What graph neural networks cannot learn: depth vs width

This paper studies the capacity limits of graph neural networks (GNN). Rather than focusing on a specific architecture, the networks considered here are those that fall within the message-passing framework, a model that encompasses several state-of-the-art networks. Two main results are presented. First, GNN are shown to be Turing universal under sufficient conditions on their depth, width, node identification, and layer expressiveness. In addition, it is discovered that GNN can lose a significant portion of their power when their depth and width is restricted. The proposed impossibility statements stem from a new technique that enables the re-purposing of seminal results from theoretical computer science. This leads to lower bounds for an array of decision, optimization, and estimation problems involving graphs. Strikingly, several of these problems are deemed impossible unless the product of a GNN's depth and width exceeds the graph size; this dependence remains significant even for tasks that appear simple or when considering approximation.