A Short Tutorial on The Weisfeiler-Lehman Test And Its Variants
This is an incremental pedagogical note for researchers in graph neural networks, clarifying existing WL variants without introducing new methods.
This tutorial explains the Weisfeiler-Lehman (WL) test and its variants, focusing on their role in characterizing the expressive power of graph neural networks and inspiring architecture designs, with visual examples to clarify differences between formulations.
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.