Title: Wasserstein Weisfeiler-Lehman Subtree Distance for Graph-Structured Data. Abstract: Defining a valid graph distance is a challenging task in graph machine learning because we need to consider the theoretical validity of the distance, its computational complexity, and effectiveness as a distance between graphs. Addressing the shortcomings of the popular Weisfeiler-Lehman (WL) test for the graph isomorphism problem, this paper proposes a novel distance between graph structures. More specifically, we first analyze the WL algorithm from a geometric point of view and argue that discriminating nodes based on only the consistency of categorical labels do not fully capture important structural information. Therefore, instead of using such categorical labels, we define a node distance between WL subtrees with tree edit distance and propose an efficient calculation algorithm. We then apply the proposed node distance to define a graph Wasserstein distance on tree edit embedding space exploiting Optimal Transport framework. To summarize, these two distances have been proposed at the node level and graph level, respectively. Numerical experimentations on graph classification tasks show that the proposed graph Wasserstein distance performs equally or better than conventional methods.