Generalized solutions to bounded-confidence models

TL;DR

This paper investigates the mathematical solutions of bounded-confidence models in social dynamics, addressing the existence, uniqueness, and properties of solutions under different definitions, which is crucial for understanding multi-agent interaction behaviors.

Contribution

It provides a comprehensive analysis of solution concepts for discontinuous differential equations in bounded-confidence models, establishing existence and conditions for uniqueness.

Findings

Existence of solutions is guaranteed under general assumptions.

Uniqueness may fail in some cases but holds for almost all initial conditions.

Properties of solutions depend on the chosen solution concept and initial states.

Abstract

Bounded-confidence models in social dynamics describe multi-agent systems, where each individual interacts only locally with others. Several models are written as systems of ordinary differential equations with discontinuous right-hand side: this is a direct consequence of restricting interactions to a bounded region with non-vanishing strength at the boundary. Various works in the literature analyzed properties of solutions, such as barycenter invariance and clustering. On the other side, the problem of giving a precise definition of solution, from an analytical point of view, was often overlooked. However, a rich literature proposing different concepts of solution to discontinuous differential equations is available. Using several concepts of solution, we show how existence is granted under general assumptions, while uniqueness may fail even in dimension one, but holds for almost every initial conditions. Consequently, various properties of solutions depend on the used definition and initial conditions.