Analysis and control of spectral centralities in graphs and hypergraphs

Abstract

Network science has witnessed a surge in popularity in the past two decades. This research interest has transitioned from exploring standard networks to more intricate mathematical abstractions such as multilayer networks and hypergraphs. Centrality measures remain a cornerstone of complex network theory, with an active community that proposes theoretical advancements and applies them to real-world scenarios. This thesis explores the controllability of spectral centrality measures in complex networks. This type of centrality measures is particularly important because of its analytical foundations and reduced computational cost, so much so that it currently underlies most Internet search engines. We analyze various techniques for manipulating these measures in graphs and multilayer networks, classifying the different centrality control paradigms based on the type of control exerted (structural or parametric) and the amount of control attained. We then propose novel extensions of spectral centrality measures to non-uniform and directed/heterogeneous hypergraphs, overcoming the limitations of existing methods while retaining their mathematical consistency. These extensions leverage the Perron-Frobenius theory for tensors, thereby providing analytical guarantees of existence and uniqueness. Finally, we explore the controllability of the aforementioned generalization of spectral centralities under weight adjustments. Although primarily theoretical, this work lays the groundwork for future research on controlling centrality measures and paves the way for investigating potential applications in real-world complex systems.