Exact Generalisation Error Exposes Benchmarks Skew Graph Neural Networks Success (or Failure)

Graph Neural Networks (GNNs) have become the standard method for learning from networks across fields ranging from biology to social systems, yet a principled understanding of what enables them to extract meaningful representations, or why performance varies drastically between similar models, remains elusive. These questions can be answered through the generalisation error, which measures the discrepancy between a model's predictions and the true values it is meant to recover. Although several works have derived generalisation error bounds, learning theoretical bounds are typically loose, restricted to a single architecture, and offer limited insight into what governs generalisation in practice. In this work, we take a fundamentally different approach by deriving the exact generalisation error for a broad range of linear GNNs, including convolutional, PageRank-based, and attention-based models, through the lens of signal processing. Our exact generalisation error exposes a strong benchmark bias in existing literature: commonly used datasets exhibit high alignment between node features and the graph structure, inherently favouring architectures that rely on it. We further show that the similarity between connected nodes (homophily) decisively governs which architectures are best suited for a given graph, thereby explaining how specific benchmark properties systematically shape the reported performance in the literature. Together, these results explain when and why GNNs can effectively leverage structure and feature information, supporting the reliable application of GNNs.