# Pseudo-Riemannian Graph Convolutional Networks

arXiv: https://arxiv.org/abs/2106.03134 · HTML: https://mlsanity.com/paper/2106.03134/

## Key facts

- Cited by 30 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 52 references that are in the map.
- First posted on arXiv 6 Jun 2021.

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

- [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)
- [Simplifying Graph Convolutional Networks](https://mlsanity.com/paper/1902.07153/) (2019)
- [Spectral Networks and Locally Connected Networks on Graphs](https://mlsanity.com/paper/1312.6203/) (2013)
- [Revisiting Semi-Supervised Learning with Graph Embeddings](https://mlsanity.com/paper/1603.08861/) (2016)
- [Representation Learning on Graphs with Jumping Knowledge Networks](https://mlsanity.com/paper/1806.03536/) (2018)
- [Simple and Deep Graph Convolutional Networks](https://mlsanity.com/paper/2007.02133/) (2020)
- [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)
- [Hyperbolic Graph Neural Networks](https://mlsanity.com/paper/1910.12892/) (2019)
- [Hyperspherical Variational Auto-Encoders](https://mlsanity.com/paper/1804.00891/) (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)
- [Mixed-curvature Variational Autoencoders](https://mlsanity.com/paper/1911.08411/) (2019)
- [Embedding Text in Hyperbolic Spaces](https://mlsanity.com/paper/1806.04313/) (2018)
- [Lorentzian Graph Convolutional Networks](https://mlsanity.com/paper/2104.07477/) (2021)
- [Spherical Latent Spaces for Stable Variational Autoencoders](https://mlsanity.com/paper/1808.10805/) (2018)
- [Graph Geometry Interaction Learning](https://mlsanity.com/paper/2010.12135/) (2020)
- [A Hyperbolic-to-Hyperbolic Graph Convolutional Network](https://mlsanity.com/paper/2104.06942/) (2021)
- [Spherical Text Embedding](https://mlsanity.com/paper/1911.01196/) (2019)
- [Self-Supervised Hyperboloid Representations from Logical Queries over Knowledge Graphs](https://mlsanity.com/paper/2012.13023/) (2020)
- [Network Geometry](https://mlsanity.com/paper/2001.03241/) (2020)
- [DyERNIE: Dynamic Evolution of Riemannian Manifold Embeddings for Temporal Knowledge Graph Completion](https://mlsanity.com/paper/2011.03984/) (2020)
- [ResGCN: Attention-based Deep Residual Modeling for Anomaly Detection on Attributed Networks](https://mlsanity.com/paper/2009.14738/) (2020)
- [Ultrahyperbolic Representation Learning](https://mlsanity.com/paper/2007.00211/) (2020)
- [DeepSphere: towards an equivariant graph-based spherical CNN](https://mlsanity.com/paper/1904.05146/) (2019)
- [Embedding Graphs in Lorentzian Spacetime](https://mlsanity.com/paper/1602.03103/) (2016)
- [Semi-Riemannian Manifold Optimization](https://mlsanity.com/paper/1812.07643/) (2018)

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

- [Finding the Missing-half: Graph Complementary Learning for Homophily-prone and Heterophily-prone Graphs](https://mlsanity.com/paper/2306.07608/) (2023)
- [Hyperbolic Representation Learning: Revisiting and Advancing](https://mlsanity.com/paper/2306.09118/) (2023)
- [LSEnet: Lorentz Structural Entropy Neural Network for Deep Graph Clustering](https://mlsanity.com/paper/2405.11801/) (2024)
- [DeepRicci: Self-supervised Graph Structure-Feature Co-Refinement for Alleviating Over-squashing](https://mlsanity.com/paper/2401.12780/) (2024)
- [Spiking Graph Neural Network on Riemannian Manifolds](https://mlsanity.com/paper/2410.17941/) (2024)
- [Contrastive Graph Clustering in Curvature Spaces](https://mlsanity.com/paper/2305.03555/) (2023)
- [Curve Your Attention: Mixed-Curvature Transformers for Graph Representation Learning](https://mlsanity.com/paper/2309.04082/) (2023)
- [kHGCN: Tree-likeness Modeling via Continuous and Discrete Curvature Learning](https://mlsanity.com/paper/2212.01793/) (2022)
- [Graph Sequential Neural ODE Process for Link Prediction on Dynamic and Sparse Graphs](https://mlsanity.com/paper/2211.08568/) (2022)
- [Node Embedding from Neural Hamiltonian Orbits in Graph Neural Networks](https://mlsanity.com/paper/2305.18965/) (2023)
- [HyperCore: The Core Framework for Building Hyperbolic Foundation Models with Comprehensive Modules](https://mlsanity.com/paper/2504.08912/) (2025)
- [Deeper with Riemannian Geometry: Overcoming Oversmoothing and Oversquashing for Graph Foundation Models](https://mlsanity.com/paper/2510.17457/) (2025)
- [Spectro-Riemannian Graph Neural Networks](https://mlsanity.com/paper/2502.00401/) (2025)
- [Multi-Domain Riemannian Graph Gluing for Building Graph Foundation Models](https://mlsanity.com/paper/2603.00618/) (2026)
- [CurvGAD: Leveraging Curvature for Enhanced Graph Anomaly Detection](https://mlsanity.com/paper/2502.08605/) (2025)
- [ASIL: Augmented Structural Information Learning for Deep Graph Clustering in Hyperbolic Space](https://mlsanity.com/paper/2504.09970/) (2025)
- [Contrastive Sequential Interaction Network Learning on Co-Evolving Riemannian Spaces](https://mlsanity.com/paper/2401.01243/) (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)
- [Over-Squashing in Riemannian Graph Neural Networks](https://mlsanity.com/paper/2311.15945/) (2023)
- [Manifold GCN: Diffusion-based Convolutional Neural Network for Manifold-valued Graphs](https://mlsanity.com/paper/2401.14381/) (2024)
- [Leveraging Manifold Embeddings for Enhanced Graph Transformer Representations and Learning](https://mlsanity.com/paper/2507.07335/) (2025)
- [Node Embedding from Hamiltonian Information Propagation in Graph Neural Networks](https://mlsanity.com/paper/2303.01030/) (2023)
- [Symplectic Structure-Aware Hamiltonian (Graph) Embeddings](https://mlsanity.com/paper/2309.04885/) (2023)
- [Disentangled Hyperbolic Representation Learning for Heterogeneous Graphs](https://mlsanity.com/paper/2406.10367/) (2024)
- [AHSG: Adversarial Attack on High-level Semantics in Graph Neural Networks](https://mlsanity.com/paper/2412.07468/) (2024)
- [PR-CapsNet: Pseudo-Riemannian Capsule Network with Adaptive Curvature Routing for Graph Learning](https://mlsanity.com/paper/2512.08218/) (2025)
- [Hyperbolic Graph Embeddings: a Survey and an Evaluation on Anomaly Detection](https://mlsanity.com/paper/2512.18826/) (2025)
- [Gaussian Sheaf Neural Networks](https://mlsanity.com/paper/2605.21435/) (2026)
- [Are Common Substructures Transferable? Riemannian Graph Foundation Model with Neural Vector Bundles](https://mlsanity.com/paper/2606.03270/) (2026)

## Abstract (arXiv)

Graph convolutional networks (GCNs) are powerful frameworks for learning embeddings of graph-structured data. GCNs are traditionally studied through the lens of Euclidean geometry. Recent works find that non-Euclidean Riemannian manifolds provide specific inductive biases for embedding hierarchical or spherical data. However, they cannot align well with data of mixed graph topologies. We consider a larger class of pseudo-Riemannian manifolds that generalize hyperboloid and sphere. We develop new geodesic tools that allow for extending neural network operations into geodesically disconnected pseudo-Riemannian manifolds. As a consequence, we derive a pseudo-Riemannian GCN that models data in pseudo-Riemannian manifolds of constant nonzero curvature in the context of graph neural networks. Our method provides a geometric inductive bias that is sufficiently flexible to model mixed heterogeneous topologies like hierarchical graphs with cycles. We demonstrate the representational capabilities of this method by applying it to the tasks of graph reconstruction, node classification and link prediction on a series of standard graphs with mixed topologies. Empirical results demonstrate that our method outperforms Riemannian counterparts when embedding graphs of complex topologies.