Characterization and space embedding of directed graphs trough magnetic Laplacians
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Directed graphs are essential data structures which model several real-world systems, such as social networks. However, the majority of methods and measurements recently developed in network science only deals with undirected complex networks. Due to that limitation, 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 community 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.
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