# Social Balance and the Bernoulli Equation

## TL;DR

This paper introduces a simplified analysis of a social balance model, revealing that general initial conditions lead to four factions instead of two, and characterizes their formation and sizes.

## Contribution

It provides a new, simplified analysis of social balance dynamics allowing for general initial conditions, expanding understanding beyond symmetric cases.

## Key findings

- Four factions emerge from general initial conditions.
- The model characterizes the size and nature of the factions.
- Analysis extends previous symmetric case results.

## Abstract

Since the 1940's there has been an interest in the question why social networks often give rise to two antagonistic factions. Recently a dynamical model of how and why such a balance might occur was developed. This note provides an introduction to the notion of social balance and a new (and simplified) analysis of that model. This new analysis allows us to choose general initial conditions, as opposed to the symmetric ones previously considered. We show that for general initial conditions, four factions will evolve instead of two. We characterize the four factions, and we give an idea of their relative sizes.