# How Curvature Enhance the Adaptation Power of Framelet GCNs

arXiv: https://arxiv.org/abs/2307.09768 · HTML: https://mlsanity.com/paper/2307.09768/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 50 references that are in the map.
- First posted on arXiv 19 Jul 2023.

## 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)
- [A Comprehensive Survey on Graph Neural Networks](https://mlsanity.com/paper/1901.00596/) (2019)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [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)
- [DropEdge: Towards Deep Graph Convolutional Networks on Node Classification](https://mlsanity.com/paper/1907.10903/) (2019)
- [Adaptive Universal Generalized PageRank Graph Neural Network](https://mlsanity.com/paper/2006.07988/) (2020)
- [Understanding over-squashing and bottlenecks on graphs via curvature](https://mlsanity.com/paper/2111.14522/) (2021)
- [Geometric deep learning on graphs and manifolds using mixture model CNNs](https://mlsanity.com/paper/1611.08402/) (2016)
- [MixHop: Higher-Order Graph Convolutional Architectures via Sparsified Neighborhood Mixing](https://mlsanity.com/paper/1905.00067/) (2019)
- [GRAND: Graph Neural Diffusion](https://mlsanity.com/paper/2106.10934/) (2021)
- [A Note on Over-Smoothing for Graph Neural Networks](https://mlsanity.com/paper/2006.13318/) (2020)
- [Neural Sheaf Diffusion: A Topological Perspective on Heterophily and Oversmoothing in GNNs](https://mlsanity.com/paper/2202.04579/) (2022)
- [BernNet: Learning Arbitrary Graph Spectral Filters via Bernstein Approximation](https://mlsanity.com/paper/2106.10994/) (2021)
- [Continuous Graph Neural Networks](https://mlsanity.com/paper/1912.00967/) (2019)
- [Community Detection on Networks with Ricci Flow](https://mlsanity.com/paper/1907.03993/) (2019)
- [How Framelets Enhance Graph Neural Networks](https://mlsanity.com/paper/2102.06986/) (2021)
- [p-Laplacian Based Graph Neural Networks](https://mlsanity.com/paper/2111.07337/) (2021)
- [Decimated Framelet System on Graphs and Fast G-Framelet Transforms](https://mlsanity.com/paper/2012.06922/) (2020)
- [A Magnetic Framelet-Based Convolutional Neural Network for Directed Graphs](https://mlsanity.com/paper/2210.10993/) (2022)
- [Generalized energy and gradient flow via graph framelets](https://mlsanity.com/paper/2210.04124/) (2022)
- [Graph Denoising with Framelet Regularizer](https://mlsanity.com/paper/2111.03264/) (2021)

## Built on by (most cited first, 13 of 13)

- [Over-Squashing in Graph Neural Networks: A Comprehensive survey](https://mlsanity.com/paper/2308.15568/) (2023)
- [Exposition on over-squashing problem on GNNs: Current Methods, Benchmarks and Challenges](https://mlsanity.com/paper/2311.07073/) (2023)
- [Unifying over-smoothing and over-squashing in graph neural networks: A physics informed approach and beyond](https://mlsanity.com/paper/2309.02769/) (2023)
- [PANDA: Expanded Width-Aware Message Passing Beyond Rewiring](https://mlsanity.com/paper/2406.03671/) (2024)
- [Design Your Own Universe: A Physics-Informed Agnostic Method for Enhancing Graph Neural Networks](https://mlsanity.com/paper/2401.14580/) (2024)
- [Foundations and Frontiers of Graph Learning Theory](https://mlsanity.com/paper/2407.03125/) (2024)
- [When Graph Neural Networks Meet Dynamic Mode Decomposition](https://mlsanity.com/paper/2410.05593/) (2024)
- [Unleash Graph Neural Networks from Heavy Tuning](https://mlsanity.com/paper/2405.12521/) (2024)
- [Conditional Shift-Robust Conformal Prediction for Graph Neural Network](https://mlsanity.com/paper/2405.11968/) (2024)
- [Diffusing to the Top: Boost Graph Neural Networks with Minimal Hyperparameter Tuning](https://mlsanity.com/paper/2410.05697/) (2024)
- [Graph Homophily Booster: Rethinking the Role of Discrete Features on Heterophilic Graphs](https://mlsanity.com/paper/2509.12530/) (2025)
- [Graph homophily booster: Reimagining the role of discrete features in heterophilic graph learning](https://mlsanity.com/paper/2602.07256/) (2026)
- [Geometric Evolution Graph Convolutional Networks: Enhancing Graph Representation Learning via Ricci Flow](https://mlsanity.com/paper/2603.26178/) (2026)

## Abstract (arXiv)

Graph neural network (GNN) has been demonstrated powerful in modeling graph-structured data. However, despite many successful cases of applying GNNs to various graph classification and prediction tasks, whether the graph geometrical information has been fully exploited to enhance the learning performance of GNNs is not yet well understood. This paper introduces a new approach to enhance GNN by discrete graph Ricci curvature. Specifically, the graph Ricci curvature defined on the edges of a graph measures how difficult the information transits on one edge from one node to another based on their neighborhoods. Motivated by the geometric analogy of Ricci curvature in the graph setting, we prove that by inserting the curvature information with different carefully designed transformation function zeta, several known computational issues in GNN such as over-smoothing can be alleviated in our proposed model. Furthermore, we verified that edges with very positive Ricci curvature (i.e., kappai,j ≈ 1) are preferred to be dropped to enhance model's adaption to heterophily graph and one curvature based graph edge drop algorithm is proposed. Comprehensive experiments show that our curvature-based GNN model outperforms the state-of-the-art baselines in both homophily and heterophily graph datasets, indicating the effectiveness of involving graph geometric information in GNNs.