Continuous Opinion and Discrete Actions in social dynamics: analysis of a quantized model

Opinion dynamics study the evolution of the ideas of interacting individuals, whether they are people, organizations, or entire communities. Social networks are modeled as graphs, in which every node represents an individual, paired with a real number that models its opinion, and every edge represents an interaction or a social bond. In recent years, interest in the topic has been increasing, and different models have been proposed. The present work focuses on the analysis of a quantized model, which is founded on the premise that people don't have access to the exact opinion of the individuals they are interacting with. Instead, every person can infer an approximation of their interlocutors' opinion based on the behaviour they display. The model in question is discontinuous in state and piecewise linear. This discontinuity complicates the study, and requires to redefine the concept of solution of a differential equation. To study the model, we focus on Carathéodory solutions. A phenomenon caused by the discontinuity is the presence of extended equilibria, points that attract some solutions, despite not being proper equilibrium points as we would expect in the continuous case. The first original result presented concerns the basin of attraction of such points. Then, the study focuses on specific graphs, providing two sets of original results. The first set provides analytical expressions of extended equilibria on the line and ring graphs. The second set of results is presented under the assumption of binary action, which means restricting the opinion of the nodes to the interval [0,1], and thus having only two possible behaviours, 0 and 1. In this section, we provide convergence results for the line and ring graphs, an algorithm to find extended equilibria on tree graphs, and show the presence of a Zeno point in the directed cycle graph with 3 nodes.