# Dissecting the Diffusion Process in Linear Graph Convolutional Networks

arXiv: https://arxiv.org/abs/2102.10739 · HTML: https://mlsanity.com/paper/2102.10739/

## Key facts

- Cited by 58 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 36 references that are in the map.
- First posted on arXiv 22 Feb 2021.

## Builds on (most cited first, 29 of 29)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [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 Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [DeepWalk: Online Learning of Social Representations](https://mlsanity.com/paper/1403.6652/) (2014)
- [Simplifying Graph Convolutional Networks](https://mlsanity.com/paper/1902.07153/) (2019)
- [Deep Graph Infomax](https://mlsanity.com/paper/1809.10341/) (2018)
- [Deeper Insights into Graph Convolutional Networks for Semi-Supervised Learning](https://mlsanity.com/paper/1801.07606/) (2018)
- [Predict then Propagate: Graph Neural Networks meet Personalized PageRank](https://mlsanity.com/paper/1810.05997/) (2018)
- [Revisiting Semi-Supervised Learning with Graph Embeddings](https://mlsanity.com/paper/1603.08861/) (2016)
- [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)
- [GraphSAINT: Graph Sampling Based Inductive Learning Method](https://mlsanity.com/paper/1907.04931/) (2019)
- [FastGCN: Fast Learning with Graph Convolutional Networks via Importance Sampling](https://mlsanity.com/paper/1801.10247/) (2018)
- [DeepGCNs: Can GCNs Go as Deep as CNNs?](https://mlsanity.com/paper/1904.03751/) (2019)
- [PairNorm: Tackling Oversmoothing in GNNs](https://mlsanity.com/paper/1909.12223/) (2019)
- [SIGN: Scalable Inception Graph Neural Networks](https://mlsanity.com/paper/2004.11198/) (2020)
- [Beyond Finite Layer Neural Networks: Bridging Deep Architectures and Numerical Differential Equations](https://mlsanity.com/paper/1710.10121/) (2017)
- [Graph-Bert: Only Attention is Needed for Learning Graph Representations](https://mlsanity.com/paper/2001.05140/) (2020)
- [Graph Neural Ordinary Differential Equations](https://mlsanity.com/paper/1911.07532/) (2019)
- [Graph Wavelet Neural Network](https://mlsanity.com/paper/1904.07785/) (2019)
- [Continuous Graph Neural Networks](https://mlsanity.com/paper/1912.00967/) (2019)
- [Certifiable Robustness to Graph Perturbations](https://mlsanity.com/paper/1910.14356/) (2019)
- [Graph Convolutional Networks using Heat Kernel for Semi-supervised Learning](https://mlsanity.com/paper/2007.16002/) (2020)
- [GResNet: Graph Residual Network for Reviving Deep GNNs from Suspended Animation](https://mlsanity.com/paper/1909.05729/) (2019)
- [The nonlinear heat equation on dense graphs and graph limits](https://mlsanity.com/paper/1302.5804/) (2013)

## Built on by (most cited first, 30 of 58)

- [DIFFormer: Scalable (Graph) Transformers Induced by Energy Constrained Diffusion](https://mlsanity.com/paper/2301.09474/) (2023)
- [Multivariate Time Series Forecasting with Dynamic Graph Neural ODEs](https://mlsanity.com/paper/2202.08408/) (2022)
- [Anti-Symmetric DGN: a stable architecture for Deep Graph Networks](https://mlsanity.com/paper/2210.09789/) (2022)
- [Cold Brew: Distilling Graph Node Representations with Incomplete or Missing Neighborhoods](https://mlsanity.com/paper/2111.04840/) (2021)
- [GREAD: Graph Neural Reaction-Diffusion Networks](https://mlsanity.com/paper/2211.14208/) (2022)
- [On Vanishing Gradients, Over-Smoothing, and Over-Squashing in GNNs: Bridging Recurrent and Graph Learning](https://mlsanity.com/paper/2502.10818/) (2025)
- [Towards Understanding the Generalization of Graph Neural Networks](https://mlsanity.com/paper/2305.08048/) (2023)
- [Blurring-Sharpening Process Models for Collaborative Filtering](https://mlsanity.com/paper/2211.09324/) (2022)
- [Optimization-Induced Graph Implicit Nonlinear Diffusion](https://mlsanity.com/paper/2206.14418/) (2022)
- [PC-Conv: Unifying Homophily and Heterophily with Two-fold Filtering](https://mlsanity.com/paper/2312.14438/) (2023)
- [On Oversquashing in Graph Neural Networks Through the Lens of Dynamical Systems](https://mlsanity.com/paper/2405.01009/) (2024)
- [Long Range Propagation on Continuous-Time Dynamic Graphs](https://mlsanity.com/paper/2406.02740/) (2024)
- [Reversible and irreversible bracket-based dynamics for deep graph neural networks](https://mlsanity.com/paper/2305.15616/) (2023)
- [Adaptive Message Passing: A General Framework to Mitigate Oversmoothing, Oversquashing, and Underreaching](https://mlsanity.com/paper/2312.16560/) (2023)
- [Geometric Knowledge Distillation: Topology Compression for Graph Neural Networks](https://mlsanity.com/paper/2210.13014/) (2022)
- [A Message Passing Perspective on Learning Dynamics of Contrastive Learning](https://mlsanity.com/paper/2303.04435/) (2023)
- [Laplacian Canonization: A Minimalist Approach to Sign and Basis Invariant Spectral Embedding](https://mlsanity.com/paper/2310.18716/) (2023)
- [Climate Modeling with Neural Diffusion Equations](https://mlsanity.com/paper/2111.06011/) (2021)
- [Bridging the Gap between Spatial and Spectral Domains: A Unified Framework for Graph Neural Networks](https://mlsanity.com/paper/2107.10234/) (2021)
- [Deep learning for dynamic graphs: models and benchmarks](https://mlsanity.com/paper/2307.06104/) (2023)
- [Rethinking Node-wise Propagation for Large-scale Graph Learning](https://mlsanity.com/paper/2402.06128/) (2024)
- [SA-GDA: Spectral Augmentation for Graph Domain Adaptation](https://mlsanity.com/paper/2408.09189/) (2024)
- [ResNorm: Tackling Long-tailed Degree Distribution Issue in Graph Neural Networks via Normalization](https://mlsanity.com/paper/2206.08181/) (2022)
- [Reconstructing Graph Diffusion History from a Single Snapshot](https://mlsanity.com/paper/2306.00488/) (2023)
- [A Generalized Neural Diffusion Framework on Graphs](https://mlsanity.com/paper/2312.08616/) (2023)
- [Return of ChebNet: Understanding and Improving an Overlooked GNN on Long Range Tasks](https://mlsanity.com/paper/2506.07624/) (2025)
- [Temporal Graph ODEs for Irregularly-Sampled Time Series](https://mlsanity.com/paper/2404.19508/) (2024)
- [SkipNode: On Alleviating Performance Degradation for Deep Graph Convolutional Networks](https://mlsanity.com/paper/2112.11628/) (2021)
- [Less is More: SlimG for Accurate, Robust, and Interpretable Graph Mining](https://mlsanity.com/paper/2210.04081/) (2022)
- [Generalized energy and gradient flow via graph framelets](https://mlsanity.com/paper/2210.04124/) (2022)

## Abstract (arXiv)

Graph Convolutional Networks (GCNs) have attracted more and more attentions in recent years. A typical GCN layer consists of a linear feature propagation step and a nonlinear transformation step. Recent works show that a linear GCN can achieve comparable performance to the original non-linear GCN while being much more computationally efficient. In this paper, we dissect the feature propagation steps of linear GCNs from a perspective of continuous graph diffusion, and analyze why linear GCNs fail to benefit from more propagation steps. Following that, we propose Decoupled Graph Convolution (DGC) that decouples the terminal time and the feature propagation steps, making it more flexible and capable of exploiting a very large number of feature propagation steps. Experiments demonstrate that our proposed DGC improves linear GCNs by a large margin and makes them competitive with many modern variants of non-linear GCNs.