Characterization and space embedding of directed graphs and social networks through magnetic Laplacians

Though commonly found in the real world, directed networks have received relatively less attention from the literature in which concerns their topological and dynamical characteristics. In this work, we develop a magnetic Laplacian-based framework that can be used for studying directed complex networks. More specifically, we introduce a specific heat measurement that can help to characterize the network topology. It is shown that, by using this approach, it is possible to identify the types of several networks, as well as to infer parameters underlying specific network configurations. Then, we consider the dynamics associated with the magnetic Laplacian as a means of embedding networks into a metric space, allowing the identification of mesoscopic structures in artificial networks or unravel the polarization on political blogosphere. By defining a coarse-graining procedure in this metric space, we show how to connect the specific heat measurement and the positions of nodes in this space.