In this paper, we investigated the possibility of using the magnetic Laplacian to characterize directed networks. We address the problem of characterization of network models and perform the inference of the parameters used to generate these networks under analysis. Many interesting results are obtained, including the finding that the community structure is related to rotational symmetry in the spectral measurements for a type of stochastic block model. Due the hermiticity property of the magnetic Laplacian we show here how to scale our approach to larger networks containing hundreds of thousands of nodes using the Kernel Polynomial Method (KPM), a method commonly used in condensed matter physics. Using a combination of KPM with the Wasserstein metric, we show how we can measure distances between networks, even when these networks are directed, large, and have different sizes, a hard problem that cannot be tackled by previous methods presented in the literature.

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