Generalized solutions to opinion dynamics models with discontinuities

TL;DR

This paper investigates generalized solutions to opinion dynamics models with discontinuities, comparing metric and topological interactions in terms of existence, uniqueness, and long-term behavior.

Contribution

It introduces and analyzes Caratheodory and Krasovsky solutions for discontinuous opinion models with state-dependent interactions, highlighting differences between metric and topological approaches.

Findings

Existence and uniqueness of solutions depend on interaction type.

Different asymptotic behaviors are observed for metric and topological models.

The study provides a framework for analyzing discontinuous social dynamics models.

Abstract

Social dynamics models may present discontinuities in the right-hand side of the dynamics for multiple reasons, including topology changes and quantization. Several concepts of generalized solutions for discontinuous equations are available in the literature and are useful to analyze these models. In this chapter, we study Caratheodory and Krasovsky generalized solutions for discontinuous models of opinion dynamics with state dependent interactions. We consider two definitions of "bounded confidence" interactions, which we respectively call metric and topological: in the former, individuals interact if their opinions are closer than a threshold; in the latter, individuals interact with a fixed number of nearest neighbors. We compare the dynamics produced by the two kinds of interactions, in terms of existence, uniqueness and asymptotic behavior of different types of solutions.