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Diffuse, Sample, Project: Plug-And-Play Controllable Graph Generation
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Proposes PRODIGY, a plug-and-play sampler that projects intermediate samples of any pre-trained graph diffusion model onto hard, non-differentiable constraint sets, achieving up to 100% constraint satisfaction while preserving the learned distribution.
Problem
Controllable graph generation under user-specified hard, non-differentiable constraints (e.g., edge count, atom count, molecular weight) using pre-trained diffusion models without retraining or labeled conditional data.
Approach
Augment each reverse diffusion step with a (partial, weighted) Euclidean projection onto the constraint feasible set, with closed-form or bisection-based projection operators derived via KKT conditions for many graph constraints, so the sampler works plug-and-play on any pre-trained continuous or discrete graph diffusion model.
Key results
- PRODIGY lifts constraint validity to up to 100% on non-attributed graphs (e.g., GDSS Community-small edge count 1.00) while keeping MMD changes small (ΔMMD mostly ≥ -0.02).
- On molecular graphs, PRODIGY achieves 99–100% valency-constraint validity on QM9 and ZINC250k for GDSS and DruM, with ΔFCD often improved.
- On 3D GeoLDM with a dipole-moment constraint, unconditional GeoLDM + PRODIGY matches conditionally trained models (MAE 1.10 vs 0.00 bias-adjusted) without retraining.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add projection sampling to GDSS | GDSS on Community-small Edge Count · Community-small, Edge Count | VALC: 0.30 -> 1.00 | Helped |
| Add projection sampling to EDP-GNN | EDP-GNN on Community-small Triangle Count · Community-small, Triangle Count | VALC: 0.70 -> 0.83 | Helped |
| Add projection sampling to GDSS | GDSS on Community-small Degree · Community-small, Degree | VALC: 0.60 -> 1.00 | Helped |
| Add projection sampling to DruM | DruM on QM9 Valency · QM9, Valency | VALC: 0.99 -> 0.99 | No clear effect |
| Add projection sampling to GDSS | GDSS on QM9 Atom Count · QM9, Atom Count | VALC: 0.33 -> 1.00 | Helped |
| Add projection sampling to GDSS | GDSS on QM9 Molecular Weight · QM9, Molecular Weight | VALC: 0.13 -> 0.53 | Helped |
| Add projection sampling to GDSS | GDSS on ZINC250k Valency · ZINC250k, Valency | VALC: 0.94 -> 0.99 | Helped |
| Add projection sampling to DruM | DruM on ZINC250k Atom Count · ZINC250k, Atom Count | VALC: 0.01 -> 1.00 | Helped |
| Add projection sampling to GeoLDM | GeoLDM unconditional 3D molecule generation · QM9, dipole moment constraint | MAE µdm: 1.10 (0.04) -> 0.00 (1.15) | Helped |
| Use time-based rather than distance-based projection scheduling (ablation) | PRODIGY on Community-small · Community-small graph constraints | VALC | Mixed |
Surprising findings
- Projection-based sampling can match or beat specially trained conditional models on dipole moment control without retraining.
- Molecular weight constraints achieve low validity in several models (e.g., 28.9% on EDP-GNN for QM9), which the authors attribute to projections yielding negative values.
Limitations
- Non-linear property approximators (e.g., neural property predictors) lack closed-form projections and are not supported.
- Molecular weight constraint yields low validity for several models; hyperplane projection tends to produce negative values.
- Not directly applicable to latent diffusion models since constraints are defined in graph space.
Experimental setup
Datasets: Community-small Ego-small Enzymes SBM Planar QM9 ZINC250k
Benchmarks: Community-small Ego-small Enzymes SBM Planar QM9 ZINC250k
Metrics: VALC (constraint validity) delta MMD V.U.N. Val.% Novelty % delta FCD MAE of dipole moment
Baselines: EDP-GNN GDSS DruM DiGress GeoLDM EDM DiGress+Guidance
Builds on: Mirrored Langevin Dynamics GDSS DiGress EDP-GNN DruM
Closest papers by meaning
- Graph Generation with Diffusion Mixture (ICML 2024 · similarity 0.90)
- Generative Modelling of Structurally Constrained Graphs (NeurIPS 2024 · similarity 0.89)
- Discrete-state Continuous-time Diffusion for Graph Generation (NeurIPS 2024 · similarity 0.88)
- Diffusion Twigs with Loop Guidance for Conditional Graph Generation (NeurIPS 2024 · similarity 0.88)
- Advancing Graph Generation through Beta Diffusion (ICLR 2025 · similarity 0.88)
- Graph Diffusion that can Insert and Delete (NeurIPS 2025 · similarity 0.88)
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