# Graph Diffusion that can Insert and Delete

arXiv: https://arxiv.org/abs/2506.15725 · HTML: https://mlsanity.com/paper/2506.15725/

## Key facts

- Cited by 6 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 23 of its 28 references that are in the map.
- First posted on arXiv 6 Jun 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, 23 of 23)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [FiLM: Visual Reasoning with a General Conditioning Layer](https://mlsanity.com/paper/1709.07871/) (2017)
- [Equivariant Diffusion for Molecule Generation in 3D](https://mlsanity.com/paper/2203.17003/) (2022)
- [Junction Tree Variational Autoencoder for Molecular Graph Generation](https://mlsanity.com/paper/1802.04364/) (2018)
- [DiGress: Discrete Denoising diffusion for graph generation](https://mlsanity.com/paper/2209.14734/) (2022)
- [Principal Neighbourhood Aggregation for Graph Nets](https://mlsanity.com/paper/2004.05718/) (2020)
- [GraphRNN: Generating Realistic Graphs with Deep Auto-regressive Models](https://mlsanity.com/paper/1802.08773/) (2018)
- [Graph Convolutional Policy Network for Goal-Directed Molecular Graph Generation](https://mlsanity.com/paper/1806.02473/) (2018)
- [Hierarchical Generation of Molecular Graphs using Structural Motifs](https://mlsanity.com/paper/2002.03230/) (2020)
- [Constrained Graph Variational Autoencoders for Molecule Design](https://mlsanity.com/paper/1805.09076/) (2018)
- [MiDi: Mixed Graph and 3D Denoising Diffusion for Molecule Generation](https://mlsanity.com/paper/2302.09048/) (2023)
- [Learning Multimodal Graph-to-Graph Translation for Molecular Optimization](https://mlsanity.com/paper/1812.01070/) (2018)
- [Autoregressive Diffusion Model for Graph Generation](https://mlsanity.com/paper/2307.08849/) (2023)
- [HouseDiffusion: Vector Floorplan Generation via a Diffusion Model with Discrete and Continuous Denoising](https://mlsanity.com/paper/2211.13287/) (2022)
- [A Survey on Deep Graph Generation: Methods and Applications](https://mlsanity.com/paper/2203.06714/) (2022)
- [Deep Graph Generators: A Survey](https://mlsanity.com/paper/2012.15544/) (2020)
- [Trans-Dimensional Generative Modeling via Jump Diffusion Models](https://mlsanity.com/paper/2305.16261/) (2023)
- [Physics Guided Deep Learning for Generative Design of Crystal Materials with Symmetry Constraints](https://mlsanity.com/paper/2203.14352/) (2022)
- [Beyond In-Place Corruption: Insertion and Deletion In Denoising Probabilistic Models](https://mlsanity.com/paper/2107.07675/) (2021)
- [Classifier-free graph diffusion for molecular property targeting](https://mlsanity.com/paper/2312.17397/) (2023)
- [Lift Your Molecules: Molecular Graph Generation in Latent Euclidean Space](https://mlsanity.com/paper/2406.10513/) (2024)

## Built on by (most cited first, 6 of 6)

- [Generative Molecular Morphing for Flexible-Size Design via Unbalanced Optimal Transport](https://mlsanity.com/paper/2606.07239/) (2026)
- [Graph Energy Matching: Transport-Aligned Energy-Based Modeling for Graph Generation](https://mlsanity.com/paper/2603.23398/) (2026)
- [From Single-Step Edit Response to Multi-Step Molecular Optimization](https://mlsanity.com/paper/2605.10035/) (2026)
- [Graph Forward Distribution Matching for Molecular Inverse Design](https://mlsanity.com/paper/2609.32056/) (2026)
- [When Edge Independence Fails: Joint Graph Diffusion with Latent Sociability Priors](https://mlsanity.com/paper/2610.13765/) (2026)
- [FragFlex: Fragment-Based Flexible Molecular Generation for Exploring Vast Chemical Space](https://mlsanity.com/paper/2610.15284/) (2026)

## Abstract (arXiv)

Generative models of graphs based on discrete Denoising Diffusion Probabilistic Models (DDPMs) offer a principled approach to molecular generation by systematically removing structural noise through iterative atom and bond adjustments. However, existing formulations are fundamentally limited by their inability to adapt the graph size (that is, the number of atoms) during the diffusion process, severely restricting their effectiveness in conditional generation scenarios such as property-driven molecular design, where the targeted property often correlates with the molecular size. In this paper, we reformulate the noising and denoising processes to support monotonic insertion and deletion of nodes. The resulting model, which we call GrIDDD, dynamically grows or shrinks the chemical graph during generation. GrIDDD matches or exceeds the performance of existing graph diffusion models on molecular property targeting despite being trained on a more difficult problem. Furthermore, when applied to molecular optimization, GrIDDD exhibits competitive performance compared to specialized optimization models. This work paves the way for size-adaptive molecular generation with graph diffusion.