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Spectro-Riemannian Graph Neural Networks
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Proposes CUSP, a spectral GNN on products of hyperbolic, spherical and Euclidean manifolds that uses a curvature-weighted Ollivier-Ricci Laplacian, a GPR-based filter bank, and curvature-aware attention pooling.
Problem
Existing mixed-curvature Riemannian GNNs act as low-pass filters and ignore task-specific curvature relevance, while spectral GNNs assume flat Euclidean geometry; the paper targets node classification and link prediction on homophilic and heterophilic graphs.
Approach
Combine an Ollivier-Ricci-curvature-weighted graph Laplacian (Cusp Laplacian) with a GPR-based spectral filter bank lifted to a learnable-curvature product manifold, plus attention pooling with a Bochner-based curvature positional encoding.
Key results
- CUSP achieves the best node classification F1 on all eight datasets, e.g. 94.03 on Texas and 52.98 on Squirrel, with up to 5.32% improvement over the second-best baseline (Actor).
- Ablation shows CUSP without mixed curvature (Euclidean variant) drops about 3.2% on average for node classification.
- Learned filter weights favor low-pass filters on homophilic graphs (Citeseer, PubMed) and high-order filters on heterophilic graphs (Actor, Cornell).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use CUSP instead of the strongest baseline | Node classification · Cora | F1 score: 80.19 -> 83.45 | Helped |
| Use CUSP instead of the strongest baseline | Node classification · Citeseer | F1 score: 71.26 -> 74.21 | Helped |
| Use CUSP instead of the strongest baseline | Node classification · PubMed | F1 score: 85.18 -> 87.99 | Helped |
| Use CUSP instead of the strongest baseline | Node classification · Chameleon | F1 score: 67.18 -> 70.23 | Helped |
| Use CUSP instead of the strongest baseline | Node classification · Actor | F1 score: 38.99 -> 43.91 | Helped |
| Use CUSP instead of the strongest baseline | Node classification · Squirrel | F1 score: 49.79 -> 52.98 | Helped |
| Use CUSP instead of the strongest baseline | Node classification · Texas | F1 score: 90.92 -> 94.03 | Helped |
| Use CUSP instead of the strongest baseline | Node classification · Cornell | F1 score: 87.21 -> 92.31 | Helped |
| Add mixed-curvature representations and Cusp Pooling instead of using only Euclidean representations (ablation) | CUSP node classification · All eight node classification datasets | F1 score: 3.19% average drop for CUSP_euc | Helped |
| Add the Cusp Laplacian instead of standard graph adjacency (ablation) | CUSP node classification · All eight node classification datasets | F1 score: 1.57% average drop when replaced | Helped |
| Add curvature-based positional encoding (ablation) | CUSP node classification · All eight node classification datasets | F1 score: null | Helped |
| Use Cusp Pooling rather than simple concatenation (ablation) | CUSP node classification · All eight node classification datasets | F1 score: 2.19 average gain over CUSP_pool | Helped |
Surprising findings
- Mismatched manifold signatures (e.g., spherical-dominated signatures on negatively curved graphs) cause large performance drops, showing sensitivity to signature choice.
Limitations
- ORC computation is costly (Sinkhorn quadratic), limiting scalability to large graphs.
- Manifold signature is determined heuristically from ORC, and mismatched signatures degrade performance.
Experimental setup
Datasets: Cora Citeseer PubMed Chameleon Squirrel Actor Texas Cornell
Benchmarks: Cora Citeseer PubMed Chameleon Squirrel Actor Texas Cornell
Metrics: F1 score ROC-AUC
Baselines: GCN GAT GraphSAGE HGCN HGAT kappaGCN QGCN SelfMGNN ChebyNet BernNet GPRGNN FiGURe
Builds on: GPRGNN kappa-GCN Ollivier-Ricci curvature kappa-stereographic model BernNet
Closest papers by meaning
- Motif-Aware Riemannian Graph Neural Network with Generative-Contrastive Learning (AAAI 2024 · similarity 0.89)
- Toward a Unified Geometry Understanding : Riemannian Diffusion Framework for Graph Generation and Prediction (NeurIPS 2025 · similarity 0.88)
- Discrete Curvature Graph Information Bottleneck (AAAI 2025 · similarity 0.87)
- Effective Structural Encodings via Local Curvature Profiles (ICLR 2024 · similarity 0.87)
- GIST: Gauge-Invariant Spectral Transformers for Scalable Graph Neural Operators (NeurIPS 2026 · similarity 0.86)
- Deeper with Riemannian Geometry: Overcoming Oversmoothing and Oversquashing for Graph Foundation Models (NeurIPS 2025 · similarity 0.86)
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