A Short Tutorial on The Weisfeiler-Lehman Test And Its Variants

TL;DR

This paper provides a clear, pedagogical overview of the Weisfeiler-Lehman test and its variants, explaining their differences and relevance to graph neural network architecture design.

Contribution

It clarifies the distinctions between WL and folklore-WL formulations, with visual examples, aiding understanding of their roles in GNNs.

Findings

Clarifies differences between WL and folklore-WL formulations

Visualizes example to illustrate formulation differences

Highlights relevance of WL hierarchy to GNN expressiveness

Abstract

Graph neural networks are designed to learn functions on graphs. Typically, the relevant target functions are invariant with respect to actions by permutations. Therefore the design of some graph neural network architectures has been inspired by graph-isomorphism algorithms. The classical Weisfeiler-Lehman algorithm (WL) -- a graph-isomorphism test based on color refinement -- became relevant to the study of graph neural networks. The WL test can be generalized to a hierarchy of higher-order tests, known as $k$-WL. This hierarchy has been used to characterize the expressive power of graph neural networks, and to inspire the design of graph neural network architectures. A few variants of the WL hierarchy appear in the literature. The goal of this short note is pedagogical and practical: We explain the differences between the WL and folklore-WL formulations, with pointers to existing discussions in the literature. We illuminate the differences between the formulations by visualizing an example.