# Bures-Wasserstein Flow Matching for Graph Generation arXiv: https://arxiv.org/abs/2506.14020 ยท HTML: https://mlsanity.com/paper/2506.14020/ ## Key facts - Cited by 9 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 52 of its 69 references that are in the map. - First posted on arXiv 16 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, 30 of 52) - [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) - [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022) - [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018) - [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022) - [node2vec: Scalable Feature Learning for Networks](https://mlsanity.com/paper/1607.00653/) (2016) - [Variational Graph Auto-Encoders](https://mlsanity.com/paper/1611.07308/) (2016) - [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022) - [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023) - [DiGress: Discrete Denoising diffusion for graph generation](https://mlsanity.com/paper/2209.14734/) (2022) - [A Continuous Time Framework for Discrete Denoising Models](https://mlsanity.com/paper/2205.14987/) (2022) - [GraphRNN: Generating Realistic Graphs with Deep Auto-regressive Models](https://mlsanity.com/paper/1802.08773/) (2018) - [MolGAN: An implicit generative model for small molecular graphs](https://mlsanity.com/paper/1805.11973/) (2018) - [Discrete Flow Matching](https://mlsanity.com/paper/2407.15595/) (2024) - [Generative Flows on Discrete State-Spaces: Enabling Multimodal Flows with Applications to Protein Co-Design](https://mlsanity.com/paper/2402.04997/) (2024) - [Flow Matching Guide and Code](https://mlsanity.com/paper/2412.06264/) (2024) - [GuacaMol: Benchmarking Models for De Novo Molecular Design](https://mlsanity.com/paper/1811.09621/) (2018) - [Multisample Flow Matching: Straightening Flows with Minibatch Couplings](https://mlsanity.com/paper/2304.14772/) (2023) - [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) - [Flow Matching on General Geometries](https://mlsanity.com/paper/2302.03660/) (2023) - [GEOM: Energy-annotated molecular conformations for property prediction and molecular generation](https://mlsanity.com/paper/2006.05531/) (2020) - [Efficient Graph Generation with Graph Recurrent Attention Networks](https://mlsanity.com/paper/1910.00760/) (2019) - [Score-based Continuous-time Discrete Diffusion Models](https://mlsanity.com/paper/2211.16750/) (2022) - [Understanding the Latent Space of Diffusion Models through the Lens of Riemannian Geometry](https://mlsanity.com/paper/2307.12868/) (2023) - [Dirichlet Flow Matching with Applications to DNA Sequence Design](https://mlsanity.com/paper/2402.05841/) (2024) - [SPECTRE: Spectral Conditioning Helps to Overcome the Expressivity Limits of One-shot Graph Generators](https://mlsanity.com/paper/2204.01613/) (2022) - [MiDi: Mixed Graph and 3D Denoising Diffusion for Molecule Generation](https://mlsanity.com/paper/2302.09048/) (2023) - [Variational Flow Matching for Graph Generation](https://mlsanity.com/paper/2406.04843/) (2024) - [GMNN: Graph Markov Neural Networks](https://mlsanity.com/paper/1905.06214/) (2019) ## Built on by (most cited first, 9 of 9) - [Generalized Wasserstein Flow Matching: Transport Plans, Everywhere, All at Once](https://mlsanity.com/paper/2605.08424/) (2026) - [LGDC: Latent Graph Diffusion via Spectrum-Preserving Coarsening](https://mlsanity.com/paper/2512.01190/) (2025) - [Semantic DLM+: Improving Diffusion Language Models through Bias-variance Trade-off in Transition Kernel Design](https://mlsanity.com/paper/2606.15327/) (2026) - [Gromov-Monge Flow Matching for Equivariant Graph Generation](https://mlsanity.com/paper/2608.26961/) (2026) - [From topology learning to graph generation: A unifying perspective](https://mlsanity.com/paper/2609.02286/) (2026) - [Manifold-Stable Flow Matching](https://mlsanity.com/paper/2609.35454/) (2026) - [GRFBrain: Graph-Structured Rectified Flows for EEG Dynamic Modeling](https://mlsanity.com/paper/2609.37934/) (2026) - [When Edge Independence Fails: Joint Graph Diffusion with Latent Sociability Priors](https://mlsanity.com/paper/2610.13765/) (2026) - [Your Discrete Graph Diffusion Model is Secretly Memorizing](https://mlsanity.com/paper/2610.13786/) (2026) ## Abstract (arXiv) Graph generation has emerged as a critical task in fields ranging from drug discovery to circuit design. Contemporary approaches, notably diffusion and flow-based models, have achieved solid graph generative performance through constructing a probability path that interpolates between reference and data distributions. However, these methods typically model the evolution of individual nodes and edges independently and use linear interpolations in the disjoint space of nodes/edges to build the path. This disentangled interpolation breaks the interconnected patterns of graphs, making the constructed probability path irregular and non-smooth, which causes poor training dynamics and faulty sampling convergence. To address the limitation, this paper first presents a theoretically grounded framework for probability path construction in graph generative models. Specifically, we model the joint evolution of the nodes and edges by representing graphs as connected systems parameterized by Markov random fields (MRF). We then leverage the optimal transport displacement between MRF objects to design a smooth probability path that ensures the co-evolution of graph components. Based on this, we introduce BWFlow, a flow-matching framework for graph generation that utilizes the derived optimal probability path to benefit the training and sampling algorithm design. Experimental evaluations in plain graph generation and molecule generation validate the effectiveness of BWFlow with competitive performance, better training convergence, and efficient sampling.