite size) of certain limiting cases for which analytical solutions are possible. In particular, we exploit the recently discovered equivalence between structurally balanced systems and the Ising model to derive an exact form of the edge magnetization susceptibility for systems in Heider equilibrium.