Generalized solutions to bounded-confidence models
Benedetto Piccoli, Francesco Rossi
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 (ODEs) 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 useddefinition and initial conditions.