# Depth-Adaptive Graph Neural Networks via Learnable Bakry-'Emery Curvature

arXiv: https://arxiv.org/abs/2503.01079 · HTML: https://mlsanity.com/paper/2503.01079/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 33 of its 79 references that are in the map.
- First posted on arXiv 3 Mar 2025.

## Builds on (most cited first, 30 of 33)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [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)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [A Comprehensive Survey on Graph Neural Networks](https://mlsanity.com/paper/1901.00596/) (2019)
- [Open Graph Benchmark: Datasets for Machine Learning on Graphs](https://mlsanity.com/paper/2005.00687/) (2020)
- [Simplifying Graph Convolutional Networks](https://mlsanity.com/paper/1902.07153/) (2019)
- [How Attentive are Graph Attention Networks?](https://mlsanity.com/paper/2105.14491/) (2021)
- [Link Prediction Based on Graph Neural Networks](https://mlsanity.com/paper/1802.09691/) (2018)
- [Geom-GCN: Geometric Graph Convolutional Networks](https://mlsanity.com/paper/2002.05287/) (2020)
- [TUDataset: A collection of benchmark datasets for learning with graphs](https://mlsanity.com/paper/2007.08663/) (2020)
- [Benchmarking Graph Neural Networks](https://mlsanity.com/paper/2003.00982/) (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)
- [MixHop: Higher-Order Graph Convolutional Architectures via Sparsified Neighborhood Mixing](https://mlsanity.com/paper/1905.00067/) (2019)
- [A Fair Comparison of Graph Neural Networks for Graph Classification](https://mlsanity.com/paper/1912.09893/) (2019)
- [Identity-aware Graph Neural Networks](https://mlsanity.com/paper/2101.10320/) (2021)
- [FoSR: First-order spectral rewiring for addressing oversquashing in GNNs](https://mlsanity.com/paper/2210.11790/) (2022)
- [How Powerful are K-hop Message Passing Graph Neural Networks](https://mlsanity.com/paper/2205.13328/) (2022)
- [Edge Directionality Improves Learning on Heterophilic Graphs](https://mlsanity.com/paper/2305.10498/) (2023)
- [On the Trade-off between Over-smoothing and Over-squashing in Deep Graph Neural Networks](https://mlsanity.com/paper/2212.02374/) (2022)
- [Mitigating Over-Smoothing and Over-Squashing using Augmentations of Forman-Ricci Curvature](https://mlsanity.com/paper/2309.09384/) (2023)
- [Cooperative Graph Neural Networks](https://mlsanity.com/paper/2310.01267/) (2023)
- [Deep Graph Representation Learning and Optimization for Influence Maximization](https://mlsanity.com/paper/2305.02200/) (2023)
- [Curvature Graph Neural Network](https://mlsanity.com/paper/2106.15762/) (2021)
- [Towards Deep Attention in Graph Neural Networks: Problems and Remedies](https://mlsanity.com/paper/2306.02376/) (2023)
- [Adaptive Message Passing: A General Framework to Mitigate Oversmoothing, Oversquashing, and Underreaching](https://mlsanity.com/paper/2312.16560/) (2023)
- [Ricci Curvature of the Internet Topology](https://mlsanity.com/paper/1501.04138/) (2015)
- [Breaking the Entanglement of Homophily and Heterophily in Semi-supervised Node Classification](https://mlsanity.com/paper/2312.04111/) (2023)

## Built on by (most cited first, 8 of 8)

- [Beyond Fixed Depth: Adaptive Graph Neural Networks for Node Classification Under Varying Homophily](https://mlsanity.com/paper/2511.06608/) (2025)
- [Graph Neural Diffusion via Generalized Opinion Dynamics](https://mlsanity.com/paper/2508.11249/) (2025)
- [Bridging Computational Social Science and Deep Learning: Cultural Dissemination-Inspired Graph Neural Networks](https://mlsanity.com/paper/2509.19084/) (2025)
- [A roadmap for curvature-based geometric data analysis and learning](https://mlsanity.com/paper/2510.22599/) (2025)
- [Invariant-Stratified Propagation for Expressive Graph Neural Networks](https://mlsanity.com/paper/2603.01388/) (2026)
- [Graph Navier Stokes Networks](https://mlsanity.com/paper/2605.21247/) (2026)
- [Not Just Oversmoothing: Detecting the Echo Chamber Effect in Graph Neural Networks](https://mlsanity.com/paper/2609.06521/) (2026)
- [Learning Propagation Geometry from Message-Passing Feedback](https://mlsanity.com/paper/2609.34711/) (2026)

## Abstract (arXiv)

Graph Neural Networks (GNNs) have demonstrated strong representation learning capabilities for graph-based tasks. Recent advances on GNNs leverage geometric properties, such as curvature, to enhance its representation capabilities by modeling complex connectivity patterns and information flow within graphs. However, most existing approaches focus solely on discrete graph topology, overlooking diffusion dynamics and task-specific dependencies essential for effective learning. To address this, we propose integrating Bakry-'Emery curvature, which captures both structural and task-driven aspects of information propagation. We develop an efficient, learnable approximation strategy, making curvature computation scalable for large graphs. Furthermore, we introduce an adaptive depth mechanism that dynamically adjusts message-passing layers per vertex based on its curvature, ensuring efficient propagation. Our theoretical analysis establishes a link between curvature and feature distinctiveness, showing that high-curvature vertices require fewer layers, while low-curvature ones benefit from deeper propagation. Extensive experiments on benchmark datasets validate the effectiveness of our approach, showing consistent performance improvements across diverse graph learning tasks.