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

Abstract

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.