On Size Generalization in Graph Neural Networks

Graph neural networks (GNNs) can process graphs of different sizes but their capacity to generalize across sizes is still not well understood. Size generalization is key to numerous GNN applications, from solving combinatorial optimization problems to learning in molecular biology. In such problems, obtaining labels and training on large graphs can be prohibitively expensive, but training on smaller graphs is possible. This paper puts forward the size-generalization question and characterizes important aspects of that problem theoretically and empirically. We prove that even for very simple tasks, such as counting the number of nodes or edges in a graph, GNNs do not naturally generalize to graphs of larger size. Instead, their generalization performance is closely related to the distribution of local patterns of connectivity and features and how that distribution changes from small to large graphs. Specifically, we prove that for many tasks, there are weight assignments for GNNs that can perfectly solve the task on small graphs but fail on large graphs, if there is a discrepancy between their local patterns. We further demonstrate on several tasks, that training GNNs on small graphs results in solutions which do not generalize to larger graphs. We then formalize size generalization as a domain-adaption problem and describe two learning setups where size generalization can be improved. First, as a self-supervised learning problem (SSL) over the target domain of large graphs. Second as a semi-supervised learning problem when few samples are available in the target domain. We demonstrate the efficacy of these solutions on a diverse set of benchmark graph datasets.