Behavior of Nonlinear Opinion Dynamics over Large Networks

Abstract

Opinion dynamics have been studied for decades across disciplines, with much of the theoretical literature focusing on behaviors such as consensus, polarization, and clustering. Although classic models can exhibit more complex opinion patterns in simulations, quantifying such distributions is not fully understood. To address this question, in this paper, we study the behavior of nonlinear opinion dynamics over large-scale networks using graphons, which capture the underlying network structure. In the model, agents update their opinions according to a nonlinear rule that includes saturation effects in interactions. The network is represented by random graphs generated from a graphon, and a corresponding nonlinear dynamical model is introduced over the graphon. We show that the graphon dynamics approximate the finite-dimensional system, when the network size is large. Leveraging spectral approximation results for random graphs, we further show that the equilibria of the nonlinear model can also be approximated by those of the continuum limit. This result enables a quantitative characterization of the opinion distribution based on the underlying graphon structure. The theoretical results are illustrated by numerical simulation.