# A Note on Over-Smoothing for Graph Neural Networks

arXiv: https://arxiv.org/abs/2006.13318 · HTML: https://mlsanity.com/paper/2006.13318/

## Key facts

- Cited by 249 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Jun 2020.

## Built on by (most cited first, 30 of 249)

- [A Survey on Oversmoothing in Graph Neural Networks](https://mlsanity.com/paper/2303.10993/) (2023)
- [Language is All a Graph Needs](https://mlsanity.com/paper/2308.07134/) (2023)
- [Neural Sheaf Diffusion: A Topological Perspective on Heterophily and Oversmoothing in GNNs](https://mlsanity.com/paper/2202.04579/) (2022)
- [Pure Transformers are Powerful Graph Learners](https://mlsanity.com/paper/2207.02505/) (2022)
- [On Over-Squashing in Message Passing Neural Networks: The Impact of Width, Depth, and Topology](https://mlsanity.com/paper/2302.02941/) (2023)
- [Anti-Oversmoothing in Deep Vision Transformers via the Fourier Domain Analysis: From Theory to Practice](https://mlsanity.com/paper/2203.05962/) (2022)
- [Pre-training via Denoising for Molecular Property Prediction](https://mlsanity.com/paper/2206.00133/) (2022)
- [FoSR: First-order spectral rewiring for addressing oversquashing in GNNs](https://mlsanity.com/paper/2210.11790/) (2022)
- [Revisiting Over-smoothing and Over-squashing Using Ollivier-Ricci Curvature](https://mlsanity.com/paper/2211.15779/) (2022)
- [Dirichlet Energy Constrained Learning for Deep Graph Neural Networks](https://mlsanity.com/paper/2107.02392/) (2021)
- [Understanding Oversquashing in GNNs through the Lens of Effective Resistance](https://mlsanity.com/paper/2302.06835/) (2023)
- [Position: Graph Foundation Models are Already Here](https://mlsanity.com/paper/2402.02216/) (2024)
- [DRew: Dynamically Rewired Message Passing with Delay](https://mlsanity.com/paper/2305.08018/) (2023)
- [On the Connection Between MPNN and Graph Transformer](https://mlsanity.com/paper/2301.11956/) (2023)
- [Understanding convolution on graphs via energies](https://mlsanity.com/paper/2206.10991/) (2022)
- [On Provable Benefits of Depth in Training Graph Convolutional Networks](https://mlsanity.com/paper/2110.15174/) (2021)
- [How does over-squashing affect the power of GNNs?](https://mlsanity.com/paper/2306.03589/) (2023)
- [Demystifying Oversmoothing in Attention-Based Graph Neural Networks](https://mlsanity.com/paper/2305.16102/) (2023)
- [A Fractional Graph Laplacian Approach to Oversmoothing](https://mlsanity.com/paper/2305.13084/) (2023)
- [Simple GNN Regularisation for 3D Molecular Property Prediction & Beyond](https://mlsanity.com/paper/2106.07971/) (2021)
- [AdaGNN: Graph Neural Networks with Adaptive Frequency Response Filter](https://mlsanity.com/paper/2104.12840/) (2021)
- [On Vanishing Gradients, Over-Smoothing, and Over-Squashing in GNNs: Bridging Recurrent and Graph Learning](https://mlsanity.com/paper/2502.10818/) (2025)
- [Simplicial Attention Networks](https://mlsanity.com/paper/2204.09455/) (2022)
- [Mitigating Over-Smoothing and Over-Squashing using Augmentations of Forman-Ricci Curvature](https://mlsanity.com/paper/2309.09384/) (2023)
- [Transformers Generalize DeepSets and Can be Extended to Graphs and Hypergraphs](https://mlsanity.com/paper/2110.14416/) (2021)
- [Residual Connections and Normalization Can Provably Prevent Oversmoothing in GNNs](https://mlsanity.com/paper/2406.02997/) (2024)
- [GraphControl: Adding Conditional Control to Universal Graph Pre-trained Models for Graph Domain Transfer Learning](https://mlsanity.com/paper/2310.07365/) (2023)
- [A Mechanistic Analysis of Looped Reasoning Language Models](https://mlsanity.com/paper/2604.11791/) (2026)
- [ACMP: Allen-Cahn Message Passing with Attractive and Repulsive Forces for Graph Neural Networks](https://mlsanity.com/paper/2206.05437/) (2022)
- [Modeling Multivariate Biosignals With Graph Neural Networks and Structured State Space Models](https://mlsanity.com/paper/2211.11176/) (2022)

## Abstract (arXiv)

Graph Neural Networks (GNNs) have achieved a lot of success on graph-structured data. However, it is observed that the performance of graph neural networks does not improve as the number of layers increases. This effect, known as over-smoothing, has been analyzed mostly in linear cases. In this paper, we build upon previous results citeoono2019graph to further analyze the over-smoothing effect in the general graph neural network architecture. We show when the weight matrix satisfies the conditions determined by the spectrum of augmented normalized Laplacian, the Dirichlet energy of embeddings will converge to zero, resulting in the loss of discriminative power. Using Dirichlet energy to measure "expressiveness" of embedding is conceptually clean; it leads to simpler proofs than citeoono2019graph and can handle more non-linearities.