  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.
