Graph Representation via Elements of Discrete Morse and Cobordism Theories
This paper tackles the tendency of graph diffusion models to lose global topology—such as cycles and higher-order motifs—during local denoising. It introduces MG-Diff, a framework that extracts a discrete Morse skeleton from a graph and conditions generation on this sparse topological backbone. By framing diffusion as traversing a discrete cobordism, the authors aim to anchor the generative trajectory to structurally critical events rather than treating topology as an afterthought. The promise is better preservation of connectivity and improved uncertainty quantification in spatio-temporal forecasting.
The paper offers a novel theoretical perspective by using discrete Morse theory to build a conditioning signal for graph diffusion. Its strongest asset is a concrete, deterministic skeleton extractor with a margin-based stability guarantee. However, the evaluation is limited—graph generation is shown only on a single small benchmark—and the stability theorem applies to the extractor alone, not the diffusion model. As the authors themselves caution, the result is local and does not imply global stability of the generative process.
The discrete Morse skeleton construction is algorithmically precise and can be computed efficiently in practice. The distinction between persistence diagrams and descending-manifold structure is well argued, with explicit counterexamples in Appendix J showing that identical persistence can yield different skeletons. Forecasting results on three traffic and air-quality datasets show consistent gains across MAE, RMSE, and CRPS.
The use of cobordism theory is primarily intuitive rather than formal. The paper never proves that the diffusion trajectory satisfies cobordism axioms or employs cobordism invariants to constrain generation; instead it relies on an analogy.
The empirical evaluation for graph generation is extremely narrow, with only the community-small synthetic dataset reported in the main text. Standard molecular or larger generic benchmarks are absent, and the authors acknowledge that on tree-like graphs the Morse skeleton offers no guidance because the critical edge set is empty.
The spatio-temporal comparisons use standard baselines and metrics, and the paper correctly notes that CRPS gains do not hold on AIR-GZ, where MG-Diff is second to USTD. However, there is no direct head-to-head comparison with the higher-order-guided HOG-Diff baseline on shared tasks, and the graph generation claims rest on a single small dataset with limited metrics, making it hard to judge whether the Morse guidance truly outperforms persistence-based or cell-complex methods at scale.
The deterministic extractor algorithm and several key hyperparameters are specified, but no code repository is mentioned and full architectural details are relegated to appendices not fully shown here. The Morse construction adds no per-epoch cost and is fast on ogbn-arxiv, yet the worst-case complexity is cubic in the number of nodes, which could become prohibitive for dense graphs.
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.
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