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Spatio-Spectral Graph Neural Networks
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Proposes Spatio-Spectral GNNs (S²GNNs) that combine spatial message passing with spectrally parametrized filters on a partial eigendecomposition, giving global receptive fields, tighter approximation bounds, free expressive positional encodings, and directed-graph support.
Problem
Spatial MPGNNs have receptive fields limited to the l-hop neighborhood and suffer from over-squashing, limiting long-range information exchange.
Approach
Augment spatial message passing with spectral filters parametrized directly in the Laplacian eigenvalue domain (Gaussian smearing plus linear map, windowed, on the lowest-k eigenpairs), with a permutation-equivariant spectral neural network, a Hermitian magnetic Laplacian for directed graphs, and eigenvector-based positional encodings reusing the same EVD.
Key results
- On peptides-func, S²GCN+PE reaches 0.7311 AP versus 0.7156 for the best prior graph transformer (GPS+HDSE) with about 40% fewer parameters.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| add spectral filters to GCN | GCN · peptides-func | AP: 0.6860\pm0.0050 -> 0.7275\pm0.0066 | Helped |
| add positional encodings (ablation) | S² GCN · peptides-func | AP: 0.7275\pm0.0066 -> 0.7311\boldsymbol{\pm}0.0066 | Helped |
| add positional encodings (ablation) | S² GCN · peptides-struct | MAE: 0.2467\pm0.0019 -> 0.2447\boldsymbol{\pm}0.0032 | Helped |
| add spectral filters to GAT | GAT · OGB Products test | Accuracy: 0.798\pm0.003 -> 0.811\boldsymbol{\pm}0.007 | Helped |
| add spectral filters to GAT | GAT · OGB Products test | F1: 0.347\pm0.004 -> 0.381\pm0.009 | Helped |
| add spectral filters to GCN | GCN · TPUGraphs layout | Kendall tau: 60.09 -> 65.74 | Helped |
| use more than one eigenvector instead of a single eigenvector (ablation) | S²GNN · long-range tasks | performance | Helped |
| add positional encodings to node features (ablation) | S²GNN · LR-CLUSTER | performance | Helped |
| use a directed-graph spectral filter (ablation) | S² GCN · associative recall | generalization | Helped |
| add spectral filters to spatial MPGNNs | spatial MPGNNs · peptides-func and peptides-struct | performance | Helped |
| use a virtual node alone instead of richer spectral filters (ablation) | S²GNN · LR-CLUSTER | long-range modeling | Hurt |
Surprising findings
- A single virtual node (k=1 spectral filter) is insufficient for some long-range tasks; using more eigenvectors yields substantial gains.
- The authors report that S²GNNs are data-hungry.
Limitations
- Data-hungry models according to the authors
- Injectivity of the magnetic Laplacian is only shown for binary adjacency; real-valued weights left for future work
- Choice of lambda_cut and k relies on fixed hyperparameters; adaptive strategies did not yield superior results
Experimental setup
Datasets: peptides-func peptides-struct CLUSTER LR-CLUSTER OGB Products TPUGraphs arXiv-year associative recall
Benchmarks: peptides-func peptides-struct LR-CLUSTER OGB Products TPUGraphs arXiv-year associative recall
Metrics: AP MAE accuracy F1 Kendall tau
Baselines: GCN GAT GatedGCN GRIT GPS+HDSE Graph Mamba DRew-GCN Hyena H3 Transformer TIGT MGT+WPE
Builds on: GCN Magnetic Laplacian Hyena virtual node Graph Transformer positional encodings
Closest papers by meaning
- Designing Specialized Two-Dimensional Graph Spectral Filters for Spatial-Temporal Graph Modeling (AAAI 2025 · similarity 0.92)
- SLOG: An Inductive Spectral Graph Neural Network Beyond Polynomial Filter (ICML 2024 · similarity 0.90)
- Graph Neural Networks Are More Than Filters: Revisiting and Benchmarking from A Spectral Perspective (ICLR 2025 · similarity 0.90)
- HoloNets: Spectral Convolutions do extend to Directed Graphs (ICLR 2024 · similarity 0.89)
- Over-squashing in Spatiotemporal Graph Neural Networks (NeurIPS 2025 · similarity 0.89)
- Accurate and Scalable Graph Neural Networks via Message Invariance (ICLR 2025 · similarity 0.88)
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