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Permutation Equivariant Neural Controlled Differential Equations for Dynamic Graph Representation Learning
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Proposes PENG-CDE, a permutation-equivariant Graph Neural CDE whose adjacency fusion is projected onto the 15-dimensional space of linear equivariant maps, improving interpolation/extrapolation on synthetic dynamics and achieving SOTA on TGB tgbn-genre.
Problem
Graph Neural CDEs for dynamic graphs break permutation equivariance in their adjacency fusion step, adding unnecessary parameters and limiting generalisation on dynamic graph tasks.
Approach
Project the learnable adjacency-fusion maps of Graph Neural CDEs onto the space of permutation-equivariant linear maps on edge-valued data (15 basis terms), yielding a model that is permutation equivariant and time-warp equivariant, and is the projection of the original model in the linear case.
Key results
- On tgbn-genre, PENG-CDE achieves NDCG@10 of 0.523 ± 0.017, the best among learned models and above the heuristic Moving Avg (L) baseline at 0.509.
- On heat diffusion, PENG-CDE reports relative MSE improvements over the original GN-CDE of 30.44% to 73.84% across the four graph distributions.
- On twitter-tennis test MSE, PENG-CDE achieves 0.440 ± 0.053, the lowest among compared models; on england-covid test MSE it achieves 0.913 ± 0.200.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use the 15-term permutation-equivariant fusion instead of the original GN-CDE fusion | PENG-CDE vs Original GN-CDE · Heat diffusion, Community | MSE: 73.84% | Helped |
| use the 15-term permutation-equivariant fusion instead of the original GN-CDE fusion | PENG-CDE vs Original GN-CDE · Heat diffusion, Grid | MSE: 63.70% | Helped |
| use the 15-term permutation-equivariant fusion instead of the original GN-CDE fusion | PENG-CDE vs Original GN-CDE · Heat diffusion, Power Law | MSE: 30.44% | Helped |
| use the 15-term permutation-equivariant fusion instead of the original GN-CDE fusion | PENG-CDE vs Original GN-CDE · Heat diffusion, Small World | MSE: 55.20% | Helped |
| use the 15-term permutation-equivariant fusion instead of the original GN-CDE fusion | PENG-CDE vs Original GN-CDE · Gene regulation, Community | MSE: 67.06% | Helped |
| use the 15-term permutation-equivariant fusion instead of the original GN-CDE fusion | PENG-CDE vs Original GN-CDE · Gene regulation, Grid | MSE: 45.90% | Helped |
| use the 15-term permutation-equivariant fusion instead of the original GN-CDE fusion | PENG-CDE vs Original GN-CDE · Gene regulation, Power Law | MSE: 39.71% | Helped |
| use the 15-term permutation-equivariant fusion instead of the original GN-CDE fusion | PENG-CDE vs Original GN-CDE · Gene regulation, Small World | MSE: 54.14% | Helped |
| add permutation equivariance to GN-CDE fusion | GN-CDE variants · Synthetic heat diffusion and gene regulation tasks | MSE | Helped |
| add source/target identification (ablation) | PENG-CDE · tgbn-trade | NDCG@10: 0.716 -> 0.734 | Helped |
| add source/target identification (ablation) | PENG-CDE · tgbn-genre | NDCG@10 | No clear effect |
| use PENG-CDE instead of GN-CDE | Dynamic snapshot node regression · england-covid test | MSE: 0.962 ± 0.278 -> 0.913 ± 0.200 | Helped |
Surprising findings
- Pre-multiplication fusion GN-CDE (non-equivariant) performs an order of magnitude worse than equivariant variants on synthetic tasks.
- On tgbn-trade, PENG-CDE is outperformed by TGNv2 and only matches it when source/target identification is added.
Limitations
- Quadratic memory complexity in number of nodes due to dense adjacency matrices; JAX lacks good sparse support.
- No theoretical guarantees for graph Neural CDEs (universality, generalisation) yet.
- Projection optimality proven only in the purely linear (no nonlinearity) case.
Experimental setup
Datasets: heat diffusion (synthetic) gene regulation (synthetic) wealth dynamics (synthetic) opinion dynamics (synthetic) SIR model (synthetic) england-covid twitter-tennis tgbn-trade tgbn-genre
Benchmarks: heat diffusion gene regulation england-covid twitter-tennis tgbn-trade tgbn-genre SIR classification
Metrics: MSE NDCG@10 classification accuracy
Baselines: DCRNN STIDGCN ASTGCN STG-NCDE Graph Neural ODE GN-CDE Pre Mult GN-CDE Adjacency GN-CDE Constant JODIE TGAT CAWN
Builds on: Graph Neural CDE Neural CDE Log-NCDE Neural ODE Equivariant linear maps on tensors (Maron et al.)
Closest papers by meaning
- CSG-ODE: ControlSynth Graph ODE For Modeling Complex Evolution of Dynamic Graphs (ICML 2025 · similarity 0.89)
- SEGNO: Generalizing Equivariant Graph Neural Networks with Physical Inductive Biases (ICLR 2024 · similarity 0.88)
- Hyperbolic-PDE GNN: Spectral Graph Neural Networks in the Perspective of A System of Hyperbolic Partial Differential Equations (ICML 2025 · similarity 0.87)
- Uncertainty Modeling in Graph Neural Networks via Stochastic Differential Equations (ICLR 2025 · similarity 0.86)
- Signed Graph Neural Ordinary Differential Equation for Modeling Continuous-Time Dynamics (AAAI 2024 · similarity 0.86)
- A Characterization Theorem for Equivariant Networks with Point-wise Activations (ICLR 2024 · similarity 0.86)
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