# MSGNN: A Spectral Graph Neural Network Based on a Novel Magnetic Signed Laplacian

arXiv: https://arxiv.org/abs/2209.00546 · HTML: https://mlsanity.com/paper/2209.00546/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 50 references that are in the map.
- First posted on arXiv 1 Sep 2022.

## Builds on (most cited first, 26 of 26)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [Modeling Relational Data with Graph Convolutional Networks](https://mlsanity.com/paper/1703.06103/) (2017)
- [Spectral Networks and Locally Connected Networks on Graphs](https://mlsanity.com/paper/1312.6203/) (2013)
- [Composition-based Multi-Relational Graph Convolutional Networks](https://mlsanity.com/paper/1911.03082/) (2019)
- [PyTorch Geometric Temporal: Spatiotemporal Signal Processing with Neural Machine Learning Models](https://mlsanity.com/paper/2104.07788/) (2021)
- [MagNet: A Neural Network for Directed Graphs](https://mlsanity.com/paper/2102.11391/) (2021)
- [Transferability of Spectral Graph Convolutional Neural Networks](https://mlsanity.com/paper/1907.12972/) (2019)
- [Directed Graph Convolutional Network](https://mlsanity.com/paper/2004.13970/) (2020)
- [MotifNet: a motif-based Graph Convolutional Network for directed graphs](https://mlsanity.com/paper/1802.01572/) (2018)
- [Signed Graph Attention Networks](https://mlsanity.com/paper/1906.10958/) (2019)
- [SDGNN: Learning Node Representation for Signed Directed Networks](https://mlsanity.com/paper/2101.02390/) (2021)
- [Spectral-based Graph Convolutional Network for Directed Graphs](https://mlsanity.com/paper/1907.08990/) (2019)
- [Magnetic Eigenmaps for the Visualization of Directed Networks](https://mlsanity.com/paper/1606.08266/) (2016)
- [SigMaNet: One Laplacian to Rule Them All](https://mlsanity.com/paper/2205.13459/) (2022)
- [Magnetic eigenmaps for community detection in directed networks](https://mlsanity.com/paper/1606.07359/) (2016)
- [SSSNET: Semi-Supervised Signed Network Clustering](https://mlsanity.com/paper/2110.06623/) (2021)
- [DIGRAC: Digraph Clustering Based on Flow Imbalance](https://mlsanity.com/paper/2106.05194/) (2021)
- [Clustering Signed Networks with the Geometric Mean of Laplacians](https://mlsanity.com/paper/1701.00757/) (2017)
- [Finding large balanced subgraphs in signed networks](https://mlsanity.com/paper/2002.00775/) (2020)
- [Signed Graph Neural Networks: A Frequency Perspective](https://mlsanity.com/paper/2208.07323/) (2022)
- [Regularized spectral methods for clustering signed networks](https://mlsanity.com/paper/2011.01737/) (2020)
- [Characterization and comparison of large directed graphs through the spectra of the magnetic Laplacian](https://mlsanity.com/paper/2007.03466/) (2020)
- [A Note on Markov Normalized Magnetic Eigenmaps](https://mlsanity.com/paper/1608.04418/) (2016)

## Built on by (most cited first, 28 of 28)

- [Transformers Meet Directed Graphs](https://mlsanity.com/paper/2302.00049/) (2023)
- [HoloNets: Spectral Convolutions do extend to Directed Graphs](https://mlsanity.com/paper/2310.02232/) (2023)
- [LightDiC: A Simple yet Effective Approach for Large-scale Digraph Representation Learning](https://mlsanity.com/paper/2401.11772/) (2024)
- [What Are Good Positional Encodings for Directed Graphs?](https://mlsanity.com/paper/2407.20912/) (2024)
- [Graph Learning in 4D: a Quaternion-valued Laplacian to Enhance Spectral GCNs](https://mlsanity.com/paper/2312.17361/) (2023)
- [Signed Graph Autoencoder for Explainable and Polarization-Aware Network Embeddings](https://mlsanity.com/paper/2409.10452/) (2024)
- [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)
- [Spectral GNN via Two-dimensional (2-D) Graph Convolution](https://mlsanity.com/paper/2404.04559/) (2024)
- [Improving Graph Neural Networks by Learning Continuous Edge Directions](https://mlsanity.com/paper/2410.14109/) (2024)
- [GegenNet: Spectral Convolutional Neural Networks for Link Sign Prediction in Signed Bipartite Graphs](https://mlsanity.com/paper/2508.19907/) (2025)
- [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)
- [Haar-Laplacian for directed graphs](https://mlsanity.com/paper/2411.15527/) (2024)
- [Gauge-Equivariant Graph Networks via Self-Interference Cancellation](https://mlsanity.com/paper/2511.16062/) (2025)
- [A Scalable Inter-edge Correlation Modeling in CopulaGNN for Link Sign Prediction](https://mlsanity.com/paper/2601.19175/) (2026)
- [Finsler Geometry, Graph Neural Networks, and You](https://mlsanity.com/paper/2606.17185/) (2026)
- [Nonlinear Laplacians Improve Signed-Directed Graph Learning](https://mlsanity.com/paper/2608.00836/) (2026)
- [ScaleNet: Scale Invariance Learning in Directed Graphs](https://mlsanity.com/paper/2411.08758/) (2024)
- [Toward Data-centric Directed Graph Learning: An Entropy-driven Approach](https://mlsanity.com/paper/2505.00983/) (2025)
- [Toward Robust Signed Graph Learning through Joint Input-Target Denoising](https://mlsanity.com/paper/2510.22513/) (2025)
- [Gene regulatory network inference algorithm based on spectral signed directed graph convolution](https://mlsanity.com/paper/2512.11927/) (2025)
- [Graph Learning Should Move Beyond Restrictive Views of Spectral and Message-Passing GNNs](https://mlsanity.com/paper/2602.10031/) (2026)
- [Gauge-Invariant, Parameter-Insensitive Regularization for Potential Recovery from Flow on Directed Graphs](https://mlsanity.com/paper/2607.13609/) (2026)
- [ChemHyperMag: Physics-informed magnetic hypergraph learning improves molecular ADMET prediction](https://mlsanity.com/paper/2607.18332/) (2026)
- [Can Graph Learning Learn Circuits?](https://mlsanity.com/paper/2608.08536/) (2026)
- [Recent Advances in Deep Learning-Based Drug-Target Binding Affinity Prediction](https://mlsanity.com/paper/2608.13797/) (2026)
- [Signed Graph Pre-Training and Prompt Learning](https://mlsanity.com/paper/2609.25722/) (2026)

## Abstract (arXiv)

Signed and directed networks are ubiquitous in real-world applications. However, there has been relatively little work proposing spectral graph neural networks (GNNs) for such networks. Here we introduce a signed directed Laplacian matrix, which we call the magnetic signed Laplacian, as a natural generalization of both the signed Laplacian on signed graphs and the magnetic Laplacian on directed graphs. We then use this matrix to construct a novel efficient spectral GNN architecture and conduct extensive experiments on both node clustering and link prediction tasks. In these experiments, we consider tasks related to signed information, tasks related to directional information, and tasks related to both signed and directional information. We demonstrate that our proposed spectral GNN is effective for incorporating both signed and directional information, and attains leading performance on a wide range of data sets. Additionally, we provide a novel synthetic network model, which we refer to as the Signed Directed Stochastic Block Model, and a number of novel real-world data sets based on lead-lag relationships in financial time series.