Analysis and Models for the Dynamics of Opinions and Interpersonal Relationships

This doctoral thesis focuses on the study of opinion dynamics over social networks. The treatment is made from a control system perspective: psychology and social sciences underpin the modeling, analysis and control of complex networked systems. The common thread of this work is the mathematical formalization of the micro-mechanisms of social networks in order to understand how they affect the dynamic behaviour of the network at a macro-level. This thesis develops along three main directions: in the first part we assume the topology to be fixed in a social equilibrium configurations and we study the effects of the weights given to interpersonal and personal ties on the opinion dynamics. In particular, we assume the network to be clustering balanced and we study, under some assumption on the interpersonal weights, how much each individual should be convinced about its own opinion so that agents’ opinion cluster conformably to the clusters in the topology, thus giving rise to the "k-partite consensus" phenomenon. In the second part, we take into account the case of opinion varying network topologies and we study the reaching of structural equilibria in the network. This paradigm is the most suitable when opinions and interpersonal ties evolve on time scales that are comparable in magnitude. We take into account two mechanisms according to which the interpersonal ties evolve over time: the influence mechanism and the homophily mechanism. In this context, we also study two multi-dimensional extensions of one of the pioneering models in the scientific literature related to opinion dynamics: the Hegselmann-Krause model. In the average-based model, agents compare the average opinion that they have on different topics while in the uniform-affinity model agents compare their opinions topic-wise. The first model suits better for contexts in which the topics into play are related so that, supposedly, there is not much deviation among the opinions that each agent has on the various topic. The second variation is also suitable for contexts in which the opinions are expressed on a broader variety of topics. The third and last part of this thesis is related to the study of herdability, namely the capability of a system to be driven towards the interior of the positive orthant. The study of this property becomes of interest in contexts in which (positive) thresholds come into play, such as in marketing advertisements, or electoral contexts. We will investigate under what structural properties some specific leader-follower network topologies lead to herdable systems.