Opinion Dynamics on Signed Graphs and Graphons

TL;DR

This paper uses graphon theory to analyze opinion dynamics on large signed networks, demonstrating that graphon-based models accurately approximate the behavior of large converging graphs with negative interactions.

Contribution

It introduces a novel graphon framework for opinion dynamics with negative interactions, proving existence, uniqueness, and approximation results for large networks.

Findings

Graphon models accurately approximate large network dynamics.

Existence and uniqueness of solutions for signed graphon dynamics.

New convergence results for W-random graphs under general conditions.

Abstract

In this paper, we make use of graphon theory to study opinion dynamics on large undirected networks. The opinion dynamics models that we take into consideration allow for negative interactions between the individuals, whose opinions can thus grow apart. We consider both the repelling and the opposing models of negative interactions, which have been studied in the literature. We define the repelling and the opposing dynamics on signed graphons and we show that their initial value problem solutions exist and are unique. We then show that, in a suitable sense, the graphon dynamics is a good approximation of the dynamics on large graphs that converge to a graphon. This result applies to large random graphs that are sampled according to a graphon (W-random graphs), for which we provide a new convergence result under very general assumptions.