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

TL;DR

This paper introduces effective adjacency matrices derived from deformed Laplacian operators for directed and signed graphs, enabling the use of undirected graph tools and methods.

Contribution

It defines effective adjacency matrices from deformed Laplacians, bridging directed/signed graphs with undirected graph analysis tools.

Findings

Effective matrices map complex graphs to undirected equivalents.

Hodge-Helmholtz decomposition aids in understanding operator interplay.

Facilitates application of undirected graph measures to directed/signed 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.