# Graph Representation via Elements of Discrete Morse and Cobordism Theories

arXiv: https://arxiv.org/abs/2610.01937 · HTML: https://mlsanity.com/paper/2610.01937/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 30 of its 72 references that are in the map.
- First posted on arXiv 1 Oct 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 30)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Equivariant Diffusion for Molecule Generation in 3D](https://mlsanity.com/paper/2203.17003/) (2022)
- [CSDI: Conditional Score-based Diffusion Models for Probabilistic Time Series Imputation](https://mlsanity.com/paper/2107.03502/) (2021)
- [DiGress: Discrete Denoising diffusion for graph generation](https://mlsanity.com/paper/2209.14734/) (2022)
- [Autoregressive Denoising Diffusion Models for Multivariate Probabilistic Time Series Forecasting](https://mlsanity.com/paper/2101.12072/) (2021)
- [Score-based Generative Modeling of Graphs via the System of Stochastic Differential Equations](https://mlsanity.com/paper/2202.02514/) (2022)
- [Weisfeiler and Lehman Go Topological: Message Passing Simplicial Networks](https://mlsanity.com/paper/2103.03212/) (2021)
- [DeFoG: Discrete Flow Matching for Graph Generation](https://mlsanity.com/paper/2410.04263/) (2024)
- [DiffSTG: Probabilistic Spatio-Temporal Graph Forecasting with Denoising Diffusion Models](https://mlsanity.com/paper/2301.13629/) (2023)
- [Generative Diffusion Models on Graphs: Methods and Applications](https://mlsanity.com/paper/2302.02591/) (2023)
- [Efficient and Degree-Guided Graph Generation via Discrete Diffusion Modeling](https://mlsanity.com/paper/2305.04111/) (2023)
- [A Survey on Diffusion Models for Time Series and Spatio-Temporal Data](https://mlsanity.com/paper/2404.18886/) (2024)
- [PriSTI: A Conditional Diffusion Framework for Spatiotemporal Imputation](https://mlsanity.com/paper/2302.09746/) (2023)
- [Position: Topological Deep Learning is the New Frontier for Relational Learning](https://mlsanity.com/paper/2402.08871/) (2024)
- [Quantifying Uncertainty in Deep Spatiotemporal Forecasting](https://mlsanity.com/paper/2105.11982/) (2021)
- [Conditional Diffusion Based on Discrete Graph Structures for Molecular Graph Generation](https://mlsanity.com/paper/2301.00427/) (2023)
- [Topological Approaches to Deep Learning](https://mlsanity.com/paper/1811.01122/) (2018)
- [Sparse Training of Discrete Diffusion Models for Graph Generation](https://mlsanity.com/paper/2311.02142/) (2023)
- [Learning Probabilistic Topological Representations Using Discrete Morse Theory](https://mlsanity.com/paper/2206.01742/) (2022)
- [Towards Unifying Diffusion Models for Probabilistic Spatio-Temporal Graph Learning](https://mlsanity.com/paper/2310.17360/) (2023)
- [NVDiff: Graph Generation through the Diffusion of Node Vectors](https://mlsanity.com/paper/2211.10794/) (2022)
- [Clique topology reveals intrinsic geometric structure in neural correlations](https://mlsanity.com/paper/1502.06172/) (2015)
- [Simple and Critical Iterative Denoising: A Recasting of Discrete Diffusion in Graph Generation](https://mlsanity.com/paper/2503.21592/) (2025)
- [Graph Reconstruction by Discrete Morse Theory](https://mlsanity.com/paper/1803.05093/) (2018)
- [Graph Guided Diffusion: Unified Guidance for Conditional Graph Generation](https://mlsanity.com/paper/2505.19685/) (2025)
- [Do Graph Diffusion Models Accurately Capture and Generate Substructure Distributions?](https://mlsanity.com/paper/2502.02488/) (2025)
- [A Non-Asymptotic Convergent Analysis for Scored-Based Graph Generative Model via a System of Stochastic Differential Equations](https://mlsanity.com/paper/2508.14351/) (2025)
- [Universal Distances for Extended Persistence](https://mlsanity.com/paper/2007.01834/) (2020)
- [Revisiting Accurate Geometry for Morse-Smale Complexes](https://mlsanity.com/paper/2409.05532/) (2024)
- [SBGD: Improving Graph Diffusion Generative Model via Stochastic Block Diffusion](https://mlsanity.com/paper/2508.14352/) (2025)

## Abstract (arXiv)

Topology is, by its nature and design, suited to structure that is nonlinear, multiscale, and nonstationary - however, within machine learning, its use remains largely confined to topological data analysis. We advocate that tools from low-dimensional topology which have remained almost exclusively contained within the domain of pure mathematics (such as Morse theory) offer a strong, complementary, and yet virtually unexplored perspective on the hidden structure of data-generating processes and learning tasks built upon them. Here we introduce concepts from cobordism theory and harness tools from discrete Morse theory to improve the performance of graph diffusion models through our pipeline MG-Diff. Further, we derive theoretical guarantees and sufficient conditions so that under a positive decision-gap, the Morse-theoretic tools and their application for induced diffusion guidance are stable under small perturbations. Finally, we illustrate the utility of discrete Morse theory in application to graph diffusion models for spatio-temporal graph forecasting and graph regeneration, and argue that these applications are only a small window into the part of what low-dimensional topology can offer to the field of machine learning.