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Hyperbolic Graph Diffusion Model
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Proposes HGDM, a two-stage graph generative model that encodes nodes into a Poincaré-ball hyperbolic latent space with a hyperbolic VAE and runs score-based diffusion there, using a hyperbolic graph attention layer.
Problem
Graph generation with diffusion models in Euclidean space distorts hierarchical, power-law graph structure; real-world graphs (social, molecular, traffic) have non-Euclidean hierarchical topology.
Approach
Two-stage model: a hyperbolic VAE (with HGAT layers, Fermi-Dirac link regularization and a centroid-distance decoder) maps nodes to Poincaré-ball embeddings, then two score networks run VE/VP SDE diffusion on hyperbolic node latents (Riemannian predictor-corrector sampling) and Euclidean adjacency matrices.
Key results
- On Ego-small, HGDM reaches average MMD 0.014 vs GDSS 0.017 (about 17.6% improvement), and 0.024 vs 0.046 on Community-small (47.8% improvement).
- On QM9, HGDM achieves 98.04% validity without correction, NSPDK MMD 0.002 and FCD 2.131, beating GDSS (95.79%, 0.003, 2.813).
- On ZINC250k, HGDM gets NSPDK MMD 0.016 (best) but FCD 17.69 is worse than GraphAF's 16.289 and GDSS's 16.621, and validity is 93.51% vs GDSS 95.90%.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use hyperbolic latent diffusion rather than GDSS | Graph generation · Ego-small and Community-small | Avg. MMD: 17.6%; 47.8% | Helped |
| Use hyperbolic latent diffusion rather than GDSS | Graph generation · Enzymes and Grid | Avg. MMD: 0.0%; -1.6% | Mixed |
| Use HGDM rather than GDSS* | Molecular generation · QM9 | FCD: 2.813 -> 2.131 | Helped |
| Use HGDM rather than GDSS* | Molecular generation · QM9 | Val. w/o corr. (%): 95.79 -> 98.04 | Helped |
| Use HGDM rather than GDSS* | Molecular generation · ZINC250k | NSPDK MMD: 0.019 -> 0.016 | Helped |
| Use HGDM rather than GDSS* | Molecular generation · ZINC250k | FCD: 16.621 -> 17.69 | Hurt |
| Use HGDM rather than GDSS* | Molecular generation · ZINC250k | Val. w/o corr. (%): 95.90 -> 93.51 | Hurt |
| Add a Euclidean autoencoder embedding before GDSS diffusion (ablation) | GDSS · QM9 | FCD: 2.813 -> 2.615 | Helped |
| Add a Euclidean autoencoder embedding before GDSS diffusion (ablation) | GDSS · QM9 | Val. w/o corr. (%): 95.79 -> 95.95 | Helped |
| Replace HGAT layers with HGCN layers (ablation) | HGDM · QM9 | time (s): 1.23$e^2$ -> 1.48$e^2$ | Hurt |
| Replace HGAT layers with HGCN layers (ablation) | HGDM · QM9 | FCD: 2.131 -> 2.331 | Hurt |
Surprising findings
- On Grid (high hyperbolicity, delta=9.76), HGDM does not beat GDSS on average MMD and is outperformed by GraphRNN, so hyperbolic latents help mainly on low-hyperbolicity graphs.
- Simply adding a Euclidean autoencoder (GDSS+AE) yields only marginal gains over GDSS on QM9.
Limitations
- Gains are concentrated on low-hyperbolicity datasets; performance is comparable or worse on Enzymes-average, Grid and ZINC250k FCD/validity.
- Two-stage design with a pretrained HVAE; curvature fixed at -0.01 by empirical choice rather than tuned per dataset.
- Several baseline numbers are taken from prior papers, and reported standard deviations are available only for some settings.
Experimental setup
Datasets: Ego-small Community-small Enzymes Grid QM9 ZINC250k
Benchmarks: Ego-small Community-small Enzymes Grid QM9 ZINC250k
Metrics: MMD degree MMD clustering coefficient MMD orbit count NSPDK MMD FCD validity without correction generation time
Baselines: GraphVAE DeepGMG GraphRNN GraphAF GraphDF GNF EDP-GNN GDSS MoFlow GraphEBM
Builds on: GDSS hyperbolic graph convolutional network score-based SDE Hyperbolic VAE (P-VAE)
Closest papers by meaning
- Hyperbolic Geometric Latent Diffusion Model for Graph Generation (ICML 2024 · similarity 0.91)
- Graph Generation with Diffusion Mixture (ICML 2024 · similarity 0.88)
- Advancing Graph Generation through Beta Diffusion (ICLR 2025 · similarity 0.88)
- SBGD: Improving Graph Diffusion Generative Model via Stochastic Block Diffusion (ICML 2025 · similarity 0.87)
- HYGENE: A Diffusion-Based Hypergraph Generation Method (AAAI 2025 · similarity 0.87)
- Generating Graphs via Spectral Diffusion (ICLR 2025 · similarity 0.87)
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