# Local-Global Geometric Insights for Graph Neural Networks via Entropic Curvature

arXiv: https://arxiv.org/abs/2607.22381 · HTML: https://mlsanity.com/paper/2607.22381/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 25 of its 72 references that are in the map.
- First posted on arXiv 24 Jul 2026.

## Builds on (most cited first, 25 of 25)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [Deeper Insights into Graph Convolutional Networks for Semi-Supervised Learning](https://mlsanity.com/paper/1801.07606/) (2018)
- [Revisiting Semi-Supervised Learning with Graph Embeddings](https://mlsanity.com/paper/1603.08861/) (2016)
- [Geom-GCN: Geometric Graph Convolutional Networks](https://mlsanity.com/paper/2002.05287/) (2020)
- [TUDataset: A collection of benchmark datasets for learning with graphs](https://mlsanity.com/paper/2007.08663/) (2020)
- [Understanding over-squashing and bottlenecks on graphs via curvature](https://mlsanity.com/paper/2111.14522/) (2021)
- [A Note on Over-Smoothing for Graph Neural Networks](https://mlsanity.com/paper/2006.13318/) (2020)
- [Constant Curvature Graph Convolutional Networks](https://mlsanity.com/paper/1911.05076/) (2019)
- [FoSR: First-order spectral rewiring for addressing oversquashing in GNNs](https://mlsanity.com/paper/2210.11790/) (2022)
- [Revisiting Over-smoothing and Over-squashing Using Ollivier-Ricci Curvature](https://mlsanity.com/paper/2211.15779/) (2022)
- [How Powerful are K-hop Message Passing Graph Neural Networks](https://mlsanity.com/paper/2205.13328/) (2022)
- [Community Detection on Networks with Ricci Flow](https://mlsanity.com/paper/1907.03993/) (2019)
- [Towards Understanding the Generalization of Graph Neural Networks](https://mlsanity.com/paper/2305.08048/) (2023)
- [Mitigating Over-Smoothing and Over-Squashing using Augmentations of Forman-Ricci Curvature](https://mlsanity.com/paper/2309.09384/) (2023)
- [Graph Positional Encoding via Random Feature Propagation](https://mlsanity.com/paper/2303.02918/) (2023)
- [Curvature Graph Neural Network](https://mlsanity.com/paper/2106.15762/) (2021)
- [A Manifold Perspective on the Statistical Generalization of Graph Neural Networks](https://mlsanity.com/paper/2406.05225/) (2024)
- [Configuring Random Graph Models with Fixed Degree Sequences](https://mlsanity.com/paper/1608.00607/) (2016)
- [Learning Parametrised Graph Shift Operators](https://mlsanity.com/paper/2101.10050/) (2021)
- [Curvature-based Clustering on Graphs](https://mlsanity.com/paper/2307.10155/) (2023)
- [Effective Structural Encodings via Local Curvature Profiles](https://mlsanity.com/paper/2311.14864/) (2023)
- [The Effectiveness of Curvature-Based Rewiring and the Role of Hyperparameters in GNNs Revisited](https://mlsanity.com/paper/2407.09381/) (2024)
- [Which Graph Shift Operator? A Spectral Answer to an Empirical Question](https://mlsanity.com/paper/2602.06557/) (2026)

## Abstract (arXiv)

Curvature notions on graphs, particularly Ollivier-Ricci and Forman, have emerged as powerful tools for addressing fundamental issues in Graph Neural Networks (GNNs) such as oversmoothing and oversquashing, but rely almost exclusively on local edge-level comparisons and therefore fail to certify how information actually propagates over long distances. We introduce Entropic Curvature, a global, transport-based curvature obtained by extending the Lott-Sturm-Villani framework to graphs through the displacement convexity of entropy along Wasserstein geodesics. We define a tractable Weak Entropic Curvature proxy that lower-bounds the global entropic curvature, and from it derive (i) a Poincare-type inequality controlling oversmoothing, (ii) a transport-entropy generalization bound, and (iii) an expansion paradox proving that sparsity, strong spectral expansion, and positive entropic curvature cannot coexist in large graphs, unifying oversmoothing and oversquashing as opposite ends of a single curvature spectrum. We translate the theory into three practical mechanisms, the E-Gate aggregator, the ENT structural encoding, and Midpoint-Completion Rewiring (MCR), and benchmark them against SDRF, FoSR, BORF, LCP, and Graph Ricci Flow on six node-classification benchmarks, and graph-classification.