# Uncertainty Modeling in Graph Neural Networks via Stochastic Differential Equations

arXiv: https://arxiv.org/abs/2408.16115 · HTML: https://mlsanity.com/paper/2408.16115/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 30 references that are in the map.
- First posted on arXiv 28 Aug 2024.

## Builds on (most cited first, 14 of 14)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Pitfalls of Graph Neural Network Evaluation](https://mlsanity.com/paper/1811.05868/) (2018)
- [Image-based Recommendations on Styles and Substitutes](https://mlsanity.com/paper/1506.04757/) (2015)
- [Scalable Gradients for Stochastic Differential Equations](https://mlsanity.com/paper/2001.01328/) (2020)
- [Graph Neural Ordinary Differential Equations](https://mlsanity.com/paper/1911.07532/) (2019)
- [Uncertainty Aware Semi-Supervised Learning on Graph Data](https://mlsanity.com/paper/2010.12783/) (2020)
- [Bayesian Graph Neural Networks with Adaptive Connection Sampling](https://mlsanity.com/paper/2006.04064/) (2020)
- [Graph Posterior Network: Bayesian Predictive Uncertainty for Node Classification](https://mlsanity.com/paper/2110.14012/) (2021)
- [Mat'ern Gaussian Processes on Graphs](https://mlsanity.com/paper/2010.15538/) (2020)
- [Infinitely Deep Bayesian Neural Networks with Stochastic Differential Equations](https://mlsanity.com/paper/2102.06559/) (2021)
- [Neural Latent Geometry Search: Product Manifold Inference via Gromov-Hausdorff-Informed Bayesian Optimization](https://mlsanity.com/paper/2309.04810/) (2023)
- [Continuous-Depth Neural Models for Dynamic Graph Prediction](https://mlsanity.com/paper/2106.11581/) (2021)
- [Partially Stochastic Infinitely Deep Bayesian Neural Networks](https://mlsanity.com/paper/2402.03495/) (2024)
- [Graph Neural Stochastic Differential Equations for Learning Brownian Dynamics](https://mlsanity.com/paper/2306.11435/) (2023)

## Built on by (most cited first, 8 of 8)

- [On Vanishing Gradients, Over-Smoothing, and Over-Squashing in GNNs: Bridging Recurrent and Graph Learning](https://mlsanity.com/paper/2502.10818/) (2025)
- [Graph ODEs and Beyond: A Comprehensive Survey on Integrating Differential Equations with Graph Neural Networks](https://mlsanity.com/paper/2503.23167/) (2025)
- [A Gaussian Process View on Observation Noise and Initialization in Wide Neural Networks](https://mlsanity.com/paper/2502.01556/) (2025)
- [Richer Bayesian Last Layers with Subsampled NTK Features](https://mlsanity.com/paper/2602.01279/) (2026)
- [Latent Dynamics Graph Convolutional Networks for model order reduction of parameterized time-dependent PDEs](https://mlsanity.com/paper/2601.11259/) (2026)
- [Activation-Space Uncertainty Quantification for Pretrained Networks](https://mlsanity.com/paper/2602.14934/) (2026)
- [MLGIB: Multi-Label Graph Information Bottleneck for Expressive and Robust Message Passing](https://mlsanity.com/paper/2605.13126/) (2026)
- [Hypergraph Neural Stochastic Diffusion: An SDE Framework for Uncertainty Estimation](https://mlsanity.com/paper/2607.07330/) (2026)

## Abstract (arXiv)

We propose a novel Stochastic Differential Equation (SDE) framework to address the problem of learning uncertainty-aware representations for graph-structured data. While Graph Neural Ordinary Differential Equations (GNODEs) have shown promise in learning node representations, they lack the ability to quantify uncertainty. To address this, we introduce Latent Graph Neural Stochastic Differential Equations (LGNSDE), which enhance GNODE by embedding randomness through a Bayesian prior-posterior mechanism for epistemic uncertainty and Brownian motion for aleatoric uncertainty. By leveraging the existence and uniqueness of solutions to graph-based SDEs, we prove that the variance of the latent space bounds the variance of model outputs, thereby providing theoretically sensible guarantees for the uncertainty estimates. Furthermore, we show mathematically that LGNSDEs are robust to small perturbations in the input, maintaining stability over time. Empirical results across several benchmarks demonstrate that our framework is competitive in out-of-distribution detection, robustness to noise, and active learning, underscoring the ability of LGNSDEs to quantify uncertainty reliably. Code is available at hrefhttps://github.com/Richard-Bergna/GraphNeuralSDEtextttgithub.com/Richard-Bergna/GraphNeuralSDE.