# A Hyperbolic-to-Hyperbolic Graph Convolutional Network

arXiv: https://arxiv.org/abs/2104.06942 · HTML: https://mlsanity.com/paper/2104.06942/

## Key facts

- Cited by 53 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 52 references that are in the map.
- First posted on arXiv 14 Apr 2021.

## Builds on (most cited first, 27 of 27)

- [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)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [Simplifying Graph Convolutional Networks](https://mlsanity.com/paper/1902.07153/) (2019)
- [Deep Graph Infomax](https://mlsanity.com/paper/1809.10341/) (2018)
- [Spectral Networks and Locally Connected Networks on Graphs](https://mlsanity.com/paper/1312.6203/) (2013)
- [Geom-GCN: Geometric Graph Convolutional Networks](https://mlsanity.com/paper/2002.05287/) (2020)
- [Hierarchical Graph Representation Learning with Differentiable Pooling](https://mlsanity.com/paper/1806.08804/) (2018)
- [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)
- [Dynamic Edge-Conditioned Filters in Convolutional Neural Networks on Graphs](https://mlsanity.com/paper/1704.02901/) (2017)
- [Learning Continuous Hierarchies in the Lorentz Model of Hyperbolic Geometry](https://mlsanity.com/paper/1806.03417/) (2018)
- [Hyperbolic Graph Neural Networks](https://mlsanity.com/paper/1910.12892/) (2019)
- [A Fair Comparison of Graph Neural Networks for Graph Classification](https://mlsanity.com/paper/1912.09893/) (2019)
- [Hyperbolic Image Embeddings](https://mlsanity.com/paper/1904.02239/) (2019)
- [Representation Tradeoffs for Hyperbolic Embeddings](https://mlsanity.com/paper/1804.03329/) (2018)
- [Multi-relational Poincar'e Graph Embeddings](https://mlsanity.com/paper/1905.09791/) (2019)
- [Hyperbolic Graph Attention Network](https://mlsanity.com/paper/1912.03046/) (2019)
- [From Trees to Continuous Embeddings and Back: Hyperbolic Hierarchical Clustering](https://mlsanity.com/paper/2010.00402/) (2020)
- [Tree! I am no Tree! I am a Low Dimensional Hyperbolic Embedding](https://mlsanity.com/paper/2005.03847/) (2020)
- [Knowledge Association with Hyperbolic Knowledge Graph Embeddings](https://mlsanity.com/paper/2010.02162/) (2020)
- [Robust Large-Margin Learning in Hyperbolic Space](https://mlsanity.com/paper/2004.05465/) (2020)
- [Cross Euclidean-to-Riemannian Metric Learning with Application to Face Recognition from Video](https://mlsanity.com/paper/1608.04200/) (2016)

## Built on by (most cited first, 30 of 53)

- [Hyperbolic Image Segmentation](https://mlsanity.com/paper/2203.05898/) (2022)
- [Hyperbolic Deep Learning in Computer Vision: A Survey](https://mlsanity.com/paper/2305.06611/) (2023)
- [Hyperbolic Graph Neural Networks: A Review of Methods and Applications](https://mlsanity.com/paper/2202.13852/) (2022)
- [Pseudo-Riemannian Graph Convolutional Networks](https://mlsanity.com/paper/2106.03134/) (2021)
- [Hyperbolic Fine-Tuning for Large Language Models](https://mlsanity.com/paper/2410.04010/) (2024)
- [Poincar'e ResNet](https://mlsanity.com/paper/2303.14027/) (2023)
- [Motif-aware Riemannian Graph Neural Network with Generative-Contrastive Learning](https://mlsanity.com/paper/2401.01232/) (2024)
- [Hyperbolic Deep Learning for Foundation Models: A Survey](https://mlsanity.com/paper/2507.17787/) (2025)
- [DeepRicci: Self-supervised Graph Structure-Feature Co-Refinement for Alleviating Over-squashing](https://mlsanity.com/paper/2401.12780/) (2024)
- [SA-GDA: Spectral Augmentation for Graph Domain Adaptation](https://mlsanity.com/paper/2408.09189/) (2024)
- [Learning Structured Representations with Hyperbolic Embeddings](https://mlsanity.com/paper/2412.01023/) (2024)
- [Graph Neural Networks on SPD Manifolds for Motor Imagery Classification: A Perspective from the Time-Frequency Analysis](https://mlsanity.com/paper/2211.02641/) (2022)
- [Contrastive Graph Clustering in Curvature Spaces](https://mlsanity.com/paper/2305.03555/) (2023)
- [Nested Hyperbolic Spaces for Dimensionality Reduction and Hyperbolic NN Design](https://mlsanity.com/paper/2112.03402/) (2021)
- [HGWaveNet: A Hyperbolic Graph Neural Network for Temporal Link Prediction](https://mlsanity.com/paper/2304.07302/) (2023)
- [Matrix Manifold Neural Networks++](https://mlsanity.com/paper/2405.19206/) (2024)
- [Public Wisdom Matters! Discourse-Aware Hyperbolic Fourier Co-Attention for Social-Text Classification](https://mlsanity.com/paper/2209.13017/) (2022)
- [D-HYPR: Harnessing Neighborhood Modeling and Asymmetry Preservation for Digraph Representation Learning](https://mlsanity.com/paper/2112.11734/) (2021)
- [Autoencoding Hyperbolic Representation for Adversarial Generation](https://mlsanity.com/paper/2201.12825/) (2022)
- [Hyperbolic Graph Diffusion Model](https://mlsanity.com/paper/2306.07618/) (2023)
- [A Fashion Item Recommendation Model in Hyperbolic Space](https://mlsanity.com/paper/2409.02599/) (2024)
- [Klein Model for Hyperbolic Neural Networks](https://mlsanity.com/paper/2410.16813/) (2024)
- [Continual Hyperbolic Learning of Instances and Classes](https://mlsanity.com/paper/2506.10710/) (2025)
- [Hyperbolic Busemann Neural Networks](https://mlsanity.com/paper/2602.18858/) (2026)
- [DeepHGCN: Toward Deeper Hyperbolic Graph Convolutional Networks](https://mlsanity.com/paper/2310.02027/) (2023)
- [Robust Hyperbolic Learning with Curvature-Aware Optimization](https://mlsanity.com/paper/2405.13979/) (2024)
- [Curvature Learning for Generalization of Hyperbolic Neural Networks](https://mlsanity.com/paper/2508.17232/) (2025)
- [Supervised Contrastive Learning with Structure Inference for Graph Classification](https://mlsanity.com/paper/2203.07691/) (2022)
- [Contrastive Sequential Interaction Network Learning on Co-Evolving Riemannian Spaces](https://mlsanity.com/paper/2401.01243/) (2024)
- [Three Revisits to Node-Level Graph Anomaly Detection: Outliers, Message Passing and Hyperbolic Neural Networks](https://mlsanity.com/paper/2403.04010/) (2024)

## Abstract (arXiv)

Hyperbolic graph convolutional networks (GCNs) demonstrate powerful representation ability to model graphs with hierarchical structure. Existing hyperbolic GCNs resort to tangent spaces to realize graph convolution on hyperbolic manifolds, which is inferior because tangent space is only a local approximation of a manifold. In this paper, we propose a hyperbolic-to-hyperbolic graph convolutional network (H2H-GCN) that directly works on hyperbolic manifolds. Specifically, we developed a manifold-preserving graph convolution that consists of a hyperbolic feature transformation and a hyperbolic neighborhood aggregation. The hyperbolic feature transformation works as linear transformation on hyperbolic manifolds. It ensures the transformed node representations still lie on the hyperbolic manifold by imposing the orthogonal constraint on the transformation sub-matrix. The hyperbolic neighborhood aggregation updates each node representation via the Einstein midpoint. The H2H-GCN avoids the distortion caused by tangent space approximations and keeps the global hyperbolic structure. Extensive experiments show that the H2H-GCN achieves substantial improvements on the link prediction, node classification, and graph classification tasks.