Structural Balance and Affective Polarization in Signed Networks

This project was inspired by my PTSD from Twitter wars. Traditional gossip models typically assume that all influence is cooperative, but as any podcaster will tell you, influence in real world social media is often antagonistic. I've always been fascinated by how online spaces create these signed relationships where trust, distrust, algorithmic amplification, and external information all interact to produce collective dynamics, and this project was a chance to explore that in a more rigorous way. Huge thanks to Prof. Ben Golub, who was my networks professor and always entertained my questions (naive and ill-fated as they often were). Also thanks to the MMSS program for their support through the senior thesis project.

It was interesting working with a more parsimonious model that was just as deep and rich as the math I had done previously for ML theory, but I really liked this flavor of working with a very simple, very explainable, very write-down-able model. Even though I've only been working on it for a couple of months, I've already been surprised by how much depth there is to uncover, both mathematically and sociologically.

Abstract

Classical models of opinion dynamics assume agents update by averaging over neighbors, treating all influence as cooperative. But both Bayesian learning agents and natural social phenomena involve antagonistic relationships. A small recent literature has introduced signed graphs with "opposing" and "repelling" update rules, but existing analyses rely on restrictive assumptions: structural balance, absolute row-stochasticity, and constant external signals.

My thesis asks what happens outside this restrictive case: what is the long-run behavior under time-varying external information with repelling dynamics? What breaks when we relax structural balance? I analyze linear opinion-updating rules on signed graphs using operator-theoretic tools (spectral radii, stability regions) to characterize regimes of convergence, damping, and instability. So far, I have corrected spurious claims in the literature and generalized key lemmas relating structural balance to spectral properties.