Graph neural networks (GNNs) excel in modeling relational data such as biological, social, and transportation networks, but the underpinnings of their success are not well understood. Traditional complexity measures from statistical learning theory fail to account for observed phenomena like the double descent or the impact of relational semantics on generalization error. Motivated by experimental observations of ``transductive'' double descent in key networks and datasets, we use analytical tools from statistical physics and random matrix theory to precisely characterize generalization in simple graph convolution networks on the contextual stochastic block model. Our results illuminate the nuances of learning on homophilic versus heterophilic data and predict double descent whose existence in GNNs has been questioned by recent work. We show how risk is shaped by the interplay between the graph noise, feature noise, and the number of training labels. Our findings apply beyond stylized models, capturing qualitative trends in real-world GNNs and datasets. As a case in point, we use our analytic insights to improve performance of state-of-the-art graph convolution networks on heterophilic datasets.

翻译：图神经网络（GNNs）在生物、社交和交通网络等关系型数据建模中表现出色，但其成功的内在机制尚未被充分理解。来自统计学习理论的传统复杂度指标无法解释诸如双重下降或关系语义对泛化误差的影响等观测现象。受关键网络和数据集中“转导式”双重下降实验观察的启发，我们利用统计物理和随机矩阵理论的分析工具，精确刻画了上下文随机块模型上简单图卷积网络的泛化行为。研究结果揭示了同质性与异质性数据上学习的细微差异，并预测了近年来研究质疑GNNs中存在的双重下降现象。我们展示了风险如何由图噪声、特征噪声和训练标签数量之间的相互作用所塑造。研究结论不仅适用于简化模型，还能捕捉真实世界GNNs和数据集中的定性趋势。作为例证，我们利用分析洞见提升了最先进图卷积网络在异质性数据集上的性能。