Research map / Time series / Long-term forecasting
Time-Aware Knowledge Representations of Dynamic Objects with Multidimensional Persistence
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Proposes Temporal MultiPersistence (TMP), a zigzag-plus-multipersistence topological vectorization for time-dependent graphs, fed into a GNN (TMP-Nets) for traffic, Ethereum, and ECG forecasting.
Problem
Time-dependent objects such as dynamic networks and multivariate time series lack an efficient, time-aware topological representation; multiparameter persistence is computationally expensive and lacks a complete theory, and time-indexed graphs are not nested.
Approach
Use zigzag persistence along the time direction combined with sublevel filtrations along other functions to form a bifiltration, then vectorize each slice with existing single-parameter vectorizations (landscapes, silhouettes, Betti curves, persistence images) into a multidimensional TMP tensor with stability guarantees, which a CNN+GRU+adaptive-graph-convolution model consumes.
Key results
- On Bytom and Decentraland, TMP-Nets achieves the lowest MAPE (28.77% and 22.97%), beating the best baseline Z-GCNETs by 7.89% and 3.66% relative; it is worse on Golem (29.01% vs 22.32%).
- With limited data (first 1,000 PeMSD4 snapshots), TMP-Nets RMSE is 147.57 vs 150.22 for the next best, giving a reported 1.79% relative RMSE gain on PeMSD4.
- TMP computation is reported to be up to 59.5 times faster than MP-I and 1.2 to 8.6 times faster than MP-L.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use TMP-Nets instead of Z-GCNETs | Traffic forecasting · PeMSD4, first 1,000 networks | RMSE: 153.47$\pm$0.17 -> 147.57$\pm$0.23 | Helped |
| Use TMP-Nets instead of StemGNN | Traffic forecasting · PeMSD8, first 1,000 networks | RMSE: 89.60$\pm$0.60 -> 85.86$\pm$0.64 | Helped |
| Use TMP-Nets instead of Z-GCNETs | Traffic forecasting · PeMSD4, first 2,000 networks | MAPE: 180.11$\pm$0.26 -> 170.97$\pm$0.22 | Helped |
| Use TMP-Nets instead of Z-GCNETs | Traffic forecasting · PeMSD8, first 2,000 networks | MAE: 47.11$\pm$0.20 -> 50.20$\pm$0.37 | Hurt |
| Use TMP-Nets instead of Z-GCNETs | Traffic forecasting · PeMSD8, first 2,000 networks | RMSE: 80.25$\pm$0.24 -> 80.17$\pm$0.26 | No clear effect |
| Use TMP-Nets instead of Z-GCNETs | Blockchain forecasting · Bytom | MAPE: 31.04$\pm$0.78 -> 28.77$\pm$3.30 | Helped |
| Use TMP-Nets instead of Z-GCNETs | Blockchain forecasting · Golem | MAPE: 22.32$\pm$1.42 -> 29.01$\pm$1.05 | Hurt |
| Use TMP-Nets instead of StemGNN | ECG forecasting · ECG5000 | RMSE: 0.52$\pm$0.006 -> 0.52$\pm$0.005 | No clear effect |
| Add TMP topological features to spatial graph convolution features (ablation) | TMP-Nets · PeMSD4, first 1,000 networks | RMSE: 153.75$\pm$0.22 -> {\bf 147.57}$\pm${\bf 0.23} | Helped |
| Add spatial graph convolution features to TMP topological features (ablation) | TMP-Nets · PeMSD4, first 1,000 networks | RMSE: 165.67$\pm$0.30 -> {\bf 147.57}$\pm${\bf 0.23} | Helped |
| Use TMP-Nets with time-axis zigzag persistence instead of degree-axis zigzag persistence (ablation) | TMP-Nets · Bytom | MAPE: 29.15$\pm$4.17 -> 28.77$\pm$3.30 | Helped |
| Use TMP vectorizations instead of bigraded Betti numbers (ablation) | Topological-feature forecasting model · PeMSD4, first 1,000 networks | RMSE: 151.58$\pm$0.19 -> 147.57$\pm$0.23 | Helped |
Surprising findings
- TMP-Nets underperforms Z-GCNETs and StemGNN on the Golem token (MAPE 29.01% vs 22.32%), which the authors attribute to lower assortativity (disassortative structure).
- On full PeMSD4 and PeMSD8 datasets, TMP-Nets is not best; Z-GCNETs performs best on most metrics, so the advantage is mainly in limited-data regimes.
Limitations
- Scaling to very large graphs and ultra high-dimensional or streaming data remains a challenge.
- Performance on Golem is weaker than baselines, attributed to disassortative network structure.
- Computational cost grows with the choice of filtering functions and resolution.
Experimental setup
Datasets: PeMSD4 PeMSD8 Bytom Decentraland Golem ECG5000
Benchmarks: PeMSD4 PeMSD8 Ethereum token networks (Bytom, Decentraland, Golem) ECG5000
Metrics: MAE RMSE MAPE computation time
Baselines: DCRNN STGCN GraphWaveNet AGCRN Z-GCNETs StemGNN TAMP-S2GCNets MP-I MP-L
Builds on: Z-GCNETs TAMP-S2GCNets adaptive adjacency graph convolution (AGCRN) zigzag persistence
Closest papers by meaning
- HGE: Embedding Temporal Knowledge Graphs in a Product Space of Heterogeneous Geometric Subspaces (AAAI 2024 · similarity 0.86)
- Integrating Large Language Models and Möbius Group Transformations for Temporal Knowledge Graph Embedding on the Riemann Sphere (AAAI 2025 · similarity 0.86)
- Neural Persistence Dynamics (NeurIPS 2024 · similarity 0.86)
- TGB 2.0: A Benchmark for Learning on Temporal Knowledge Graphs and Heterogeneous Graphs (NeurIPS 2024 · similarity 0.86)
- Continuous-Time Graph Representation with Sequential Survival Process (AAAI 2024 · similarity 0.85)
- Topological Periodicity Test (TopPT) via Confidence Bound of Time-Delay Embeddings (NeurIPS 2026 · similarity 0.85)
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