# SigMaNet: One Laplacian to Rule Them All

arXiv: https://arxiv.org/abs/2205.13459 · HTML: https://mlsanity.com/paper/2205.13459/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 53 references that are in the map.
- First posted on arXiv 26 May 2022.

## Builds on (most cited first, 18 of 18)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [A Comprehensive Survey on Graph Neural Networks](https://mlsanity.com/paper/1901.00596/) (2019)
- [Dynamic Graph CNN for Learning on Point Clouds](https://mlsanity.com/paper/1801.07829/) (2018)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [Predict then Propagate: Graph Neural Networks meet Personalized PageRank](https://mlsanity.com/paper/1810.05997/) (2018)
- [MagNet: A Neural Network for Directed Graphs](https://mlsanity.com/paper/2102.11391/) (2021)
- [Directed Graph Convolutional Network](https://mlsanity.com/paper/2004.13970/) (2020)
- [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)
- [Discrete Signal Processing on Graphs: Sampling Theory](https://mlsanity.com/paper/1503.05432/) (2015)
- [Exploiting Social Network Structure for Person-to-Person Sentiment Analysis](https://mlsanity.com/paper/1409.2450/) (2014)
- [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)
- [SSSNET: Semi-Supervised Signed Network Clustering](https://mlsanity.com/paper/2110.06623/) (2021)
- [Distributed Representations of Signed Networks](https://mlsanity.com/paper/1702.06819/) (2017)

## Built on by (most cited first, 24 of 24)

- [DiffAssemble: A Unified Graph-Diffusion Model for 2D and 3D Reassembly](https://mlsanity.com/paper/2402.19302/) (2024)
- [MSGNN: A Spectral Graph Neural Network Based on a Novel Magnetic Signed Laplacian](https://mlsanity.com/paper/2209.00546/) (2022)
- [Geometric Scattering on Measure Spaces](https://mlsanity.com/paper/2208.08561/) (2022)
- [What Are Good Positional Encodings for Directed Graphs?](https://mlsanity.com/paper/2407.20912/) (2024)
- [Commute Graph Neural Networks](https://mlsanity.com/paper/2407.01635/) (2024)
- [Signed Directed Graph Contrastive Learning with Laplacian Augmentation](https://mlsanity.com/paper/2301.05163/) (2023)
- [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)
- [Sheaves Reloaded: A Directional Awakening](https://mlsanity.com/paper/2506.02842/) (2025)
- [HyperMagNet: A Magnetic Laplacian based Hypergraph Neural Network](https://mlsanity.com/paper/2402.09676/) (2024)
- [GegenNet: Spectral Convolutional Neural Networks for Link Sign Prediction in Signed Bipartite Graphs](https://mlsanity.com/paper/2508.19907/) (2025)
- [Learning Disentangled Representations in Signed Directed Graphs without Social Assumptions](https://mlsanity.com/paper/2307.03077/) (2023)
- [Trustworthiness-Driven Graph Convolutional Networks for Signed Network Embedding](https://mlsanity.com/paper/2309.00816/) (2023)
- [A Benchmark on Directed Graph Representation Learning in Hardware Designs](https://mlsanity.com/paper/2410.06460/) (2024)
- [DLGNet: Hyperedge Classification through Directed Line Graphs for Chemical Reactions](https://mlsanity.com/paper/2410.06969/) (2024)
- [Maps from Motion (MfM): Generating 2D Semantic Maps from Sparse Multi-view Images](https://mlsanity.com/paper/2411.12620/) (2024)
- [Haar-Laplacian for directed graphs](https://mlsanity.com/paper/2411.15527/) (2024)
- [LineMVGNN: Anti-Money Laundering with Line-Graph-Assisted Multi-View Graph Neural Networks](https://mlsanity.com/paper/2603.23584/) (2026)
- [Nonlinear Laplacians Improve Signed-Directed Graph Learning](https://mlsanity.com/paper/2608.00836/) (2026)
- [Rethinking Link Prediction for Directed Graphs](https://mlsanity.com/paper/2502.05724/) (2025)
- [Directed Semi-Simplicial Learning with Applications to Brain Activity Decoding](https://mlsanity.com/paper/2505.17939/) (2025)
- [Geometry-Induced Diffusion on Graphs: A Learnable Weighted Laplacian for Spectral GNNs](https://mlsanity.com/paper/2602.18141/) (2026)
- [Signed Graph Pre-Training and Prompt Learning](https://mlsanity.com/paper/2609.25722/) (2026)
- [PANDA: Prior-guided Attentional Dual-path Architecture](https://mlsanity.com/paper/2610.13635/) (2026)

## Abstract (arXiv)

This paper introduces SigMaNet, a generalized Graph Convolutional Network (GCN) capable of handling both undirected and directed graphs with weights not restricted in sign nor magnitude. The cornerstone of SigMaNet is the Sign-Magnetic Laplacian (Lσ), a new Laplacian matrix that we introduce ex novo in this work. Lσ allows us to bridge a gap in the current literature by extending the theory of spectral GCNs to (directed) graphs with both positive and negative weights. Lσ exhibits several desirable properties not enjoyed by other Laplacian matrices on which several state-of-the-art architectures are based, among which encoding the edge direction and weight in a clear and natural way that is not negatively affected by the weight magnitude. Lσ is also completely parameter-free, which is not the case of other Laplacian operators such as, e.g., the Magnetic Laplacian. The versatility and the performance of our proposed approach is amply demonstrated via computational experiments. Indeed, our results show that, for at least a metric, SigMaNet achieves the best performance in 15 out of 21 cases and either the first- or second-best performance in 21 cases out of 21, even when compared to architectures that are either more complex or that, due to being designed for a narrower class of graphs, should -- but do not -- achieve a better performance.