# Permutation Equivariant Neural Controlled Differential Equations for Dynamic Graph Representation Learning

arXiv: https://arxiv.org/abs/2506.20324 · HTML: https://mlsanity.com/paper/2506.20324/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 41 of its 67 references that are in the map.
- First posted on arXiv 25 Jun 2025.

## Builds on (most cited first, 30 of 41)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [Spatio-Temporal Graph Convolutional Networks: A Deep Learning Framework for Traffic Forecasting](https://mlsanity.com/paper/1709.04875/) (2017)
- [Graph Neural Networks for Social Recommendation](https://mlsanity.com/paper/1902.07243/) (2019)
- [Temporal Graph Networks for Deep Learning on Dynamic Graphs](https://mlsanity.com/paper/2006.10637/) (2020)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [Inductive Representation Learning on Temporal Graphs](https://mlsanity.com/paper/2002.07962/) (2020)
- [Predicting Dynamic Embedding Trajectory in Temporal Interaction Networks](https://mlsanity.com/paper/1908.01207/) (2019)
- [Invariant and Equivariant Graph Networks](https://mlsanity.com/paper/1812.09902/) (2018)
- [On Neural Differential Equations](https://mlsanity.com/paper/2202.02435/) (2022)
- [Graph Neural Network for Traffic Forecasting: A Survey](https://mlsanity.com/paper/2101.11174/) (2021)
- [GRAND: Graph Neural Diffusion](https://mlsanity.com/paper/2106.10934/) (2021)
- [Inductive Representation Learning in Temporal Networks via Causal Anonymous Walks](https://mlsanity.com/paper/2101.05974/) (2021)
- [Towards Better Dynamic Graph Learning: New Architecture and Unified Library](https://mlsanity.com/paper/2303.13047/) (2023)
- [Graph Neural Ordinary Differential Equations](https://mlsanity.com/paper/1911.07532/) (2019)
- [Equinox: neural networks in JAX via callable PyTrees and filtered transformations](https://mlsanity.com/paper/2111.00254/) (2021)
- [Do We Really Need Complicated Model Architectures For Temporal Networks?](https://mlsanity.com/paper/2302.11636/) (2023)
- [Temporal Graph Benchmark for Machine Learning on Temporal Graphs](https://mlsanity.com/paper/2307.01026/) (2023)
- [Graph Neural Controlled Differential Equations for Traffic Forecasting](https://mlsanity.com/paper/2112.03558/) (2021)
- [PyTorch Geometric Temporal: Spatiotemporal Signal Processing with Neural Machine Learning Models](https://mlsanity.com/paper/2104.07788/) (2021)
- [PDE-GCN: Novel Architectures for Graph Neural Networks Motivated by Partial Differential Equations](https://mlsanity.com/paper/2108.01938/) (2021)
- [Theoretical Foundations of Deep Selective State-Space Models](https://mlsanity.com/paper/2402.19047/) (2024)
- [TCL: Transformer-based Dynamic Graph Modelling via Contrastive Learning](https://mlsanity.com/paper/2105.07944/) (2021)
- [Understanding convolution on graphs via energies](https://mlsanity.com/paper/2206.10991/) (2022)
- [GREAD: Graph Neural Reaction-Diffusion Networks](https://mlsanity.com/paper/2211.14208/) (2022)
- [word2vec, node2vec, graph2vec, X2vec: Towards a Theory of Vector Embeddings of Structured Data](https://mlsanity.com/paper/2003.12590/) (2020)

## Built on by (most cited first, 3 of 3)

- [On the Convergence and Size Transferability of Continuous-depth Graph Neural Networks](https://mlsanity.com/paper/2510.03923/) (2025)
- [Universal Time Series Generation with Neural Controlled Differential Equations](https://mlsanity.com/paper/2605.28507/) (2026)
- [Faithful Embeddings of Irregular and Asynchronous Data for Online Log-NCDEs](https://mlsanity.com/paper/2605.30213/) (2026)

## Abstract (arXiv)

Dynamic graphs exhibit complex temporal dynamics due to the interplay between evolving node features and changing network structures. Recently, Graph Neural Controlled Differential Equations (Graph Neural CDEs) successfully adapted Neural CDEs from paths on Euclidean domains to paths on graph domains. Building on this foundation, we introduce Permutation Equivariant Neural Graph CDEs, which project Graph Neural CDEs onto permutation equivariant function spaces. This significantly reduces the model's parameter count without compromising representational power, resulting in more efficient training and improved generalisation. We empirically demonstrate the advantages of our approach through experiments on simulated dynamical systems and real-world tasks, showing improved performance in both interpolation and extrapolation scenarios.