# Continuous Graph Neural Networks

arXiv: https://arxiv.org/abs/1912.00967 · HTML: https://mlsanity.com/paper/1912.00967/

## Key facts

- Cited by 114 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Dec 2019.

## Built on by (most cited first, 30 of 114)

- [GRAND: Graph Neural Diffusion](https://mlsanity.com/paper/2106.10934/) (2021)
- [A Survey on Oversmoothing in Graph Neural Networks](https://mlsanity.com/paper/2303.10993/) (2023)
- [Neural Sheaf Diffusion: A Topological Perspective on Heterophily and Oversmoothing in GNNs](https://mlsanity.com/paper/2202.04579/) (2022)
- [Spatial-Temporal Graph ODE Networks for Traffic Flow Forecasting](https://mlsanity.com/paper/2106.12931/) (2021)
- [Graph-Coupled Oscillator Networks](https://mlsanity.com/paper/2202.02296/) (2022)
- [Shift-Robust GNNs: Overcoming the Limitations of Localized Graph Training Data](https://mlsanity.com/paper/2108.01099/) (2021)
- [Beltrami Flow and Neural Diffusion on Graphs](https://mlsanity.com/paper/2110.09443/) (2021)
- [Multivariate Time Series Forecasting with Dynamic Graph Neural ODEs](https://mlsanity.com/paper/2202.08408/) (2022)
- [On the Unreasonable Effectiveness of Feature propagation in Learning on Graphs with Missing Node Features](https://mlsanity.com/paper/2111.12128/) (2021)
- [Understanding convolution on graphs via energies](https://mlsanity.com/paper/2206.10991/) (2022)
- [Dissecting the Diffusion Process in Linear Graph Convolutional Networks](https://mlsanity.com/paper/2102.10739/) (2021)
- [GREAD: Graph Neural Reaction-Diffusion Networks](https://mlsanity.com/paper/2211.14208/) (2022)
- [Hyperbolic Variational Graph Neural Network for Modeling Dynamic Graphs](https://mlsanity.com/paper/2104.02228/) (2021)
- [How Powerful is Graph Convolution for Recommendation?](https://mlsanity.com/paper/2108.07567/) (2021)
- [Gradient Gating for Deep Multi-Rate Learning on Graphs](https://mlsanity.com/paper/2210.00513/) (2022)
- [Nonlocal Kernel Network (NKN): a Stable and Resolution-Independent Deep Neural Network](https://mlsanity.com/paper/2201.02217/) (2022)
- [A Fractional Graph Laplacian Approach to Oversmoothing](https://mlsanity.com/paper/2305.13084/) (2023)
- [ACMP: Allen-Cahn Message Passing with Attractive and Repulsive Forces for Graph Neural Networks](https://mlsanity.com/paper/2206.05437/) (2022)
- [Optimization-Induced Graph Implicit Nonlinear Diffusion](https://mlsanity.com/paper/2206.14418/) (2022)
- [Physics-Informed Machine Learning for Modeling and Control of Dynamical Systems](https://mlsanity.com/paper/2306.13867/) (2023)
- [On Oversquashing in Graph Neural Networks Through the Lens of Dynamical Systems](https://mlsanity.com/paper/2405.01009/) (2024)
- [Language Knowledge-Assisted Representation Learning for Skeleton-Based Action Recognition](https://mlsanity.com/paper/2305.12398/) (2023)
- [Semi-supervised Domain Adaptation in Graph Transfer Learning](https://mlsanity.com/paper/2309.10773/) (2023)
- [Reversible and irreversible bracket-based dynamics for deep graph neural networks](https://mlsanity.com/paper/2305.15616/) (2023)
- [From Continuous Dynamics to Graph Neural Networks: Neural Diffusion and Beyond](https://mlsanity.com/paper/2310.10121/) (2023)
- [Geometer: Graph Few-Shot Class-Incremental Learning via Prototype Representation](https://mlsanity.com/paper/2205.13954/) (2022)
- [Continual Learning on Graphs: Challenges, Solutions, and Opportunities](https://mlsanity.com/paper/2402.11565/) (2024)
- [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)
- [TIDE: Time Derivative Diffusion for Deep Learning on Graphs](https://mlsanity.com/paper/2212.02483/) (2022)

## Abstract (arXiv)

This paper builds on the connection between graph neural networks and traditional dynamical systems. We propose continuous graph neural networks (CGNN), which generalise existing graph neural networks with discrete dynamics in that they can be viewed as a specific discretisation scheme. The key idea is how to characterise the continuous dynamics of node representations, i.e. the derivatives of node representations, w.r.t. time. Inspired by existing diffusion-based methods on graphs (e.g. PageRank and epidemic models on social networks), we define the derivatives as a combination of the current node representations, the representations of neighbors, and the initial values of the nodes. We propose and analyse two possible dynamics on graphs---including each dimension of node representations (a.k.a. the feature channel) change independently or interact with each other---both with theoretical justification. The proposed continuous graph neural networks are robust to over-smoothing and hence allow us to build deeper networks, which in turn are able to capture the long-range dependencies between nodes. Experimental results on the task of node classification demonstrate the effectiveness of our proposed approach over competitive baselines.