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

Raffaele Paolino, Sohir Maskey, Pascal Welke, Gitta Kutyniok

Keywords: Graph Neural Networks, Weisfeiler-Leman (WL) Test, Homomorphism Counting, Theory and Expressivity in GNNs, Cactus Graphs

TL;DR: We introduce GNNs that can count cycles and homomorphisms of cactus graphs, surpassing the limitations of existing GNNs while being scalable on real-world graphs.

Abstract: 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 extends 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 $r$-$\ell$MPNN on several synthetic datasets and demonstrate the scalability and strong performance on various real-world datasets, particularly on sparse graphs.

Primary Area: Graph neural networks

Submission Number: 16891