Dynamics of Social Balance on Networks: The Emergence of Multipolar Societies

TL;DR

This paper explores how societies with friendly and hostile relations evolve into stable multipolar configurations, revealing how initial conditions and the nature of negative relations influence societal polarization.

Contribution

It introduces a model with variable energy for negative triads and demonstrates how initial link density and triad tension lead to different societal structures, including multipolar states.

Findings

Final society state depends on initial friendly link density and triad tension.

Lower triad tension favors multipolar societies over bipolar or unipolar.

A mean-field approximation estimates critical initial densities consistent with simulations.

Abstract

Within the context of social balance theory, much attention has been paid to the attainment and stability of unipolar or bipolar societies. However, multipolar societies are commonplace in the real world, despite the fact that the mechanism of their emergence is much less explored. Here, we investigate the evolution of a society of interacting agents with friendly (positive) and enmity (negative) relations into a final stable multipolar state. Triads are assigned energy according to the degree of tension they impose on the network. Agents update their connections in order to decrease the total energy (tension) of the system, on average. Our approach is to consider a variable energy $\epsilon\in[0,1]$ for triads which are entirely made of negative relations. We show that the final state of the system depends on the initial density of the friendly links $\rho_0$. For initial densities greater than an $\epsilon$ dependent threshold $\rho^c_0(\epsilon)$ unipolar (paradise) state is reached. However, for $\rho_0 \leq \rho^c_0(\epsilon)$ multi-polar and bipolar states can emerge. We observe that the number of stable final poles increases with decreasing $\epsilon$ where the first transition from bipolar to multipolar society occurs at $\epsilon^*\approx 0.67$. We end the paper by providing a mean-field calculation that provides an estimate for the critical ($\epsilon$ dependent) initial positive link density, which is consistent with our simulations.