Social dynamics with quantized states: simulations and analytical results.

Over the years, interest in studying the evolution of social dynamics over graphs has greatly increased: different models and types of dynamics have been proposed to analyze how the nodes of a graph interact. These nodes can be conceivable as human individuals interacting, animals, cells or any other element whose behavior can be understood if placed inside a network and analysed through some sort of dynamics on this network. This thesis aims at reviewing many of the models proposed to describe opinion formation under some social influence, hence analysing a recent model of social influence in which the real opinions are concealed to the other nodes of the network and only seen through a filter, a quantizer. Indeed, in many situations, individuals may not be able to fully show their opinion which can only be assessed through their displayed behaviors (social networks posts, people followed, tweets, etc.). The difficulty in this last dynamics arises as the dynamics involves a discontinuous vector field coming from the quantization of the individual opinions. First of all, the problem of how to generalize solutions for these discontinuous dynamics will be addressed, then some convergence properties will be analyzed with the aim of analytically extending current results to different graphs. Simulating these dynamics has got a great part in the analysis: numerical simulations may lead to some results and may for instance suggest convergence properties but they might also fail in capturing all the possible solutions. In our case numerical methods have shown solutions compatible with the theory which had not been found analytically. These solutions show periodic patterns and may not converge to consensus and in general may not converge at all. Generalised solutions which have been considered are Carathéodory and Krasovskii, the latter requiring some theory about differential inclusions, has been briefly retrieved in the dissertation. The study about the dynamics follows a microscopic approach, whether it must be stated that another approach to social dynamics is often considered: the macroscopic one, in which the study doesn’t focus on the single individual’s opinion evolution. In the macroscopic setting, the evolution of some statistic upon the entire network (the total mean of vote preference for instance) is studied and not the single individual votes.