Relationship between Heider links and Ising spins

TL;DR

This paper establishes a theoretical equivalence between the Heider model with an external field and the Ising model in the context of structural balance, revealing a phase transition in social systems.

Contribution

It demonstrates that balanced Heider relations can be represented as Ising spins, linking social balance theory to statistical physics models.

Findings

Heider model with external field is equivalent to the Ising model in the limit of structural balance.

Balanced Heider states undergo a phase transition similar to the Ising model.

The critical social field value maximizes edge magnetization fluctuations.

Abstract

We show that the Heider model with an external field is equivalent, in the limit of structural balance, to the Ising model with nearest-neighbor interactions without an external field. More precisely, we claim that the signs of the Heider relations that maintain structural equilibrium in the system can be represented as nearest neighbor Ising spin products. We demonstrate this explicitly for a complete graph and provide a general argument for an arbitrary graph. A consequence of the equivalence is that the system of balanced Heider states undergoes a phase transition, inherited from the Ising model, at a critical value of the social field at which the fluctuations of edge magnetization are maximal.