# Ollivier-Ricci Curvature for Hypergraphs: A Unified Framework

arXiv: https://arxiv.org/abs/2210.12048 · HTML: https://mlsanity.com/paper/2210.12048/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 61 references that are in the map.
- First posted on arXiv 21 Oct 2022.

## Builds on (most cited first, 17 of 17)

- [Understanding over-squashing and bottlenecks on graphs via curvature](https://mlsanity.com/paper/2111.14522/) (2021)
- [Fast Computation of Wasserstein Barycenters](https://mlsanity.com/paper/1310.4375/) (2013)
- [UniGNN: a Unified Framework for Graph and Hypergraph Neural Networks](https://mlsanity.com/paper/2105.00956/) (2021)
- [Clustering in graphs and hypergraphs with categorical edge labels](https://mlsanity.com/paper/1910.09943/) (2019)
- [Structural Patterns and Generative Models of Real-world Hypergraphs](https://mlsanity.com/paper/2006.07060/) (2020)
- [SPONGE: A generalized eigenproblem for clustering signed networks](https://mlsanity.com/paper/1904.08575/) (2019)
- [Characterizing Complex Networks with Forman-Ricci Curvature and Associated Geometric Flows](https://mlsanity.com/paper/1607.08654/) (2016)
- [Hypernetwork Science via High-Order Hypergraph Walks](https://mlsanity.com/paper/1906.11295/) (2019)
- [Hypergraph Motifs: Concepts, Algorithms, and Discoveries](https://mlsanity.com/paper/2003.01853/) (2020)
- [Wasserstein Exponential Kernels](https://mlsanity.com/paper/2002.01878/) (2020)
- [Forman-Ricci Curvature for Hypergraphs](https://mlsanity.com/paper/1811.07825/) (2018)
- [Forman's Ricci curvature - From networks to hypernetworks](https://mlsanity.com/paper/1810.07749/) (2018)
- [THyMe+: Temporal Hypergraph Motifs and Fast Algorithms for Exact Counting](https://mlsanity.com/paper/2109.08341/) (2021)
- [Finding Bipartite Components in Hypergraphs](https://mlsanity.com/paper/2205.02771/) (2022)
- [Curvature of Hypergraphs via Multi-Marginal Optimal Transport](https://mlsanity.com/paper/1803.08584/) (2018)
- [All the World's a (Hyper)Graph: A Data Drama](https://mlsanity.com/paper/2206.08225/) (2022)
- [Wasserstein Soft Label Propagation on Hypergraphs: Algorithm and Generalization Error Bounds](https://mlsanity.com/paper/1809.01833/) (2018)

## Built on by (most cited first, 12 of 12)

- [Curvature Filtrations for Graph Generative Model Evaluation](https://mlsanity.com/paper/2301.12906/) (2023)
- [On the Expressivity of Persistent Homology in Graph Learning](https://mlsanity.com/paper/2302.09826/) (2023)
- [CliquePH: Higher-Order Information for Graph Neural Networks through Persistent Homology on Clique Graphs](https://mlsanity.com/paper/2409.08217/) (2024)
- [HGFormer: Topology-Aware Vision Transformer with HyperGraph Learning](https://mlsanity.com/paper/2504.02440/) (2025)
- [Tackling Over-smoothing on Hypergraphs: A Ricci Flow-guided Neural Diffusion Approach](https://mlsanity.com/paper/2603.15696/) (2026)
- [Accelerated Evaluation of Ollivier-Ricci Curvature Lower Bounds: Bridging Theory and Computation](https://mlsanity.com/paper/2405.13302/) (2024)
- [Heterophily-Agnostic Hypergraph Neural Networks with Riemannian Local Exchanger](https://mlsanity.com/paper/2603.00599/) (2026)
- [Hypergraph clustering using Ricci curvature: an edge transport perspective](https://mlsanity.com/paper/2412.15695/) (2024)
- [Characterizing Physician Referral Networks with Ricci Curvature](https://mlsanity.com/paper/2408.16022/) (2024)
- [Enhancing the Utility of Higher-Order Information in Relational Learning](https://mlsanity.com/paper/2502.09570/) (2025)
- [Lower Ricci Curvature for Hypergraphs](https://mlsanity.com/paper/2506.03943/) (2025)
- [A roadmap for curvature-based geometric data analysis and learning](https://mlsanity.com/paper/2510.22599/) (2025)

## Abstract (arXiv)

Bridging geometry and topology, curvature is a powerful and expressive invariant. While the utility of curvature has been theoretically and empirically confirmed in the context of manifolds and graphs, its generalization to the emerging domain of hypergraphs has remained largely unexplored. On graphs, the Ollivier-Ricci curvature measures differences between random walks via Wasserstein distances, thus grounding a geometric concept in ideas from probability theory and optimal transport. We develop ORCHID, a flexible framework generalizing Ollivier-Ricci curvature to hypergraphs, and prove that the resulting curvatures have favorable theoretical properties. Through extensive experiments on synthetic and real-world hypergraphs from different domains, we demonstrate that ORCHID curvatures are both scalable and useful to perform a variety of hypergraph tasks in practice.