# ACE-HGNN: Adaptive Curvature Exploration Hyperbolic Graph Neural Network

arXiv: https://arxiv.org/abs/2110.07888 · HTML: https://mlsanity.com/paper/2110.07888/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 39 references that are in the map.
- First posted on arXiv 15 Oct 2021.

## Builds on (most cited first, 19 of 19)

- [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)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Hyperbolic Neural Networks](https://mlsanity.com/paper/1805.09112/) (2018)
- [Hyperbolic Graph Convolutional Neural Networks](https://mlsanity.com/paper/1910.12933/) (2019)
- [Learning Continuous Hierarchies in the Lorentz Model of Hyperbolic Geometry](https://mlsanity.com/paper/1806.03417/) (2018)
- [Enhancing Graph Neural Network-based Fraud Detectors against Camouflaged Fraudsters](https://mlsanity.com/paper/2008.08692/) (2020)
- [Hyperbolic Graph Neural Networks](https://mlsanity.com/paper/1910.12892/) (2019)
- [Poincar'e GloVe: Hyperbolic Word Embeddings](https://mlsanity.com/paper/1810.06546/) (2018)
- [Multi-relational Poincar'e Graph Embeddings](https://mlsanity.com/paper/1905.09791/) (2019)
- [Constant Curvature Graph Convolutional Networks](https://mlsanity.com/paper/1911.05076/) (2019)
- [SUGAR: Subgraph Neural Network with Reinforcement Pooling and Self-Supervised Mutual Information Mechanism](https://mlsanity.com/paper/2101.08170/) (2021)
- [MotifNet: a motif-based Graph Convolutional Network for directed graphs](https://mlsanity.com/paper/1802.01572/) (2018)
- [Hyperbolic Variational Graph Neural Network for Modeling Dynamic Graphs](https://mlsanity.com/paper/2104.02228/) (2021)
- [Reinforced Neighborhood Selection Guided Multi-Relational Graph Neural Networks](https://mlsanity.com/paper/2104.07886/) (2021)
- [GraLSP: Graph Neural Networks with Local Structural Patterns](https://mlsanity.com/paper/1911.07675/) (2019)
- [Higher-Order Attribute-Enhancing Heterogeneous Graph Neural Networks](https://mlsanity.com/paper/2104.07892/) (2021)

## Built on by (most cited first, 25 of 25)

- [Hyperbolic Graph Neural Networks: A Review of Methods and Applications](https://mlsanity.com/paper/2202.13852/) (2022)
- [A Self-supervised Mixed-curvature Graph Neural Network](https://mlsanity.com/paper/2112.05393/) (2021)
- [Position-aware Structure Learning for Graph Topology-imbalance by Relieving Under-reaching and Over-squashing](https://mlsanity.com/paper/2208.08302/) (2022)
- [A Self-supervised Riemannian GNN with Time Varying Curvature for Temporal Graph Learning](https://mlsanity.com/paper/2208.14073/) (2022)
- [Hyperbolic Geometric Graph Representation Learning for Hierarchy-imbalance Node Classification](https://mlsanity.com/paper/2304.05059/) (2023)
- [Motif-aware Riemannian Graph Neural Network with Generative-Contrastive Learning](https://mlsanity.com/paper/2401.01232/) (2024)
- [LSEnet: Lorentz Structural Entropy Neural Network for Deep Graph Clustering](https://mlsanity.com/paper/2405.11801/) (2024)
- [Reinforcement learning on graphs: A survey](https://mlsanity.com/paper/2204.06127/) (2022)
- [Position: Beyond Euclidean -- Foundation Models Should Embrace Non-Euclidean Geometries](https://mlsanity.com/paper/2504.08896/) (2025)
- [Curvature Graph Generative Adversarial Networks](https://mlsanity.com/paper/2203.01604/) (2022)
- [SINCERE: Sequential Interaction Networks representation learning on Co-Evolving RiEmannian manifolds](https://mlsanity.com/paper/2305.03883/) (2023)
- [Robust Hyperbolic Learning with Curvature-Aware Optimization](https://mlsanity.com/paper/2405.13979/) (2024)
- [Discrete Curvature Graph Information Bottleneck](https://mlsanity.com/paper/2412.19993/) (2024)
- [Curvature Learning for Generalization of Hyperbolic Neural Networks](https://mlsanity.com/paper/2508.17232/) (2025)
- [Heterogeneous Social Event Detection via Hyperbolic Graph Representations](https://mlsanity.com/paper/2302.10362/) (2023)
- [Hyp-UML: Hyperbolic Image Retrieval with Uncertainty-aware Metric Learning](https://mlsanity.com/paper/2310.08390/) (2023)
- [Contrastive Sequential Interaction Network Learning on Co-Evolving Riemannian Spaces](https://mlsanity.com/paper/2401.01243/) (2024)
- [Hyperbolic Graph Representation Learning: A Tutorial](https://mlsanity.com/paper/2211.04050/) (2022)
- [Torsion Graph Neural Networks](https://mlsanity.com/paper/2306.13541/) (2023)
- [Curvature-enhanced Graph Convolutional Network for Biomolecular Interaction Prediction](https://mlsanity.com/paper/2306.13699/) (2023)
- [Hyperbolic Graph Embeddings: a Survey and an Evaluation on Anomaly Detection](https://mlsanity.com/paper/2512.18826/) (2025)
- [Zero-shot Generalizable Graph Anomaly Detection with Mixture of Riemannian Experts](https://mlsanity.com/paper/2602.06859/) (2026)
- [Topology-Adaptive Hyperbolic Graph Attention Networks Guided by the Hyperbolic Sombor Index](https://mlsanity.com/paper/2609.32275/) (2026)
- [Learning Propagation Geometry from Message-Passing Feedback](https://mlsanity.com/paper/2609.34711/) (2026)
- [Temporal Geometry of Deep Networks: Hyperbolic Representations of Training Dynamics for Intrinsic Explainability](https://mlsanity.com/paper/2610.03000/) (2026)

## Abstract (arXiv)

Graph Neural Networks (GNNs) have been widely studied in various graph data mining tasks. Most existingGNNs embed graph data into Euclidean space and thus are less effective to capture the ubiquitous hierarchical structures in real-world networks. Hyperbolic Graph Neural Networks(HGNNs) extend GNNs to hyperbolic space and thus are more effective to capture the hierarchical structures of graphs in node representation learning. In hyperbolic geometry, the graph hierarchical structure can be reflected by the curvatures of the hyperbolic space, and different curvatures can model different hierarchical structures of a graph. However, most existing HGNNs manually set the curvature to a fixed value for simplicity, which achieves a suboptimal performance of graph learning due to the complex and diverse hierarchical structures of the graphs. To resolve this problem, we propose an Adaptive Curvature Exploration Hyperbolic Graph NeuralNetwork named ACE-HGNN to adaptively learn the optimal curvature according to the input graph and downstream tasks. Specifically, ACE-HGNN exploits a multi-agent reinforcement learning framework and contains two agents, ACE-Agent andHGNN-Agent for learning the curvature and node representations, respectively. The two agents are updated by a NashQ-leaning algorithm collaboratively, seeking the optimal hyperbolic space indexed by the curvature. Extensive experiments on multiple real-world graph datasets demonstrate a significant and consistent performance improvement in model quality with competitive performance and good generalization ability.