The Weisfeiler-Lehman Distance: Reinterpretation and Connection with GNNs
This work provides theoretical insights for graph machine learning researchers, but it is incremental as it builds on existing distance concepts.
The paper reinterprets the Weisfeiler-Lehman distance using stochastic processes to make it more intuitive and connects it to Message Passing Neural Networks, discussing implications for Lipschitz properties and universal approximation results.
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.