Dynamics of social balance on networks: The emergence of multipolar societies

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 to decrease the total energy (tension) of the system, on average. Our approach is to consider a variable energy epsilon is an element of [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), a unipolar (paradise) state is reached. However, for rho(0) <= rho(c)(0)(epsilon), multipolar 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* approximate to 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.