# Principled Latent Diffusion for Graphs via Laplacian Autoencoders

arXiv: https://arxiv.org/abs/2601.13780 · HTML: https://mlsanity.com/paper/2601.13780/

## Key facts

- Cited by 4 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 56 of its 70 references that are in the map.
- First posted on arXiv 20 Jan 2026.

## 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 56)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Scalable Diffusion Models with Transformers](https://mlsanity.com/paper/2212.09748/) (2022)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Scaling Rectified Flow Transformers for High-Resolution Image Synthesis](https://mlsanity.com/paper/2403.03206/) (2024)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Structured Denoising Diffusion Models in Discrete State-Spaces](https://mlsanity.com/paper/2107.03006/) (2021)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Variational Graph Auto-Encoders](https://mlsanity.com/paper/1611.07308/) (2016)
- [Strategies for Pre-training Graph Neural Networks](https://mlsanity.com/paper/1905.12265/) (2019)
- [Discrete Diffusion Modeling by Estimating the Ratios of the Data Distribution](https://mlsanity.com/paper/2310.16834/) (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)
- [Analog Bits: Generating Discrete Data using Diffusion Models with Self-Conditioning](https://mlsanity.com/paper/2208.04202/) (2022)
- [Graph Neural Networks with Learnable Structural and Positional Representations](https://mlsanity.com/paper/2110.07875/) (2021)
- [Discrete Flow Matching](https://mlsanity.com/paper/2407.15595/) (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)
- [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)
- [DIFUSCO: Graph-based Diffusion Solvers for Combinatorial Optimization](https://mlsanity.com/paper/2302.08224/) (2023)
- [Position-aware Graph Neural Networks](https://mlsanity.com/paper/1906.04817/) (2019)
- [Sign and Basis Invariant Networks for Spectral Graph Representation Learning](https://mlsanity.com/paper/2202.13013/) (2022)
- [SPECTRE: Spectral Conditioning Helps to Overcome the Expressivity Limits of One-shot Graph Generators](https://mlsanity.com/paper/2204.01613/) (2022)
- [Edge Directionality Improves Learning on Heterophilic Graphs](https://mlsanity.com/paper/2305.10498/) (2023)
- [DeFoG: Discrete Flow Matching for Graph Generation](https://mlsanity.com/paper/2410.04263/) (2024)

## Built on by (most cited first, 4 of 4)

- [From topology learning to graph generation: A unifying perspective](https://mlsanity.com/paper/2609.02286/) (2026)
- [Efficient Graph Generation via Direct Prediction and Flow Matching](https://mlsanity.com/paper/2610.05397/) (2026)
- [Latent Flow Matching for Molecular Graph Generation](https://mlsanity.com/paper/2610.06468/) (2026)
- [When Edge Independence Fails: Joint Graph Diffusion with Latent Sociability Priors](https://mlsanity.com/paper/2610.13765/) (2026)

## Abstract (arXiv)

Graph diffusion models achieve state-of-the-art performance in graph generation but suffer from quadratic complexity in the number of nodes -- and much of their capacity is wasted modeling the absence of edges in sparse graphs. Inspired by latent diffusion in other modalities, a natural idea is to compress graphs into a low-dimensional latent space and perform diffusion in that space. However, unlike images or text, graph generation requires nearly lossless reconstruction, as even a single error in decoding an adjacency matrix can render the entire sample invalid. This challenge has remained largely unaddressed. We propose LG-Flow, a latent graph diffusion framework that directly overcomes these obstacles. A permutation-equivariant autoencoder maps nodes to fixed-dimensional embeddings that enable near-lossless reconstruction of both undirected graphs and DAGs. The dimensionality of this latent representation scales linearly with the number of nodes, thereby removing the quadratic adjacency-space bottleneck in the diffusion process and enabling the training of substantially larger generative backbones. In this latent space, we train a Diffusion Transformer with flow matching, enabling efficient and expressive graph generation. Our approach achieves competitive results against state-of-the-art graph diffusion models while delivering up to a 1000× speed-up. Our code is available at https://github.com/asiraudin/LG-Flow .