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HoloNets: Spectral Convolutions do extend to Directed Graphs
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Extends spectral graph convolutions to directed graphs by replacing the graph Fourier transform with the holomorphic functional calculus, yielding HoloNets (FaberNet for heterophilic node classification, Dir-ResolvNet for resolution-stable graph regression).
Problem
Spectral convolutional networks rely on a graph Fourier transform, which requires self-adjoint (normal) operators, so they cannot be defined on directed graphs with generic non-self-adjoint characteristic operators.
Approach
Define spectral filters as holomorphic functions of non-self-adjoint characteristic operators via the holomorphic functional calculus (Dunford-Taylor integral), with Faber polynomial or resolvent bases, and combine forward and backward filters in a HoloNet layer.
Key results
- FaberNet achieves 76.71% on Squirrel and 92.24% on Roman-Empire, new state of the art over Dir-GNN (75.13, 91.3) on the five heterophilic datasets.
- Dir-ResolvNet reaches 17.12 kcal/mol MAE on QM7 versus 39.95 for the best directed baseline (DiGCN).
- On coarse-grained QM7, Dir-ResolvNet achieves 27.34 kcal/mol MAE while DiGCN reaches 6672.71, a gap of up to about 240x over baselines.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use FaberNet's directed spectral convolutions for heterophilic node classification | Node classification · Squirrel | accuracy: 75.13 ± 1.95 -> 76.71 ± 1.92 | Helped |
| Use FaberNet's directed spectral convolutions for heterophilic node classification | Node classification · Chameleon | accuracy: 79.74 ± 1.40 -> 80.33 ± 1.19 | Helped |
| Use FaberNet's directed spectral convolutions for heterophilic node classification | Node classification · Arxiv-year | accuracy: 63.97 ± 0.30 -> 64.62 ± 1.01 | Helped |
| Use FaberNet's directed spectral convolutions for heterophilic node classification | Node classification · Snap-patents | accuracy: 73.95 ± 0.05 -> 75.10 ± 0.03 | Helped |
| Use FaberNet's directed spectral convolutions for heterophilic node classification | Node classification · Roman-Empire | accuracy: 91.3 ± 0.46 -> 92.24 ± 0.43 | Helped |
| Use Dir-ResolvNet instead of DiGCN | Directed molecular graph regression · QM7 | MAE (kcal/mol): 39.95 -> 17.12 | Helped |
| Use Dir-ResolvNet instead of DirGNN on coarse-grained graphs | Directed molecular graph regression · QM7 coarse | MAE (kcal/mol): 195.64 -> 27.34 | Helped |
| Use Dir-ResolvNet graph-level features as resolution scale increases | Dir-ResolvNet · QM7 hydrogen-deflection perturbation | feature-vector convergence | Helped |
| Use baseline models' graph-level features as resolution scale increases | Directed graph baselines · QM7 hydrogen-deflection perturbation | feature-vector convergence | Hurt |
Surprising findings
- Complex-valued weights help on Squirrel and Chameleon, but MagNet's complex weights alone do not explain the gap, suggesting the filter bank and operator matter more.
Limitations
- Dir-ResolvNet evaluated only on a single molecular dataset (QM7) with modified edge weights, so generality of the graph-regression results is limited.
- Stability theorems hold only in the limit of large weight scale c; empirical convergence is shown only numerically.
- Faber polynomial basis is only approximately minimax and requires a bounding circle around the spectrum, so it is not optimal in practice.
Experimental setup
Datasets: Squirrel Chameleon Arxiv-year Snap-patents Roman-Empire QM7
Benchmarks: Squirrel Chameleon Arxiv-year Snap-patents Roman-Empire QM7 QM7coarse
Metrics: accuracy mean absolute error
Baselines: MLP GCN H2GCN GPR-GNN LINKX FSGNN ACM-GCN GloGNN Gradient Gating DiGCN MagNet Dir-GNN
Builds on: ChebNet GPR-GNN Dir-GNN MagNet DiGCN
Closest papers by meaning
- Position: Spectral GNNs Rely Less on Graph Fourier Basis than Conceived (ICML 2025 · similarity 0.91)
- A General Graph Spectral Wavelet Convolution via Chebyshev Order Decomposition (ICML 2025 · similarity 0.90)
- Unifying Homophily and Heterophily for Spectral Graph Neural Networks via Triple Filter Ensembles (NeurIPS 2024 · similarity 0.90)
- 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 0.89)
- Spatio-Spectral Graph Neural Networks (NeurIPS 2024 · similarity 0.89)
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