Structural Balance via Gradient Flows over Signed Graphs

TL;DR

This paper introduces a new class of dynamic models for structural balance in signed graphs, based on gradient flows of a dissonance energy function, linking social influence with energy landscape minimization.

Contribution

It proposes a novel gradient flow framework for modeling structural balance, connecting social influence dynamics with energy minimization techniques.

Findings

Analysis of critical points of the dissonance function

Mathematical characterization of transient behaviors

Numerical simulations illustrating model dynamics

Abstract

Structural balance is a classic property of signed graphs satisfying Heider's seminal axioms. Mathematical sociologists have studied balance theory since its inception in the 1040s. Recent research has focused on the development of dynamic models explaining the emergence of structural balance. In this paper, we introduce a novel class of parsimonious dynamic models for structural balance based on an interpersonal influence process. Our proposed models are gradient flows of an energy function, called the dissonance function, which captures the cognitive dissonance arising from violations of Heider's axioms. Thus, we build a new connection with the literature on energy landscape minimization. This gradient flow characterization allows us to study the transient and asymptotic behaviors of our model. We provide mathematical and numerical results describing the critical points of the dissonance function.