# Ricci Curvature of the Internet Topology

arXiv: https://arxiv.org/abs/1501.04138 · HTML: https://mlsanity.com/paper/1501.04138/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Jan 2015.

## Built on by (most cited first, 22 of 22)

- [Revisiting Over-smoothing and Over-squashing Using Ollivier-Ricci Curvature](https://mlsanity.com/paper/2211.15779/) (2022)
- [Position-aware Structure Learning for Graph Topology-imbalance by Relieving Under-reaching and Over-squashing](https://mlsanity.com/paper/2208.08302/) (2022)
- [Curvature Graph Neural Network](https://mlsanity.com/paper/2106.15762/) (2021)
- [kHGCN: Tree-likeness Modeling via Continuous and Discrete Curvature Learning](https://mlsanity.com/paper/2212.01793/) (2022)
- [Curvature Graph Generative Adversarial Networks](https://mlsanity.com/paper/2203.01604/) (2022)
- [Depth-Adaptive Graph Neural Networks via Learnable Bakry-'Emery Curvature](https://mlsanity.com/paper/2503.01079/) (2025)
- [Augmenting the User-Item Graph with Textual Similarity Models](https://mlsanity.com/paper/2109.09358/) (2021)
- [Knowledge Graph Completion with Mixed Geometry Tensor Factorization](https://mlsanity.com/paper/2504.02589/) (2025)
- [The Post-GCN Decade Revisited: Curvature-Stratified Evaluation of Relational Learning](https://mlsanity.com/paper/2606.06397/) (2026)
- [Accelerated Evaluation of Ollivier-Ricci Curvature Lower Bounds: Bridging Theory and Computation](https://mlsanity.com/paper/2405.13302/) (2024)
- [Analyzing Neural Network Robustness Using Graph Curvature](https://mlsanity.com/paper/2410.19607/) (2024)
- [RicciFlowRec: A Geometric Root Cause Recommender Using Ricci Curvature on Financial Graphs](https://mlsanity.com/paper/2508.09334/) (2025)
- [Alleviating neighbor bias: augmenting graph self-supervise learning with structural equivalent positive samples](https://mlsanity.com/paper/2212.04365/) (2022)
- [Hypergraph clustering using Ricci curvature: an edge transport perspective](https://mlsanity.com/paper/2412.15695/) (2024)
- [Curvature-enhanced Graph Convolutional Network for Biomolecular Interaction Prediction](https://mlsanity.com/paper/2306.13699/) (2023)
- [CTAGE: Curvature-Based Topology-Aware Graph Embedding for Learning Molecular Representations](https://mlsanity.com/paper/2307.13275/) (2023)
- [Ironing the Graphs: Toward a Correct Geometric Analysis of Large-Scale Graphs](https://mlsanity.com/paper/2407.21609/) (2024)
- [Characterizing Physician Referral Networks with Ricci Curvature](https://mlsanity.com/paper/2408.16022/) (2024)
- [Lower Ricci Curvature for Hypergraphs](https://mlsanity.com/paper/2506.03943/) (2025)
- [Ontology Neural Networks for Topologically Conditioned Constraint Satisfaction](https://mlsanity.com/paper/2601.05304/) (2026)
- [Post-Training Neural Network Pruning using Graph Curvature](https://mlsanity.com/paper/2601.16366/) (2026)
- [Sobolev--Ricci Curvature](https://mlsanity.com/paper/2603.12652/) (2026)

## Abstract (arXiv)

Analysis of Internet topologies has shown that the Internet topology has negative curvature, measured by Gromov's "thin triangle condition", which is tightly related to core congestion and route reliability. In this work we analyze the discrete Ricci curvature of the Internet, defined by Ollivier, Lin, etc. Ricci curvature measures whether local distances diverge or converge. It is a more local measure which allows us to understand the distribution of curvatures in the network. We show by various Internet data sets that the distribution of Ricci cuvature is spread out, suggesting the network topology to be non-homogenous. We also show that the Ricci curvature has interesting connections to both local measures such as node degree and clustering coefficient, global measures such as betweenness centrality and network connectivity, as well as auxilary attributes such as geographical distances. These observations add to the richness of geometric structures in complex network theory.