Uncertainty Estimation on Graphs with Structure Informed Stochastic Partial Differential Equations

TL;DR

The paper investigates uncertainty estimation on graphs under distributional shifts and proposes Structure Informed Graph SPDE (SISPDE), a physics-inspired framework that injects spatially correlated noise into graph neural ODEs via a Matérn Gaussian process on the graph. By adopting a Φ-Wiener process to shape the noise covariance with controllable smoothness parameters $ν$ and $κ$, the method captures spatial-temporal uncertainty patterns beyond local neighborhoods and adapts to graphs with varying label informativeness. The authors establish theoretical results for existence of mild solutions, provide an efficient Chebyshev-based kernel approximation, and demonstrate state-of-the-art or near-state-of-the-art OOD detection performance across eight graph datasets, including challenging heterophilic ones. The work offers a principled, controllable uncertainty modeling approach with practical scalability considerations and opens directions for Bayesian extensions and broader SPDE models in graph learning.

Abstract

Graph Neural Networks have achieved impressive results across diverse network modeling tasks, but accurately estimating uncertainty on graphs remains difficult, especially under distributional shifts. Unlike traditional uncertainty estimation, graph-based uncertainty must account for randomness arising from both the graph's structure and its label distribution, which adds complexity. In this paper, making an analogy between the evolution of a stochastic partial differential equation (SPDE) driven by Matern Gaussian Process and message passing using GNN layers, we present a principled way to design a novel message passing scheme that incorporates spatial-temporal noises motivated by the Gaussian Process approach to SPDE. Our method simultaneously captures uncertainty across space and time and allows explicit control over the covariance kernel smoothness, thereby enhancing uncertainty estimates on graphs with both low and high label informativeness. Our extensive experiments on Out-of-Distribution (OOD) detection on graph datasets with varying label informativeness demonstrate the soundness and superiority of our model to existing approaches.

Figures (4)

• Figure 1: (a) Graph Matérn kernel with varying degrees of smoothness $\nu$: for low $\nu$, the Gaussian random field is rough, with higher variance for each node and higher correlation between nodes. Red links are high correlation edges that do not exist in the original graph. (b) Our proposed Structure Informed SPDE (SISPDE) to incorporate spatial correlations between node uncertainty (section \ref{['sec:method']}): Gaussian noises between nodes are correlated according to the Matérn Gaussian Random Field.
• Figure 2: (a) Average rank of metrics for model on all graph datasets in Table \ref{['tab:ood_detection_heterophily']}; (b) Comparison of smoothness parameter $\nu$ on each dataset. Here we plot the average $\nu$ over the three cases (label, structure, feature) for each dataset.
• Figure 3: Average rank for low LI datasets. Some baselines perform better than GNN models.
• Figure 4: Runtime for one forward pass in seconds for all the models across all the datasets.

Theorems & Definitions (18)

• Proposition 1
• Definition 1
• Definition 2
• Theorem 1
• Theorem 2
• Theorem 3
• Remark 1
• Theorem 4: Chebyshev Approximation of Matérn kernel
• Definition 3: Brownian Motion
• Definition 4: Trace-Class Operators