# Ollivier-Ricci curvature convergence in random geometric graphs

arXiv: https://arxiv.org/abs/2008.01209 · HTML: https://mlsanity.com/paper/2008.01209/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Aug 2020.

## Built on by (most cited first, 11 of 11)

- [Revisiting Over-smoothing and Over-squashing Using Ollivier-Ricci Curvature](https://mlsanity.com/paper/2211.15779/) (2022)
- [The Autodidactic Universe](https://mlsanity.com/paper/2104.03902/) (2021)
- [Continuum Limits of Ollivier's Ricci Curvature on data clouds: pointwise consistency and global lower bounds](https://mlsanity.com/paper/2307.02378/) (2023)
- [Hypergraph clustering using Ricci curvature: an edge transport perspective](https://mlsanity.com/paper/2412.15695/) (2024)
- [Neural Feature Geometry Evolves as Discrete Ricci Flow](https://mlsanity.com/paper/2509.22362/) (2025)
- [Discrete scalar curvature as a weighted sum of Ollivier-Ricci curvatures](https://mlsanity.com/paper/2510.04936/) (2025)
- [Diffusion Operator Geometry of Feedforward Representations](https://mlsanity.com/paper/2605.01107/) (2026)
- [Non-Parametric Manifold Learning](https://mlsanity.com/paper/2107.08089/) (2021)
- [A roadmap for curvature-based geometric data analysis and learning](https://mlsanity.com/paper/2510.22599/) (2025)
- [Sobolev--Ricci Curvature](https://mlsanity.com/paper/2603.12652/) (2026)
- [From Direction to Magnitude: How Multimodal Instruction-Tuning Reorganizes the Geometric Encoding of Identity-Specifying Prompts in Transformer Hidden States](https://mlsanity.com/paper/2607.09842/) (2026)

## Abstract (arXiv)

Connections between continuous and discrete worlds tend to be elusive. One example is curvature. Even though there exist numerous nonequivalent definitions of graph curvature, none is known to converge in any limit to any traditional definition of curvature of a Riemannian manifold. Here we show that Ollivier curvature of random geometric graphs in any Riemannian manifold converges in the continuum limit to Ricci curvature of the underlying manifold, but only if the definition of Ollivier graph curvature is properly generalized to apply to mesoscopic graph neighborhoods. This result establishes the first rigorous link between a definition of curvature applicable to networks and a traditional definition of curvature of smooth spaces.