We propose a framework for the visualization of directed networks relying on the eigenfunctions of the magnetic Laplacian, called here Magnetic Eigenmaps. The magnetic Laplacian is a complex deformation of the well-known combinatorial Laplacian. Features such as density of links and directionality patterns are revealed by plotting the phases of the first magnetic eigenvectors. An interpretation of the magnetic eigenvectors is given in connection with the angular synchronization problem. Illustrations of our method are given for both artificial and real networks.

Communities in directed networks have often been characterized as regions with a high density of links, or as sets of nodes with certain patterns of connection. Our approach for community detection combines the optimization of a quality function and a spectral clustering of a deformation of the combinatorial Laplacian, the so-called magnetic Laplacian. The eigenfunctions of the magnetic Laplacian, which we call magnetic eigenmaps, incorporate structural information. Hence, using the magnetic eigenmaps, dense communities including directed cycles can be revealed as well as "role" communities in networks with a running flow, usually discovered thanks to mixture models. Furthermore, in the spirit of the Markov stability method, an approach for studying communities at different energy levels in the network is put forward, based on a quantum mechanical system at finite temperature.

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https://repositorio.uam.es Esta es la versión de autor del artículo publicado en: This is an author produced version of a paper published in: