The Weisfeiler-Lehman Distance: Reinterpretation and Connection with GNNs

TL;DR

This paper offers a new stochastic process-based interpretation of the Weisfeiler-Lehman distance, linking it to GNNs, graph isomorphism testing, and the Gromov-Wasserstein distance, enhancing understanding of graph comparison methods.

Contribution

It introduces a novel stochastic process perspective on the WL distance, connecting it to GNNs and other graph distances, and discusses implications for neural network properties.

Findings

WL distance has the same discriminative power as WL graph isomorphism test.

Connections established between WL distance and Message Passing Neural Networks.

Implications for Lipschitz continuity and universal approximation in GNNs.

Abstract

In this paper, we present a novel interpretation of the so-called Weisfeiler-Lehman (WL) distance, introduced by Chen et al. (2022), using concepts from stochastic processes. The WL distance aims at comparing graphs with node features, has the same discriminative power as the classic Weisfeiler-Lehman graph isomorphism test and has deep connections to the Gromov-Wasserstein distance. This new interpretation connects the WL distance to the literature on distances for stochastic processes, which also makes the interpretation of the distance more accessible and intuitive. We further explore the connections between the WL distance and certain Message Passing Neural Networks, and discuss the implications of the WL distance for understanding the Lipschitz property and the universal approximation results for these networks.