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While message passing Graph Neural Networks (GNNs) have become increasinglypopular architectures for learning with graphs, recent works have revealedimportant shortcomings in their expressive power. In response, severalhigher-order GNNs have been proposed that substantially increase the expressivepower, albeit at a large computational cost. Motivated by this gap, we explorealternative strategies and lower bounds. In particular, we analyze a newrecursive pooling technique of local neighborhoods that allows differenttradeoffs of computational cost and expressive power. First, we prove that thismodel can count subgraphs of size $k$, and thereby overcomes a known limitationof low-order GNNs. Second, we show how recursive pooling can exploit sparsityto reduce the computational complexity compared to the existing higher-orderGNNs. More generally, we provide a (near) matching information-theoretic lowerbound for counting subgraphs with graph representations that pool overrepresentations of derived (sub-)graphs. We also discuss lower bounds on timecomplexity.

Counting Substructures with Higher-Order Graph Neural Networks: Possibility and Impossibility Results