# Magnetic eigenmaps for community detection in directed networks

arXiv: https://arxiv.org/abs/1606.07359 · HTML: https://mlsanity.com/paper/1606.07359/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Jun 2016.

## Built on by (most cited first, 22 of 22)

- [MagNet: A Neural Network for Directed Graphs](https://mlsanity.com/paper/2102.11391/) (2021)
- [A Survey of Community Detection Approaches: From Statistical Modeling to Deep Learning](https://mlsanity.com/paper/2101.01669/) (2021)
- [Transformers Meet Directed Graphs](https://mlsanity.com/paper/2302.00049/) (2023)
- [MSGNN: A Spectral Graph Neural Network Based on a Novel Magnetic Signed Laplacian](https://mlsanity.com/paper/2209.00546/) (2022)
- [SigMaNet: One Laplacian to Rule Them All](https://mlsanity.com/paper/2205.13459/) (2022)
- [Geometric Scattering on Measure Spaces](https://mlsanity.com/paper/2208.08561/) (2022)
- [A Magnetic Framelet-Based Convolutional Neural Network for Directed Graphs](https://mlsanity.com/paper/2210.10993/) (2022)
- [LightDiC: A Simple yet Effective Approach for Large-scale Digraph Representation Learning](https://mlsanity.com/paper/2401.11772/) (2024)
- [Signed Graph Neural Networks: A Frequency Perspective](https://mlsanity.com/paper/2208.07323/) (2022)
- [What Are Good Positional Encodings for Directed Graphs?](https://mlsanity.com/paper/2407.20912/) (2024)
- [Finsler Multi-Dimensional Scaling: Manifold Learning for Asymmetric Dimensionality Reduction and Embedding](https://mlsanity.com/paper/2503.18010/) (2025)
- [MGC: A Complex-Valued Graph Convolutional Network for Directed Graphs](https://mlsanity.com/paper/2110.07570/) (2021)
- [Graph Learning in 4D: a Quaternion-valued Laplacian to Enhance Spectral GCNs](https://mlsanity.com/paper/2312.17361/) (2023)
- [Structural Balance and Random Walks on Complex Networks with Complex Weights](https://mlsanity.com/paper/2307.01813/) (2023)
- [HyperMagNet: A Magnetic Laplacian based Hypergraph Neural Network](https://mlsanity.com/paper/2402.09676/) (2024)
- [A Graph Convolution for Signed Directed Graphs](https://mlsanity.com/paper/2208.11511/) (2022)
- [Toward Effective Digraph Representation Learning: A Magnetic Adaptive Propagation based Approach](https://mlsanity.com/paper/2501.11817/) (2025)
- [A Benchmark on Directed Graph Representation Learning in Hardware Designs](https://mlsanity.com/paper/2410.06460/) (2024)
- [Spectral Clustering for Directed Graphs via Likelihood Estimation on Stochastic Block Models](https://mlsanity.com/paper/2403.19516/) (2024)
- [Beyond symmetrization: effective adjacency matrices and renormalization for (un)singed directed graphs](https://mlsanity.com/paper/2406.01517/) (2024)
- [Converting Transformers into DGNNs Form](https://mlsanity.com/paper/2502.00585/) (2025)
- [ChemHyperMag: Physics-informed magnetic hypergraph learning improves molecular ADMET prediction](https://mlsanity.com/paper/2607.18332/) (2026)

## Abstract (arXiv)

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, that 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.