Weisfeiler and Leman Go Loopy: A New Hierarchy for Graph Representational Learning

We introduce $r$-loopy Weisfeiler-Leman ($r$-$\ell$WL), a novel hierarchy of graph isomorphism tests and a corresponding GNN framework, $r$-$\ell$MPNNs, that can count cycles up to length $r+2$. Most notably, we show that $r$-$\ell$WL can count homomorphisms of cactus graphs. This strictly extends classical 1-WL, which can only count homomorphisms of trees and, in fact, is incomparable to $k$-WL for any fixed $k$. We empirically validate the expressive and counting power of the proposed $r$-$\ell$MPNNs on several synthetic datasets and present state-of-the-art predictive performance on various real-world datasets.