# Graph Learning in 4D: a Quaternion-valued Laplacian to Enhance Spectral GCNs

arXiv: https://arxiv.org/abs/2312.17361 · HTML: https://mlsanity.com/paper/2312.17361/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 44 references that are in the map.
- First posted on arXiv 28 Dec 2023.

## Builds on (most cited first, 22 of 22)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [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)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Predict then Propagate: Graph Neural Networks meet Personalized PageRank](https://mlsanity.com/paper/1810.05997/) (2018)
- [Gated Graph Sequence Neural Networks](https://mlsanity.com/paper/1511.05493/) (2015)
- [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)
- [Quaternion Recurrent Neural Networks](https://mlsanity.com/paper/1806.04418/) (2018)
- [MSGNN: A Spectral Graph Neural Network Based on a Novel Magnetic Signed Laplacian](https://mlsanity.com/paper/2209.00546/) (2022)
- [Exploiting Social Network Structure for Person-to-Person Sentiment Analysis](https://mlsanity.com/paper/1409.2450/) (2014)
- [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)
- [PyTorch Geometric Signed Directed: A Software Package on Graph Neural Networks for Signed and Directed Graphs](https://mlsanity.com/paper/2202.10793/) (2022)
- [Quaternion Graph Neural Networks](https://mlsanity.com/paper/2008.05089/) (2020)
- [A Graph Convolution for Signed Directed Graphs](https://mlsanity.com/paper/2208.11511/) (2022)

## Built on by (most cited first, 4 of 4)

- [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)
- [ScaleNet: Scale Invariance Learning in Directed Graphs](https://mlsanity.com/paper/2411.08758/) (2024)

## Abstract (arXiv)

We introduce QuaterGCN, a spectral Graph Convolutional Network (GCN) with quaternion-valued weights at whose core lies the Quaternionic Laplacian, a quaternion-valued Laplacian matrix by whose proposal we generalize two widely-used Laplacian matrices: the classical Laplacian (defined for undirected graphs) and the complex-valued Sign-Magnetic Laplacian (proposed to handle digraphs with weights of arbitrary sign). In addition to its generality, our Quaternionic Laplacian is the only Laplacian to completely preserve the topology of a digraph, as it can handle graphs and digraphs containing antiparallel pairs of edges (digons) of different weights without reducing them to a single (directed or undirected) edge as done with other Laplacians. Experimental results show the superior performance of QuaterGCN compared to other state-of-the-art GCNs, particularly in scenarios where the information the digons carry is crucial to successfully address the task at hand.