Why does your graph neural network fail on some graphs? Insights from exact generalisation error

2025-09-12

AI Summary

This paper investigates the fundamental causes underlying the disparate generalization performance of Graph Neural Networks (GNNs) across different graphs. Method: For the transductive learning setting, we establish the first unified signal-processing analytical framework and derive exact closed-form expressions for the generalization error of convolutional-, PageRank-, and attention-based GNNs. Contribution/Results: We theoretically prove that strong generalization is achievable only when node features are sufficiently aligned with graph structure—i.e., when homophily is high—and that structural-feature misalignment constitutes the key mechanism behind sharp performance degradation. This result explains why identical GNN architectures exhibit markedly divergent behavior on heterogeneous graphs and provides an interpretable, computationally tractable theoretical foundation for principled model selection and generalization prediction.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Graph Neural Networks (GNNs) are widely used in learning on graph-structured data, yet a principled understanding of why they succeed or fail remains elusive. While prior works have examined architectural limitations such as over-smoothing and over-squashing, these do not explain what enables GNNs to extract meaningful representations or why performance varies drastically between similar architectures. These questions are related to the role of generalisation: the ability of a model to make accurate predictions on unlabelled data. Although several works have derived generalisation error bounds for GNNs, these are typically loose, restricted to a single architecture, and offer limited insight into what governs generalisation in practice. In this work, we take a different approach by deriving the exact generalisation error for GNNs in a transductive fixed-design setting through the lens of signal processing. From this viewpoint, GNNs can be interpreted as graph filter operators that act on node features via the graph structure. By focusing on linear GNNs while allowing non-linearity in the graph filters, we derive the first exact generalisation error for a broad range of GNNs, including convolutional, PageRank-based, and attention-based models. The exact characterisation of the generalisation error reveals that only the aligned information between node features and graph structure contributes to generalisation. Furthermore, we quantify the effect of homophily on generalisation. Our work provides a framework that explains when and why GNNs can effectively leverage structural and feature information, offering practical guidance for model selection.

Problem

• Analyzing exact generalization error for GNNs in transductive settings
• Identifying why GNNs succeed or fail on specific graph structures
• Quantifying how feature-structure alignment and homophily affect generalization

Innovation

• Exact generalization error analysis via signal processing
• Graph filter operators interpret GNNs as linear models
• Quantifies homophily effect on generalization performance