# Adaptive Riemannian Graph Neural Networks

arXiv: https://arxiv.org/abs/2508.02600 · HTML: https://mlsanity.com/paper/2508.02600/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 52 references that are in the map.
- First posted on arXiv 4 Aug 2025.

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

- [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)
- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [Open Graph Benchmark: Datasets for Machine Learning on Graphs](https://mlsanity.com/paper/2005.00687/) (2020)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Geom-GCN: Geometric Graph Convolutional Networks](https://mlsanity.com/paper/2002.05287/) (2020)
- [Adaptive Universal Generalized PageRank Graph Neural Network](https://mlsanity.com/paper/2006.07988/) (2020)
- [Multi-scale Attributed Node Embedding](https://mlsanity.com/paper/1909.13021/) (2019)
- [Understanding over-squashing and bottlenecks on graphs via curvature](https://mlsanity.com/paper/2111.14522/) (2021)
- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (2018)
- [Beyond Low-frequency Information in Graph Convolutional Networks](https://mlsanity.com/paper/2101.00797/) (2021)
- [A Riemannian Network for SPD Matrix Learning](https://mlsanity.com/paper/1608.04233/) (2016)
- [Geoopt: Riemannian Optimization in PyTorch](https://mlsanity.com/paper/2005.02819/) (2020)
- [Constant Curvature Graph Convolutional Networks](https://mlsanity.com/paper/1911.05076/) (2019)
- [Self-supervised Learning on Graphs: Deep Insights and New Direction](https://mlsanity.com/paper/2006.10141/) (2020)
- [Neural Manifold Ordinary Differential Equations](https://mlsanity.com/paper/2006.10254/) (2020)
- [Hyperbolic Graph Attention Network](https://mlsanity.com/paper/1912.03046/) (2019)
- [Community Detection on Networks with Ricci Flow](https://mlsanity.com/paper/1907.03993/) (2019)
- [Hyperbolic Variational Graph Neural Network for Modeling Dynamic Graphs](https://mlsanity.com/paper/2104.02228/) (2021)
- [Mitigating Over-Smoothing and Over-Squashing using Augmentations of Forman-Ricci Curvature](https://mlsanity.com/paper/2309.09384/) (2023)
- [Metric Learning with Adaptive Density Discrimination](https://mlsanity.com/paper/1511.05939/) (2015)
- [GraphMoRE: Mitigating Topological Heterogeneity via Mixture of Riemannian Experts](https://mlsanity.com/paper/2412.11085/) (2024)
- [PyTorch Geometric Signed Directed: A Software Package on Graph Neural Networks for Signed and Directed Graphs](https://mlsanity.com/paper/2202.10793/) (2022)

## Built on by (most cited first, 1 of 1)

- [Learning Propagation Geometry from Message-Passing Feedback](https://mlsanity.com/paper/2609.34711/) (2026)

## Abstract (arXiv)

Graph data often exhibits complex geometric heterogeneity, where structures with varying local curvature, such as tree-like hierarchies and dense communities, coexist within a single network. Existing geometric GNNs, which embed graphs into single fixed-curvature manifolds or discrete product spaces, struggle to capture this diversity. We introduce Adaptive Riemannian Graph Neural Networks (ARGNN), a novel framework that learns a continuous and anisotropic Riemannian metric tensor field over the graph. It allows each node to determine its optimal local geometry, enabling the model to fluidly adapt to the graph's structural landscape. Our core innovation is an efficient parameterization of the node-wise metric tensor, specializing to a learnable diagonal form that captures directional geometric information while maintaining computational tractability. To ensure geometric regularity and stable training, we integrate a Ricci flow-inspired regularization that smooths the learned manifold. Theoretically, we establish the rigorous geometric evolution convergence guarantee for ARGNN and provide a continuous generalization that unifies prior fixed or mixed-curvature GNNs. Empirically, our method demonstrates superior performance on both homophilic and heterophilic benchmark datasets with the ability to capture diverse structures adaptively. Moreover, the learned geometries both offer interpretable insights into the underlying graph structure and empirically corroborate our theoretical analysis.