Weisfeiler and Leman Go Loopy: A New Hierarchy for Graph
  Representational Learning
Raffaele Paolino Sohir Maskey Pascal Welke Gitta Kutyniok
We introduce $r$-loopy Weisfeiler-Leman ($r$-$\ell{}$WL), a novel hierarchy
of graph isomorphism tests and a corresponding GNN framework, $r$-$\ell{}$MPNN,
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 $r$-$\ell{}$MPNN on several
synthetic datasets and present state-of-the-art predictive performance on
various real-world datasets. The code is available at
https://github.com/RPaolino/loopy