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Deeper with Riemannian Geometry: Overcoming Oversmoothing and Oversquashing for Graph Foundation Models
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Proposes GBN, a message-passing network that treats node updates as a Robin-type nonhomogeneous boundary-value heat problem on local subgraphs, to jointly mitigate oversquashing and oversmoothing without graph rewriting.
Problem
Oversmoothing and oversquashing in MPNNs, which global fixes (graph rewriting that increases the spectral gap) address only at high cost and with structural edits that hurt other regions.
Approach
Reinterpret MPNN as a neural solver of the graph heat equation and introduce a local nonhomogeneous Robin boundary condition on subgraph boundary nodes, whose learned coefficients adaptively regulate message flow at bottlenecks, while the source term from boundary values counters oversmoothing.
Key results
- GBN achieves best ACC on CS (95.78), Texas (85.01), Wisconsin (86.78), RomanEmpire (89.83) and Ratings (53.51), and is best or runner-up on the rest of the 7 datasets (per reported table).
- GBN shows no performance degradation at depths beyond 256 layers, while GCN and GAT show exponential Dirichlet energy decay.
- Ablation: removing both gamma and beta (i.e., disabling boundary terms) reduces GBN to roughly GCN-level accuracy (e.g., Texas 62.16 vs 85.01 for GBN).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use GBN instead of the strongest listed baseline, BORF | Node classification · WikiCS | ACC (%): 85.89 ± 0.28 -> 86.21 ± 0.39 | Helped |
| use GBN instead of the strongest listed baseline, BORF | Node classification · Computer | ACC (%): 91.76 ± 0.63 -> 91.33 ± 0.32 | Hurt |
| use GBN instead of the strongest listed baseline, UniFilter | Node classification · CS | ACC (%): 95.25 ± 0.11 -> 95.78 ± 0.21 | Helped |
| use GBN instead of the strongest listed baseline, G²-GCN | Node classification · Texas | ACC (%): 84.86 ± 5.43 -> 85.01 ± 6.51 | Helped |
| use GBN instead of the strongest listed baseline, G²-GCN | Node classification · Wisconsin | ACC (%): 86.12 ± 4.51 -> 86.78 ± 3.84 | Helped |
| use GBN instead of the strongest listed baseline, Graph-Mamba | Node classification · RomanEmpire | ACC (%): 88.36 ± 0.06 -> 89.83 ± 0.46 | Helped |
| use GBN instead of the strongest listed baseline, VCR-Graphormer | Node classification · Ratings | ACC (%): 53.31 ± 0.01 -> 53.51 ± 0.88 | Helped |
| add learnable boundary coefficients instead of fixed constants (ablation) | GBN · CS | ACC (%): 94.14 ± 0.26 -> 95.78 ± 0.21 | Helped |
| add external inputs (γᵢ) (ablation) | GBN · CS | ACC (%): 92.26 ± 0.58 -> 95.78 ± 0.21 | Helped |
| add boundary interactions (βᵢ) (ablation) | GBN · CS | ACC (%): 93.77 ± 0.36 -> 95.78 ± 0.21 | Helped |
| add external inputs and boundary interactions (γᵢ, βᵢ) (ablation) | GBN · CS | ACC (%): 91.71 ± 0.34 -> 95.78 ± 0.21 | Helped |
Surprising findings
- Increasing the spectral gap globally can cause gradient vanishing and faster oversmoothing, challenging the default graph-rewriting rationale.
Limitations
- Theory considers undirected graphs only; directed graphs left to future work.
- Theoretical results rely on continuous manifold analogies and approximations (e.g., first-order heat kernel, small time step) that may not hold exactly for discrete graphs.
Experimental setup
Datasets: WikiCS Computer CS Texas Wisconsin RomanEmpire Ratings
Benchmarks: WikiCS Computer CS Texas Wisconsin RomanEmpire Ratings graph transfer (long-range)
Metrics: ACC MSE Dirichlet energy
Baselines: GCN GAT GIN HGCN GREAD DR UniFilter ProxyGap MPNN+VN GraphNormv2 BORF G^2-GCN
Builds on: GCN graph rewriting (BORF) Robin boundary condition Cheeger-Buser inequality G^2-GCN
Closest papers by meaning
- Understanding Virtual Nodes: Oversquashing and Node Heterogeneity (ICLR 2025 · similarity 0.88)
- The Effectiveness of Curvature-Based Rewiring and the Role of Hyperparameters in GNNs Revisited (ICLR 2025 · similarity 0.88)
- On Oversquashing in Graph Neural Networks Through the Lens of Dynamical Systems (AAAI 2025 · similarity 0.87)
- Discrete Curvature Graph Information Bottleneck (AAAI 2025 · similarity 0.87)
- A Signed Graph Approach to Understanding and Mitigating Oversmoothing (NeurIPS 2025 · similarity 0.87)
- Accurate and Scalable Graph Neural Networks via Message Invariance (ICLR 2025 · similarity 0.87)
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