# Curvature Graph Neural Network

arXiv: https://arxiv.org/abs/2106.15762 · HTML: https://mlsanity.com/paper/2106.15762/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- Builds on 41 of its 57 references that are in the map.
- First posted on arXiv 30 Jun 2021.

## Builds on (most cited first, 30 of 41)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [A Comprehensive Survey on Graph Neural Networks](https://mlsanity.com/paper/1901.00596/) (2019)
- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [node2vec: Scalable Feature Learning for Networks](https://mlsanity.com/paper/1607.00653/) (2016)
- [Simplifying Graph Convolutional Networks](https://mlsanity.com/paper/1902.07153/) (2019)
- [Graph Contrastive Learning with Augmentations](https://mlsanity.com/paper/2010.13902/) (2020)
- [Spectral Networks and Locally Connected Networks on Graphs](https://mlsanity.com/paper/1312.6203/) (2013)
- [Deeper Insights into Graph Convolutional Networks for Semi-Supervised Learning](https://mlsanity.com/paper/1801.07606/) (2018)
- [Pitfalls of Graph Neural Network Evaluation](https://mlsanity.com/paper/1811.05868/) (2018)
- [Predict then Propagate: Graph Neural Networks meet Personalized PageRank](https://mlsanity.com/paper/1810.05997/) (2018)
- [Revisiting Semi-Supervised Learning with Graph Embeddings](https://mlsanity.com/paper/1603.08861/) (2016)
- [Gated Graph Sequence Neural Networks](https://mlsanity.com/paper/1511.05493/) (2015)
- [Convolutional Networks on Graphs for Learning Molecular Fingerprints](https://mlsanity.com/paper/1509.09292/) (2015)
- [Graph Neural Networks for Social Recommendation](https://mlsanity.com/paper/1902.07243/) (2019)
- [DropEdge: Towards Deep Graph Convolutional Networks on Node Classification](https://mlsanity.com/paper/1907.10903/) (2019)
- [Measuring and Relieving the Over-smoothing Problem for Graph Neural Networks from the Topological View](https://mlsanity.com/paper/1909.03211/) (2019)
- [Geometric deep learning on graphs and manifolds using mixture model CNNs](https://mlsanity.com/paper/1611.08402/) (2016)
- [GCC: Graph Contrastive Coding for Graph Neural Network Pre-Training](https://mlsanity.com/paper/2006.09963/) (2020)
- [Hyperbolic Graph Convolutional Neural Networks](https://mlsanity.com/paper/1910.12933/) (2019)
- [KGAT: Knowledge Graph Attention Network for Recommendation](https://mlsanity.com/paper/1905.07854/) (2019)
- [Community Detection and Stochastic Block Models](https://mlsanity.com/paper/1703.10146/) (2017)
- [Improved Semantic Representations From Tree-Structured Long Short-Term Memory Networks](https://mlsanity.com/paper/1503.00075/) (2015)
- [Wiki-CS: A Wikipedia-Based Benchmark for Graph Neural Networks](https://mlsanity.com/paper/2007.02901/) (2020)
- [Deep Convolutional Networks on Graph-Structured Data](https://mlsanity.com/paper/1506.05163/) (2015)
- [DeepInf: Social Influence Prediction with Deep Learning](https://mlsanity.com/paper/1807.05560/) (2018)

## Built on by (most cited first, 24 of 24)

- [Dynamic Graph Representation Learning with Neural Networks: A Survey](https://mlsanity.com/paper/2304.05729/) (2023)
- [Rewiring Networks for Graph Neural Network Training Using Discrete Geometry](https://mlsanity.com/paper/2207.08026/) (2022)
- [STGC-GNNs: A GNN-based traffic prediction framework with a spatial-temporal Granger causality graph](https://mlsanity.com/paper/2210.16789/) (2022)
- [Depth-Adaptive Graph Neural Networks via Learnable Bakry-'Emery Curvature](https://mlsanity.com/paper/2503.01079/) (2025)
- [AllSpark: A Multimodal Spatio-Temporal General Intelligence Model with Ten Modalities via Language as a Reference Framework](https://mlsanity.com/paper/2401.00546/) (2023)
- [Enhancing Graph Representation Learning with Localized Topological Features](https://mlsanity.com/paper/2501.09178/) (2025)
- [LSTTN: A Long-Short Term Transformer-based Spatio-temporal Neural Network for Traffic Flow Forecasting](https://mlsanity.com/paper/2403.16495/) (2024)
- [The Post-GCN Decade Revisited: Curvature-Stratified Evaluation of Relational Learning](https://mlsanity.com/paper/2606.06397/) (2026)
- [Node-Specific Space Selection via Localized Geometric Hyperbolicity in Graph Neural Networks](https://mlsanity.com/paper/2303.01724/) (2023)
- [Analyzing Neural Network Robustness Using Graph Curvature](https://mlsanity.com/paper/2410.19607/) (2024)
- [Efficient Curvature-aware Graph Network](https://mlsanity.com/paper/2511.01443/) (2025)
- [Locally temporal-spatial pattern learning with graph attention mechanism for EEG-based emotion recognition](https://mlsanity.com/paper/2208.11087/) (2022)
- [Graph Neural Networks with a Distribution of Parametrized Graphs](https://mlsanity.com/paper/2310.16401/) (2023)
- [SeFi-CD: A Semantic First Change Detection Paradigm That Can Detect Any Change You Want](https://mlsanity.com/paper/2407.09874/) (2024)
- [Dynamic Graph Structure Learning via Resistance Curvature Flow](https://mlsanity.com/paper/2601.08149/) (2026)
- [Geometric Flow enhanced Graph Coarsening](https://mlsanity.com/paper/2609.14962/) (2026)
- [Curvature-enhanced Graph Convolutional Network for Biomolecular Interaction Prediction](https://mlsanity.com/paper/2306.13699/) (2023)
- [Unsupervised Graph Anomaly Detection via Multi-Hypersphere Heterophilic Graph Learning](https://mlsanity.com/paper/2503.12037/) (2025)
- [Lower Ricci Curvature for Hypergraphs](https://mlsanity.com/paper/2506.03943/) (2025)
- [Reconciling Attribute and Structural Anomalies for Improved Graph Anomaly Detection](https://mlsanity.com/paper/2506.23469/) (2025)
- [A roadmap for curvature-based geometric data analysis and learning](https://mlsanity.com/paper/2510.22599/) (2025)
- [Geometric Evolution Graph Convolutional Networks: Enhancing Graph Representation Learning via Ricci Flow](https://mlsanity.com/paper/2603.26178/) (2026)
- [Local-Global Geometric Insights for Graph Neural Networks via Entropic Curvature](https://mlsanity.com/paper/2607.22381/) (2026)
- [Geometric Structures on Graphs: a Holonomy-Based Discretization of Curvature](https://mlsanity.com/paper/2608.22453/) (2026)

## Abstract (arXiv)

Graph neural networks (GNNs) have achieved great success in many graph-based tasks. Much work is dedicated to empowering GNNs with the adaptive locality ability, which enables measuring the importance of neighboring nodes to the target node by a node-specific mechanism. However, the current node-specific mechanisms are deficient in distinguishing the importance of nodes in the topology structure. We believe that the structural importance of neighboring nodes is closely related to their importance in aggregation. In this paper, we introduce discrete graph curvature (the Ricci curvature) to quantify the strength of structural connection of pairwise nodes. And we propose Curvature Graph Neural Network (CGNN), which effectively improves the adaptive locality ability of GNNs by leveraging the structural property of graph curvature. To improve the adaptability of curvature to various datasets, we explicitly transform curvature into the weights of neighboring nodes by the necessary Negative Curvature Processing Module and Curvature Normalization Module. Then, we conduct numerous experiments on various synthetic datasets and real-world datasets. The experimental results on synthetic datasets show that CGNN effectively exploits the topology structure information, and the performance is improved significantly. CGNN outperforms the baselines on 5 dense node classification benchmark datasets. This study deepens the understanding of how to utilize advanced topology information and assign the importance of neighboring nodes from the perspective of graph curvature and encourages us to bridge the gap between graph theory and neural networks.