Uncertainty Estimation on Graphs with Structure Informed Stochastic Partial Differential Equations (2506.06907v1)

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.

• The paper introduces an SPDE-driven message passing mechanism that models spatial correlations to enhance uncertainty estimation in GNNs.
• It demonstrates improved out-of-distribution detection through extensive experiments on diverse graph datasets with varying label informativeness.
• The approach provides theoretical insights by linking the Matérn Gaussian Process to control covariance kernel smoothness, boosting model reliability.

Uncertainty Estimation on Graphs using Structure-Informed SPDEs: A Technical Overview

The paper "Uncertainty Estimation on Graphs with Structure-Informed Stochastic Partial Differential Equations" addresses the challenge of estimating uncertainty in Graph Neural Networks (GNNs), especially under distributional shifts. GNNs have been highly effective in modeling various network tasks but often struggle with uncertainty estimation when graph structures and label distributions vary. The paper introduces a novel approach by making an analogy between stochastic partial differential equations (SPDEs) based on the Matérn Gaussian Process and a new message passing scheme in GNNs.

Key Contributions

1. SPDE-Driven Message Passing:
   • The authors present a structured way to construct GNN message passing using spatially-correlated noises inspired by SPDEs.
   • This approach facilitates capturing uncertainty across both spatial and temporal dimensions, offering explicit control over the smoothness of the covariance kernel. Such control is crucial for accommodating varying degrees of label informativeness across graphs.
2. Experimental Validation:
   • Extensive experiments on Out-of-Distribution (OOD) detection across multiple graph datasets, characterized by different levels of label informativeness, demonstrate the superiority of the proposed model compared to existing methods.
   • The authors claim enhanced performance in uncertainty estimation, supported by improved metrics in experimental scenarios.
3. Theoretical Insights:
   • The paper provides a detailed theoretical framework, including conditions for the existence and uniqueness of solutions to the underlying SPDEs.
   • The inclusion of the Matérn Gaussian Process in SPDE-driven GNNs introduces a method to regulate smoothness in the covariance kernel, thus enriching the expressivity of traditional approaches.

Implications and Future Directions

The application of SPDEs to graph-based model uncertainty estimation presents significant implications for both practical and theoretical advancements in AI. Practically, this could improve GNN reliability in domains like healthcare and finance, where safety-critical applications demand robust uncertainty quantification. Theoretically, adopting a Gaussian Process perspective aligns graph learning more closely with established continuous mathematical models, potentially improving our understanding and design of graph learning systems.

Future developments could explore the incorporation of Bayesian frameworks to refine uncertainty estimation further, addressing complexities in graph substructures. Also, larger classes of SPDE models, including fractional Brownian motions, might be tailored for improved scalability and application across more extensive datasets.

Conclusion

This paper contributes a novel SPDE-inspired method for uncertainty estimation in GNNs, emphasizing the importance of modeling dependencies between node uncertainties. By allowing explicit control over noise smoothness and leveraging spatial correlations, the study significantly advances the capabilities of GNNs in capturing uncertainties, paving the way for more accurate predictions in complex graph domains.