# Learning Propagation Geometry from Message-Passing Feedback

arXiv: https://arxiv.org/abs/2609.34711 · HTML: https://mlsanity.com/paper/2609.34711/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 31 of its 49 references that are in the map.
- First posted on arXiv 28 Sep 2026.

## Builds on (most cited first, 30 of 31)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (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)
- [Open Graph Benchmark: Datasets for Machine Learning on Graphs](https://mlsanity.com/paper/2005.00687/) (2020)
- [MoleculeNet: A Benchmark for Molecular Machine Learning](https://mlsanity.com/paper/1703.00564/) (2017)
- [Pitfalls of Graph Neural Network Evaluation](https://mlsanity.com/paper/1811.05868/) (2018)
- [Predict then Propagate: Graph Neural Networks meet Personalized PageRank](https://mlsanity.com/paper/1810.05997/) (2018)
- [Representation Learning on Graphs with Jumping Knowledge Networks](https://mlsanity.com/paper/1806.03536/) (2018)
- [Geom-GCN: Geometric Graph Convolutional Networks](https://mlsanity.com/paper/2002.05287/) (2020)
- [Image-based Recommendations on Styles and Substitutes](https://mlsanity.com/paper/1506.04757/) (2015)
- [Graph Transformer Networks](https://mlsanity.com/paper/1911.06455/) (2019)
- [MixHop: Higher-Order Graph Convolutional Architectures via Sparsified Neighborhood Mixing](https://mlsanity.com/paper/1905.00067/) (2019)
- [Hyperbolic Graph Convolutional Neural Networks](https://mlsanity.com/paper/1910.12933/) (2019)
- [Graph Structure Learning for Robust Graph Neural Networks](https://mlsanity.com/paper/2005.10203/) (2020)
- [Towards Deeper Graph Neural Networks](https://mlsanity.com/paper/2007.09296/) (2020)
- [Hyperbolic Graph Neural Networks](https://mlsanity.com/paper/1910.12892/) (2019)
- [Iterative Deep Graph Learning for Graph Neural Networks: Better and Robust Node Embeddings](https://mlsanity.com/paper/2006.13009/) (2020)
- [Neural Sheaf Diffusion: A Topological Perspective on Heterophily and Oversmoothing in GNNs](https://mlsanity.com/paper/2202.04579/) (2022)
- [NodeFormer: A Scalable Graph Structure Learning Transformer for Node Classification](https://mlsanity.com/paper/2306.08385/) (2023)
- [ACE-HGNN: Adaptive Curvature Exploration Hyperbolic Graph Neural Network](https://mlsanity.com/paper/2110.07888/) (2021)
- [Adaptive Message Passing: A General Framework to Mitigate Oversmoothing, Oversquashing, and Underreaching](https://mlsanity.com/paper/2312.16560/) (2023)
- [Motif-aware Riemannian Graph Neural Network with Generative-Contrastive Learning](https://mlsanity.com/paper/2401.01232/) (2024)
- [SGAC: A Graph Neural Network Framework for Imbalanced and Structure-Aware AMP Classification](https://mlsanity.com/paper/2412.16276/) (2024)
- [ProtoMol: Enhancing Molecular Property Prediction via Prototype-Guided Multimodal Learning](https://mlsanity.com/paper/2510.16824/) (2025)
- [Improving the Expressiveness of K-hop Message-Passing GNNs by Injecting Contextualized Substructure Information](https://mlsanity.com/paper/2406.19244/) (2024)
- [USBD: Universal Structural Basis Distillation for Source-Free Graph Domain Adaptation](https://mlsanity.com/paper/2602.08431/) (2026)
- [Depth-Adaptive Graph Neural Networks via Learnable Bakry-'Emery Curvature](https://mlsanity.com/paper/2503.01079/) (2025)
- [A General Graph Spectral Wavelet Convolution via Chebyshev Order Decomposition](https://mlsanity.com/paper/2405.13806/) (2024)
- [Node-Specific Space Selection via Localized Geometric Hyperbolicity in Graph Neural Networks](https://mlsanity.com/paper/2303.01724/) (2023)

## Abstract (arXiv)

Learning local geometry enables graph neural networks (GNNs) to adapt how they compare and integrate neighborhood information. However, estimating geometry from aggregated representations can overlook variation among individual messages and dependencies across feature dimensions. We propose GeoF, a recurrent framework that jointly evolves node features and propagation geometry through message-passing feedback. Each node maintains a local symmetric positive-definite geometry, initialized from a structure-aware prototype atlas and parameterized in block log-triangular coordinates. At each step, the geometry determines neighborhood weights, while triangular frame transport maps transformed source messages into the target node's local coordinates before aggregation. Weighted second-order statistics of residuals between aligned messages and the transformed target state capture directional variation and within-block dependencies, yielding a geometric update target. A shared controller learns complementary corrections through task supervision. A bounded log-triangular update combines these corrections, the target, and the previous geometric state while preserving positive definiteness. The geometry governs subsequent propagation, closing the feedback loop. With parameters shared across recurrent steps, task-specific readouts support node classification, link prediction, and graph classification. Experiments on benchmark datasets show that GeoF consistently outperforms state-of-the-art GNN baselines.