Statistical Mechanics of Generalization In Graph Convolution Networks

Graph neural networks (GNN) have become the default machine learning model for relational datasets, including protein interaction networks, biological neural networks, and scientific collaboration graphs. We use tools from statistical physics and random matrix theory to precisely characterize generalization in simple graph convolution networks on the contextual stochastic block model. The derived curves are phenomenologically rich: they explain the distinction between learning on homophilic and heterophilic graphs and they predict double descent whose existence in GNNs has been questioned by recent work. Our results are the first to accurately explain the behavior not only of a stylized graph learning model but also of complex GNNs on messy real-world datasets. To wit, we use our analytic insights about homophily and heterophily to improve performance of state-of-the-art graph neural networks on several heterophilic benchmarks by a simple addition of negative self-loop filters.