# Lower Ricci Curvature for Hypergraphs

arXiv: https://arxiv.org/abs/2506.03943 · HTML: https://mlsanity.com/paper/2506.03943/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 22 of its 59 references that are in the map.
- First posted on arXiv 4 Jun 2025.

## Builds on (most cited first, 22 of 22)

- [Understanding over-squashing and bottlenecks on graphs via curvature](https://mlsanity.com/paper/2111.14522/) (2021)
- [Geometric deep learning on graphs and manifolds using mixture model CNNs](https://mlsanity.com/paper/1611.08402/) (2016)
- [Contact patterns in a high school: a comparison between data collected using wearable sensors, contact diaries and friendship surveys](https://mlsanity.com/paper/1506.03645/) (2015)
- [Graph Geometry Interaction Learning](https://mlsanity.com/paper/2010.12135/) (2020)
- [Generative hypergraph clustering: from blockmodels to modularity](https://mlsanity.com/paper/2101.09611/) (2021)
- [Mitigating Over-Smoothing and Over-Squashing using Augmentations of Forman-Ricci Curvature](https://mlsanity.com/paper/2309.09384/) (2023)
- [A Self-supervised Mixed-curvature Graph Neural Network](https://mlsanity.com/paper/2112.05393/) (2021)
- [Clustering in graphs and hypergraphs with categorical edge labels](https://mlsanity.com/paper/1910.09943/) (2019)
- [Curvature Graph Neural Network](https://mlsanity.com/paper/2106.15762/) (2021)
- [Ricci Curvature of the Internet Topology](https://mlsanity.com/paper/1501.04138/) (2015)
- [Curvature-based Clustering on Graphs](https://mlsanity.com/paper/2307.10155/) (2023)
- [Co-citation and Co-authorship Networks of Statisticians](https://mlsanity.com/paper/2204.11194/) (2022)
- [Ollivier-Ricci Curvature for Hypergraphs: A Unified Framework](https://mlsanity.com/paper/2210.12048/) (2022)
- [Forman-Ricci Curvature for Hypergraphs](https://mlsanity.com/paper/1811.07825/) (2018)
- [Network Geometry Inference using Common Neighbors](https://mlsanity.com/paper/1502.05578/) (2015)
- [Measuring Road Network Topology Vulnerability by Ricci Curvature](https://mlsanity.com/paper/1811.05743/) (2018)
- [Tensor Entropy for Uniform Hypergraphs](https://mlsanity.com/paper/1912.09624/) (2019)
- [The temporal dynamics of group interactions in higher-order social networks](https://mlsanity.com/paper/2306.09967/) (2023)
- [Relational hyperevent models for the coevolution of coauthoring and citation networks](https://mlsanity.com/paper/2308.01722/) (2023)
- [Lower Ricci Curvature for Efficient Community Detection](https://mlsanity.com/paper/2401.10124/) (2024)
- [Analyzing Neural Network Robustness Using Graph Curvature](https://mlsanity.com/paper/2410.19607/) (2024)
- [Diversities and the Geometry of Hypergraphs](https://mlsanity.com/paper/1312.5408/) (2013)

## Abstract (arXiv)

Networks with higher-order interactions, prevalent in biological, social, and information systems, are naturally represented as hypergraphs, yet their structural complexity poses fundamental challenges for geometric characterization. While curvature-based methods offer powerful insights in graph analysis, existing extensions to hypergraphs suffer from critical trade-offs: combinatorial approaches such as Forman-Ricci curvature capture only coarse features, whereas geometric methods like Ollivier-Ricci curvature offer richer expressivity but demand costly optimal transport computations. To address these challenges, we introduce hypergraph lower Ricci curvature (HLRC), a novel curvature metric defined in closed form that achieves a principled balance between interpretability and efficiency. Evaluated across diverse synthetic and real-world hypergraph datasets, HLRC consistently reveals meaningful higher-order organization, distinguishing intra- from inter-community hyperedges, uncovering latent semantic labels, tracking temporal dynamics, and supporting robust clustering of hypergraphs based on global structure. By unifying geometric sensitivity with algorithmic simplicity, HLRC provides a versatile foundation for hypergraph analytics, with broad implications for tasks including node classification, anomaly detection, and generative modeling in complex systems.