Quantum processes for structural analysis

Abstract

Many real systems can be modelled as networks, being characterized by a set of items and links between them. Systems taking the form of networks, also called graphs, appear in a wide range of scenarios, varying from biological to technological domains. Illustrative examples abound and include neural networks, protein-protein interactions, metabolic reaction networks, social networks, coauthorship and citation relations, road maps, financial market stock correlations and the World Wide Web. In the last decade network theory has proven to be a very useful instrument to model the structure of systems, albeit not sufficient to cover all issues in the scope of structural analysis. For this reason it has arisen the need of drawing on ideas from fields such as physics which actually helped in gaining new insight for a relevant class of problems. In this thesis, we address matters encountering in graph structural analysis by exploiting new approaches based on quantum processes and the von Neumann entropy. In particular, we focus on the characterization aspects of graphs concerning structural properties, as well as on processes underlying network evolution. We commence by investigating spectral generative models for learning structural representations. Then we move on to quantum models, specifically quantum walks, and the von Neumann entropy characterization. Finally, we introduce a novel thermodynamic method to model time evolving networks.