Response of the competitive balance model to the external field

TL;DR

This paper explores how external influences affect the dynamics of the competitive balance model, revealing phase transitions, hysteresis, and community resistance, with implications for understanding social and political networks.

Contribution

It provides a mean-field analytical framework and Monte Carlo validation for the competitive balance model under external fields, highlighting symmetry breaking and system responses.

Findings

External field breaks system symmetry.

System response varies with temperature, showing paramagnetic or ferromagnetic behavior.

Hysteresis observed in the ferromagnetic regime.

Abstract

The competitive balance model was proposed as an extension of the structural balance theory, aiming to account for heterogeneities observed in real-world networks. In this model, different paradigms lead to form different friendship and enmity. As an example, friendship or enmity between countries can have a political or religious basis. The suggested Hamiltonian is symmetrical between paradigms. Our analyses show that a balanced state can be achieved if just one paradigm prevails in the network and the paradigm shift is possible only by imposing an external field. In this paper, we investigate the influence of the external field on the evolution of the network. We drive the mean-field solutions of the model and verify the accuracy of our analytical solutions by performing Monte-Carlo simulations. We observe that the external field breaks the symmetry of the system. The response of the system to this external field, contingent upon temperature, can be either paramagnetic or ferromagnetic. We observed a hysteresis behavior in the ferromagnetic regime. Once communities are formed based on a certain paradigm, then they resist change. We found that to avoid wasting energy we need to know the level of stochastic behavior in the network. Analogous to magnetic systems, we observe that susceptibility adheres to Curie's law.