  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.
