The Dynamics and Statistics of Public Opinion

Abstract

How do individual opinions, shaped through everyday interactions, give rise to large-scale patterns in public discourse and democratic outcomes? This thesis explores that question through two domains where opinions are both formed and revealed: online social networks and electoral systems. The first part addresses polarization in digital environments, driven by homophilic interactions and algorithmic reinforcement. To mitigate this, a simple intervention -- the random nudge -- is introduced, enabling occasional exposure beyond one's echo chamber. Simulations across various opinion dynamics models show that even a small probability of such encounters can significantly reduce polarization, breaking echo chambers while preserving diversity and autonomy. The second part examines the statistical structure of electoral competition. Using a newly assembled dataset spanning 34 countries and multiple spatial resolutions, we uncover a striking universal pattern: when margins of victory are normalized by local turnout and rescaled by their national average, the distribution collapses onto a single curve across democracies. To explain this, the Random Voting Model (RVM) is proposed -- a minimal, parameter-free stochastic model that reproduces the universal curve and accurately predicts scaled distributions of margins, winner, and runner-up votes. These models go beyond description: the random nudge suggests a viable strategy to reduce polarization online, while deviations from RVM predictions signal potential electoral anomalies, as shown in case studies from Ethiopia and Belarus. Together, these findings reveal simple, robust statistical principles underlying complex collective decisions -- and offer new tools to better understand and strengthen democracy.