Graph Neural Networks Can (Often) Count Substructures

Message passing graph neural networks (GNNs) are known to have limited expressive power in their ability to distinguish some non-isomorphic graphs. Because of this, it is well known that they are unable to detect or count arbitrary graph substructures (i.e., solving the subgraph isomorphism problem), a task that is of great importance for several types of graph-structured data. However, we observe that GNNs are in fact able to count graph patterns quite accurately across several real-world graph datasets. Motivated by this observation, we provide an analysis of the subgraph-counting capabilities of GNNs beyond the worst case, deriving several sufficient conditions for GNNs to be able to count subgraphs and, more importantly, to be able to sample-efficiently learn to count subgraphs. Moreover, we develop novel dynamic programming algorithms for solving the subgraph isomorphism problem on restricted classes of pattern and target graphs, and show that message-passing GNNs can efficiently simulate these dynamic programs. Finally, we empirically validate that our sufficient conditions for GNNs to count subgraphs hold on many real-world datasets, providing a theoretically-grounded explanation to our motivating observations.