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When Graph Neural Networks Meet Dynamic Mode Decomposition
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Connects GNN feature propagation to Koopman theory and Dynamic Mode Decomposition (DMD), proposing DMD-GNNs that use low-rank DMD eigenmodes as spectral filters for node classification, long-range, spatio-temporal and link prediction tasks.
Problem
GNN feature propagation is viewed as diffusion dynamics, but the dynamical-systems tools for analyzing these dynamics (Koopman operator, DMD) have not been exploited to design GNNs, particularly to capture nonlinear node interactions with a low-rank linear operator.
Approach
Apply DMD to snapshots of GNN feature dynamics (e.g., GCN, SGC, ACMP-style diffusion-reaction) to estimate a low-rank operator whose DMD modes and eigenvalues serve as a spectral filter, yielding DMD-GNNs; theoretically relate DMD outputs to a slow spectral submanifold of the underlying dynamics.
Key results
- DMD-SGC reaches 84.1% on Cora and DMD++ reaches 92.6% on Texas and 91.9% on Wisconsin in node classification, among the best in Table 1.
- On COCO-SP, DMD++ achieves 0.091 macro-F1 versus 0.079 for GCN, while on PascalVOC-SP DMD-GCN is best at 0.243 (roughly tied with GCN's 0.238 baseline range).
- DMD-GNNs reach best or near-best MSE on Chickenpox (DMD++ 0.910) and Covid (DMD-ACMP 0.760), and DMD-GNNs use fewer than 40 spectral parameters on heterophilic graphs.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use DMD-enhanced GNNs instead of the strongest reported baseline | Node classification · Cora, Pubmed, OGB-arXiv | accuracy: 83.8±0.9 -> 84.1±0.4; 79.8±0.8 -> 81.2±1.5; 74.1±2.3 -> 75.5±0.8 | Helped |
| Use DMD++ instead of the strongest reported baseline | Node classification · Texas, Wisconsin, Cornell | accuracy: 92.3±0.7 -> 92.6±3.4; 91.8±3.1 -> 91.9±2.6; 91.3±0.7 -> 91.4±1.7 | No clear effect |
| Use DMD-GNNs instead of the strongest reported baseline | Node classification · Citeseer | accuracy: 75.9±0.6 -> 73.2±0.4 | Hurt |
| Use DMD++ instead of GCN | Long-range node classification · COCO-SP | Macro-F1: 0.079±0.025 -> 0.091±0.009 | Helped |
| Use DMD-GCN instead of GCN | Long-range node classification · PascalVOC-SP | Macro-F1: 0.238±0.016 -> 0.243±0.014 | Helped |
| Use DMD-GCN instead of the strongest reported baseline | Spatial-temporal prediction · Covid | MSE: 0.861(±0.045) -> 0.760(±0.025) | Helped |
| Use DMD++ instead of the strongest reported baseline | Spatial-temporal prediction · Chickenpox | MSE: 0.916(±0.014) -> 0.910(±0.002) | Helped |
| Use DMD++ instead of the strongest reported baseline | Spatial-temporal prediction · WikiMath | MSE: 0.826 (±0.070) -> 0.805 (±0.042) | Helped |
| Use a simpler initial dynamic (GCN) rather than other DMD-GNN initial dynamics | DMD-GNN link prediction encoder · Cora, Citeseer, Chameleon, Squirrel | accuracy | Mixed |
| Use DMD-GCN rather than the strongest reported baseline | Link prediction encoder · Squirrel | accuracy: 75.3±0.3 -> 78.8±0.4 | Helped |
| Use DMD-GCN rather than the strongest reported baseline | Link prediction encoder · Chameleon | accuracy: 82.1±0.4 -> 76.1±0.4 | Hurt |
| Use DMD-SGC rather than DMD-GCN as the initial dynamic (ablation) | DMD-GNN node classifier · Homophilic graphs | accuracy | Mixed |
Surprising findings
- MLP performs comparably to GNNs on link prediction, and simpler initial dynamics (GCN) give better link prediction accuracy for DMD-GNNs.
- On heterophilic graphs DMD-GNNs need fewer than 40 learnable spectral parameters.
Limitations
- Gains on PascalVOC-SP are limited, attributed to few free parameters when feature dimension is below class count.
- Hyperparameters such as truncation rate and ACMP B matrix require dataset-dependent choices, and DMD-ACMP was excluded from LRGB experiments due to hyperparameter variance.
- Theoretical result is informal and relies on strong assumptions (symmetric operator, fixed point at origin, rank conditions).
Experimental setup
Datasets: Cora Citeseer Pubmed Texas Wisconsin Cornell OGB-arXiv PascalVOC-SP COCO-SP Chameleon Squirrel Computer
Benchmarks: Cora Citeseer Pubmed Texas Wisconsin Cornell OGB-arXiv PascalVOC-SP COCO-SP Chickenpox Covid WikiMath
Metrics: accuracy macro F1 MSE
Baselines: MLP GCN GAT GPRGNN H2GCN GraphSAGE GRAND++ Graph Framelets APPNP ChebNet
Builds on: Dynamic Mode Decomposition Koopman operator theory APPNP GRAND++ ACMP
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