# Composable Score-based Graph Diffusion Model for Multi-Conditional Molecular Generation

arXiv: https://arxiv.org/abs/2509.09451 · HTML: https://mlsanity.com/paper/2509.09451/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 37 of its 49 references that are in the map.
- First posted on arXiv 11 Sep 2025.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 30 of 37)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Photorealistic Text-to-Image Diffusion Models with Deep Language Understanding](https://mlsanity.com/paper/2205.11487/) (2022)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [On Calibration of Modern Neural Networks](https://mlsanity.com/paper/1706.04599/) (2017)
- [Arbitrary Style Transfer in Real-time with Adaptive Instance Normalization](https://mlsanity.com/paper/1703.06868/) (2017)
- [Structured Denoising Diffusion Models in Discrete State-Spaces](https://mlsanity.com/paper/2107.03006/) (2021)
- [MoleculeNet: A Benchmark for Molecular Machine Learning](https://mlsanity.com/paper/1703.00564/) (2017)
- [Discrete Diffusion Modeling by Estimating the Ratios of the Data Distribution](https://mlsanity.com/paper/2310.16834/) (2023)
- [Compositional Visual Generation with Composable Diffusion Models](https://mlsanity.com/paper/2206.01714/) (2022)
- [DiGress: Discrete Denoising diffusion for graph generation](https://mlsanity.com/paper/2209.14734/) (2022)
- [GeoDiff: a Geometric Diffusion Model for Molecular Conformation Generation](https://mlsanity.com/paper/2203.02923/) (2022)
- [A Continuous Time Framework for Discrete Denoising Models](https://mlsanity.com/paper/2205.14987/) (2022)
- [Beyond temperature scaling: Obtaining well-calibrated multiclass probabilities with Dirichlet calibration](https://mlsanity.com/paper/1910.12656/) (2019)
- [Discrete Flow Matching](https://mlsanity.com/paper/2407.15595/) (2024)
- [Guiding a Diffusion Model with a Bad Version of Itself](https://mlsanity.com/paper/2406.02507/) (2024)
- [Score-based Generative Modeling of Graphs via the System of Stochastic Differential Equations](https://mlsanity.com/paper/2202.02514/) (2022)
- [Molecular Sets (MOSES): A Benchmarking Platform for Molecular Generation Models](https://mlsanity.com/paper/1811.12823/) (2018)
- [Permutation Invariant Graph Generation via Score-Based Generative Modeling](https://mlsanity.com/paper/2003.00638/) (2020)
- [Applying Guidance in a Limited Interval Improves Sample and Distribution Quality in Diffusion Models](https://mlsanity.com/paper/2404.07724/) (2024)
- [Fr'echet ChemNet Distance: A metric for generative models for molecules in drug discovery](https://mlsanity.com/paper/1803.09518/) (2018)
- [Score-based Continuous-time Discrete Diffusion Models](https://mlsanity.com/paper/2211.16750/) (2022)
- [Sample Efficiency Matters: A Benchmark for Practical Molecular Optimization](https://mlsanity.com/paper/2206.12411/) (2022)
- [Concrete Score Matching: Generalized Score Matching for Discrete Data](https://mlsanity.com/paper/2211.00802/) (2022)
- [Self-Rectifying Diffusion Sampling with Perturbed-Attention Guidance](https://mlsanity.com/paper/2403.17377/) (2024)
- [Exploring Chemical Space with Score-based Out-of-distribution Generation](https://mlsanity.com/paper/2206.07632/) (2022)
- [Inverse design of 3d molecular structures with conditional generative neural networks](https://mlsanity.com/paper/2109.04824/) (2021)
- [DeFoG: Discrete Flow Matching for Graph Generation](https://mlsanity.com/paper/2410.04263/) (2024)

## Built on by (most cited first, 2 of 2)

- [MolPaQ: Modular Quantum-Classical Patch Learning for Interpretable Molecular Generation](https://mlsanity.com/paper/2604.08575/) (2026)
- [Graph Forward Distribution Matching for Molecular Inverse Design](https://mlsanity.com/paper/2609.32056/) (2026)

## Abstract (arXiv)

Controllable molecular graph generation is essential for material and drug discovery, where generated molecules must satisfy diverse property constraints. While recent advances in graph diffusion models have improved generation quality, their effectiveness in multi-conditional settings remains limited due to reliance on joint conditioning or continuous relaxations that compromise fidelity. To address these limitations, we propose Composable Score-based Graph Diffusion model (CSGD), the first model that extends score matching to discrete graphs via concrete scores, enabling flexible and principled manipulation of conditional guidance. Building on this foundation, we introduce two score-based techniques: Composable Guidance (CoG), which allows fine-grained control over arbitrary subsets of conditions during sampling, and Probability Calibration (PC), which adjusts estimated transition probabilities to mitigate train-test mismatches. Empirical results on four molecular datasets show that CSGD achieves state-of-the-art performance, with a 15.3% average improvement in controllability over prior methods, while maintaining high validity and distributional fidelity. Our findings highlight the practical advantages of score-based modeling for discrete graph generation and its capacity for flexible, multi-property molecular design.