Network systems: Social networks, Epidemics, Optimization and Contraction Theory
In this thesis, I will first present mathematical models that explain the evolution of interpersonal relationships in a social network, represented by a signed graph, converging to structures that have a long history in sociology - namely, structural and clustering balance. Then, I will present a simple model for the evolution of opinions over signed graphs, including the aforementioned special structures. Then, I will present an important phenomenon that occurs on the susceptible-infected-susceptible (SIS) model of epidemics: the emergence of a new epidemic domain of bistability when higher-order interaction among individuals are considered on the contact network. Then, I will present an algorithm for the computation of Wasserstein barycenters, and show a connection with the theory of opinion dynamics. Finally, the last part of this thesis is devoted to the study and application of contraction theory, an important tool that certifies incremental stability. We study its expansion to dynamical systems on Hilbert spaces, as well as its application to various optimization problems and settings.