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Signed Graph Neural Ordinary Differential Equation for Modeling Continuous-Time Dynamics
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Proposes SGODE, a signed-graph neural ODE that learns a signed coefficient matrix K (positive minus negative relations) plus a self-trend term, and plugs it into NDCN, STG-NCDE and DCRNN for continuous dynamics and traffic forecasting.
Problem
Existing GNN-ODE models for continuous-time dynamics use unsigned graphs and ignore signed (positive/negative) inter-node relations, and learned graph structures are hard to optimize, overfit, or are costly with ODE solvers.
Approach
Replace the normalized adjacency in GNN-ODEs with a learnable signed coefficient matrix K = sigma(E1E2^T) - sigma(E3E4^T) built from node embeddings, plus a self-trend term B(t) that stabilizes the ODE dynamics; the same design is embedded into NDCN, STG-NCDE and DCRNN.
Key results
- On METR-LA 15min, SGODE-RNN achieves MAE 2.54 and RMSE 4.74, versus MegaCRN 2.60 / 4.81.
- On PeMSD8, SGODE-RNN reaches MAE 14.55 versus GTS 14.82 and MegaCRN 14.91.
- On synthetic interpolation (Table of MAPE), SGODEv2 gets the best MAPE on most graphs, e.g. 1.7% on the Community graph for heat diffusion versus NDCN 3.7%.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| replace DCRNN with SGODE-RNN | DCRNN · METR-LA 15min | MAE: 2.60 -> 2.54 | Helped |
| replace DCRNN with SGODE-RNN | DCRNN · PEMS-BAY 15min | MAE: 1.30 -> 1.29 | Helped |
| replace DCRNN with SGODE-RNN | DCRNN · PeMSD4 | MAE: 18.92 -> 18.81 | Helped |
| replace DCRNN with SGODE-RNN | DCRNN · PeMSD8 | MAE: 14.82 -> 14.55 | Helped |
| add intermediate and initial-state feature extraction (ablation) | SGODE-RNN · PeMSD4 | MAE: 19.32 -> 18.81 | Helped |
| add intermediate and initial-state feature extraction (ablation) | SGODE-RNN · PeMSD8 | MAE: 15.07 -> 14.55 | Helped |
| add intermediate and initial-state feature extraction (ablation) | SGODE-RNN · METR-LA | MAE: 2.92 -> 2.88 | Helped |
| add intermediate and initial-state feature extraction (ablation) | SGODE-RNN · PEMS-BAY | MAE: 1.57 -> 1.55 | Helped |
| add trend B (ablation) | SGODE-RNN · PeMSD4 | MAE: 19.01 -> 18.81 | Helped |
| add trend B (ablation) | SGODE-RNN · PeMSD8 | MAE: 15.18 -> 14.55 | Helped |
| add trend B (ablation) | SGODE-RNN · METR-LA | MAE: 2.95 -> 2.88 | Helped |
| add trend B (ablation) | SGODE-RNN · PEMS-BAY | MAE: 1.58 -> 1.55 | Helped |
Surprising findings
- On PeMSD4, the MAE, MAPE and RMSE rankings of GTS and SGODE-RNN disagree, which the authors attribute to near-zero values in the data.
- NDCN with the true ground-truth graph is marginally better than SGODE variants on some synthetic dynamics, which the authors attribute to the data being generated from the true graphs.
Limitations
- For large graphs, learning the signed graph is difficult and may overfit to local optima (stated in conclusions).
- The authors note that the SGODE gain over NDCN on some synthetic dynamics is marginal and that the benefit of signed information on PEMS-BAY is small.
Experimental setup
Datasets: METR-LA PEMS-BAY PeMSD4 PeMSD8 synthetic heat diffusion, mutualistic interaction, gene regulatory dynamics on grid/random/power-law/small-world/communi
Benchmarks: METR-LA PEMS-BAY PeMSD4 PeMSD8 synthetic continuous-time dynamics
Metrics: MAE RMSE MAPE
Baselines: NDCN No-graph Adaptive-NDCN GTS-NDCN HA ARIMA VAR SVR FNN LSTM DCRNN GraphWaveNet
Builds on: NDCN STG-NCDE DCRNN GTS
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