# On the Convergence and Size Transferability of Continuous-depth Graph Neural Networks

arXiv: https://arxiv.org/abs/2510.03923 · HTML: https://mlsanity.com/paper/2510.03923/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 49 of its 91 references that are in the map.
- First posted on arXiv 4 Oct 2025.

## Builds on (most cited first, 30 of 49)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Open Graph Benchmark: Datasets for Machine Learning on Graphs](https://mlsanity.com/paper/2005.00687/) (2020)
- [Spectral Networks and Locally Connected Networks on Graphs](https://mlsanity.com/paper/1312.6203/) (2013)
- [Revisiting Semi-Supervised Learning with Graph Embeddings](https://mlsanity.com/paper/1603.08861/) (2016)
- [Geom-GCN: Geometric Graph Convolutional Networks](https://mlsanity.com/paper/2002.05287/) (2020)
- [Multi-scale Attributed Node Embedding](https://mlsanity.com/paper/1909.13021/) (2019)
- [Graph Transformer Networks](https://mlsanity.com/paper/1911.06455/) (2019)
- [Lipschitz regularity of deep neural networks: analysis and efficient estimation](https://mlsanity.com/paper/1805.10965/) (2018)
- [Graph Signal Processing: Overview, Challenges and Applications](https://mlsanity.com/paper/1712.00468/) (2017)
- [GRAND: Graph Neural Diffusion](https://mlsanity.com/paper/2106.10934/) (2021)
- [Graph Neural Ordinary Differential Equations](https://mlsanity.com/paper/1911.07532/) (2019)
- [Dissecting Neural ODEs](https://mlsanity.com/paper/2002.08071/) (2020)
- [Graph Neural Controlled Differential Equations for Traffic Forecasting](https://mlsanity.com/paper/2112.03558/) (2021)
- [Stability Properties of Graph Neural Networks](https://mlsanity.com/paper/1905.04497/) (2019)
- [Graph-Coupled Oscillator Networks](https://mlsanity.com/paper/2202.02296/) (2022)
- [Graphon Neural Networks and the Transferability of Graph Neural Networks](https://mlsanity.com/paper/2006.03548/) (2020)
- [Continuous Graph Neural Networks](https://mlsanity.com/paper/1912.00967/) (2019)
- [Transferability of Spectral Graph Convolutional Neural Networks](https://mlsanity.com/paper/1907.12972/) (2019)
- [Convergence and Stability of Graph Convolutional Networks on Large Random Graphs](https://mlsanity.com/paper/2006.01868/) (2020)
- [Theory of Graph Neural Networks: Representation and Learning](https://mlsanity.com/paper/2204.07697/) (2022)
- [GREAD: Graph Neural Reaction-Diffusion Networks](https://mlsanity.com/paper/2211.14208/) (2022)
- [Transferability of Graph Neural Networks: an Extended Graphon Approach](https://mlsanity.com/paper/2109.10096/) (2021)
- [Lipschitz Normalization for Self-Attention Layers with Application to Graph Neural Networks](https://mlsanity.com/paper/2103.04886/) (2021)
- [Graphon Signal Processing](https://mlsanity.com/paper/2003.05030/) (2020)
- [Interpretable Stability Bounds for Spectral Graph Filters](https://mlsanity.com/paper/2102.09587/) (2021)
- [Do Residual Neural Networks discretize Neural Ordinary Differential Equations?](https://mlsanity.com/paper/2205.14612/) (2022)
- [On the Universality of Graph Neural Networks on Large Random Graphs](https://mlsanity.com/paper/2105.13099/) (2021)

## Built on by (most cited first, 1 of 1)

- [Zero-Shot Size Transfer for Neural ODEs on Sparse Random Graphs: Graphon Limits and Adjoint Convergence](https://mlsanity.com/paper/2606.26662/) (2026)

## Abstract (arXiv)

Continuous-depth graph neural networks, also known as Graph Neural Differential Equations (GNDEs), combine the structural inductive bias of Graph Neural Networks (GNNs) with the continuous-depth architecture of Neural ODEs, offering a scalable and principled framework for modeling dynamics on graphs. In this paper, we present a rigorous convergence analysis of GNDEs with time-varying parameters in the infinite-node limit, providing theoretical insights into their size transferability. To this end, we introduce Graphon Neural Differential Equations (Graphon-NDEs) as the infinite-node limit of GNDEs and establish their well-posedness. Leveraging tools from graphon theory and dynamical systems, we prove the trajectory-wise convergence of GNDE solutions to Graphon-NDE solutions. Moreover, we derive explicit convergence rates under two deterministic graph sampling regimes: (1) weighted graphs sampled from smooth graphons, and (2) unweighted graphs sampled from 0,1-valued (discontinuous) graphons. We further establish size transferability bounds, providing theoretical justification for the practical strategy of transferring GNDE models trained on moderate-sized graphs to larger, structurally similar graphs without retraining. Numerical experiments using synthetic and real data support our theoretical findings.