Beyond symmetrization: effective adjacency matrices and renormalization for (un)singed directed graphs

Abstract: To address the peculiarities of directed and/or signed graphs, new Laplacian operators have emerged. For instance, in the case of directionality, we encounter the magnetic operator, dilation (which is underexplored), operators based on random walks, and so forth. The definition of these new operators leads to the need for new studies and concepts, and consequently, the development of new computational tools. But is this really necessary? In this work, we define the concept of effective adjacency matrices that arise from the definition of deformed Laplacian operators such as magnetic, dilation, and signal. These effective matrices allow mapping generic graphs to a family of unsigned, undirected graphs, enabling the application of the well-explored toolkit of measures, machine learning methods, and renormalization groups of undirected graphs. To explore the interplay between deformed operators and effective matrices, we show how the Hodge-Helmholtz decomposition can assist us in navigating this complexity.