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

We introduce $r$-loopy Weisfeiler-Leman ($r$-$ $WL), a novel hierarchy of graph isomorphism tests and a corresponding GNN framework, $r$-$ $MPNN, that can count cycles up to length $r+2$. Most notably, we show that $r$-$ $WL can count homomorphisms of cactus graphs. This extends 1-WL, which can only count homomorphisms of trees and, in fact, we prove that $r$-$ $WL is incomparable to $k$-WL for any fixed $k$. We empirically validate the expressive and counting power of $r$-$ $MPNN on several synthetic datasets and demonstrate the scalability and strong performance on various real-world datasets, particularly on sparse graphs. Our code is available.