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Graph Learning in 4D: A Quaternion-Valued Laplacian to Enhance Spectral GCNs
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Proposes the Quaternionic Laplacian, a quaternion-valued graph Laplacian that preserves digon topology with arbitrary weights and signs, and QuaterGCN, a spectral GCN built on it with quaternion weights, evaluated on directed/signed graph node and edge prediction tasks.
Problem
Spectral GCNs cannot represent digons (antiparallel edge pairs) with asymmetric weights and arbitrary signs on digraphs without collapsing them into single edges, losing topological information.
Approach
A quaternion-valued Hermitian Laplacian whose imaginary components encode asymmetric digons, generalizing the classical Laplacian and the Sign-Magnetic Laplacian, used in a spectral GCN with quaternion-valued weights and Chebyshev order-1 convolution.
Key results
- QuaterGCN is best in 9 of 10 node classification settings, with an average gain of 68.27% over the second-best model.
- On three-class edge prediction, QuaterGCN outperforms other methods on 7 of 9 datasets, with an average gain of 0.98% over the second-best.
- On four/five-class edge prediction over signed datasets, QuaterGCN is superior in 8 of 10 cases, with an average improvement of 1.14% over the second-best.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the Quaternionic Laplacian in place of the classical real-valued Laplacian | QuaterGCN versus QGNN · Node classification across reported datasets | accuracy: 309% | Helped |
| Use the Quaternionic Laplacian in place of the classical real-valued Laplacian | QuaterGCN versus QGNN · 3CEP across reported datasets | accuracy: 25.58% | Helped |
| Use QuaterGCN rather than the strongest competing model | Node classification · All node-classification datasets | accuracy: 68.27% | Helped |
| Use QuaterGCN rather than the strongest competing model | Node classification · Di500 | accuracy: 175% | Helped |
| Use QuaterGCN rather than the strongest competing model | Node classification · Di150 | accuracy: 28.19% | Helped |
| Use QuaterGCN rather than the strongest competing model | 3CEP · Reported 3CEP datasets | accuracy: 0.98% | Helped |
| Use QuaterGCN rather than the strongest competing model | 4/5CEP · Reported 4CEP and 5CEP datasets | accuracy: 1.14% | Helped |
| Use QuaterGCN rather than the strongest competing model | 3CEP · Di500, δ = 0.2 | accuracy | No clear effect |
| Increase digon percentage δ | simpler methods designed for undirected graphs · Di150 3CEP | accuracy | Helped |
| Represent asymmetric digons with the Quaternionic Laplacian rather than collapsing them to a single edge | QuaterGCN · Graphs containing asymmetric digons | accuracy | Helped |
Surprising findings
- Simpler undirected-graph methods improve relative to digraph methods as the fraction of digons increases on Di150 when digon weight differences are small.
- On the 3CEP task, every method performs close to random on Di500, so those results were omitted from the main text.
Limitations
- Quaternion operations are costlier (up to 64x per scalar product) than real or complex ones.
- Gains on 3CEP and 4/5CEP are small (around 1%), and the paper notes smaller magnitude differences on edge tasks.
- Extension to multi-relational and temporal graphs is left for future work.
Experimental setup
Datasets: Telegram Bitcoin-Alpha Bitcoin-OTC WikiRfa Slashdot Epinions DBSM Di150 Di500
Benchmarks: Telegram Bitcoin-Alpha Bitcoin-OTC WikiRfa Slashdot Epinions DBSM Di150 Di500
Metrics: accuracy
Baselines: ChebNet GCN QGNN APPNP SAGE GIN GAT SSSNET DGCN DiGraph DiGCL MagNet
Builds on: ChebNet MagNet SigMaNet QGNN
Closest papers by meaning
- SigMaNet: One Laplacian to Rule Them All (AAAI 2023 · similarity 0.91)
- Position: Spectral GNNs Rely Less on Graph Fourier Basis than Conceived (ICML 2025 · similarity 0.87)
- Generalizing CNNs to graphs with learnable neighborhood quantization (NeurIPS 2024 · similarity 0.87)
- Enhancing Spectral GNNs: From Topology and Perturbation Perspectives (ICML 2025 · similarity 0.87)
- Asymmetric Learning for Spectral Graph Neural Networks (AAAI 2025 · similarity 0.86)
- HoloNets: Spectral Convolutions do extend to Directed Graphs (ICLR 2024 · similarity 0.86)
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