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

arXiv: https://arxiv.org/abs/2406.01517 · HTML: https://mlsanity.com/paper/2406.01517/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 41 references that are in the map.
- First posted on arXiv 3 Jun 2024.

## Builds on (most cited first, 11 of 11)

- [Multi-scale Attributed Node Embedding](https://mlsanity.com/paper/1909.13021/) (2019)
- [Edge Directionality Improves Learning on Heterophilic Graphs](https://mlsanity.com/paper/2305.10498/) (2023)
- [Magnetic Eigenmaps for the Visualization of Directed Networks](https://mlsanity.com/paper/1606.08266/) (2016)
- [Magnetic eigenmaps for community detection in directed networks](https://mlsanity.com/paper/1606.07359/) (2016)
- [A physical model for efficient ranking in networks](https://mlsanity.com/paper/1709.09002/) (2017)
- [Complex networks with complex weights](https://mlsanity.com/paper/2212.06257/) (2022)
- [Characterization and comparison of large directed graphs through the spectra of the magnetic Laplacian](https://mlsanity.com/paper/2007.03466/) (2020)
- [Go with the Flow? A Large-Scale Analysis of Health Care Delivery Networks in the United States Using Hodge Theory](https://mlsanity.com/paper/2110.09637/) (2021)
- [On Spectral Analysis of Directed Signed Graphs](https://mlsanity.com/paper/1612.08102/) (2016)
- [Directed Network Laplacians and Random Graph Models](https://mlsanity.com/paper/2107.03026/) (2021)
- [Unraveling the graph structure of tabular data through Bayesian and spectral analysis](https://mlsanity.com/paper/2110.01421/) (2021)

## Abstract (arXiv)

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.