Analysis of a continuous-time model of structural balance

TL;DR

This paper analyzes a continuous-time matrix model of social balance, proving that networks evolve into either harmonious or divided states, with initial friendliness levels determining the outcome.

Contribution

It provides a rigorous proof that only two outcomes are possible in the model and derives a formula linking initial conditions to final faction membership.

Findings

Networks end in harmony or division, no other outcomes.

Initial friendliness influences whether conflict or harmony emerges.

Closed-form expression for faction membership based on initial conditions.

Abstract

It is not uncommon for certain social networks to divide into two opposing camps in response to stress. This happens, for example, in networks of political parties during winner-takes-all elections, in networks of companies competing to establish technical standards, and in networks of nations faced with mounting threats of war. A simple model for these two-sided separations is the dynamical system dX/dt = X^2 where X is a matrix of the friendliness or unfriendliness between pairs of nodes in the network. Previous simulations suggested that only two types of behavior were possible for this system: either all relationships become friendly, or two hostile factions emerge. Here we prove that for generic initial conditions, these are indeed the only possible outcomes. Our analysis yields a closed-form expression for faction membership as a function of the initial conditions, and implies that the initial amount of friendliness in large social networks (started from random initial conditions) determines whether they will end up in intractable conflict or global harmony.