Graph Neural Networks Can (Often) Count Substructures
Paolo Pellizzoni Till Hendrik Schulz Karsten Borgwardt
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.
graph neural networks, subgraphs, expressivity
We provide a theoretical analysis of the subgraph-counting capabilities of graph neural networks beyond the worst case.