Weisfeiler-Leman Indistinguishability of Graphons

Jan Böker

Abstract: The color refinement algorithm is mainly known as a heuristic method for graph isomorphism testing. It has surprising but natural characterizations in terms of, for example, homomorphism counts from trees and solutions to a system of linear equations. Grebík and Rocha (2021) have recently shown that color refinement and some of its characterizations generalize to graphons, a natural notion for the limit of a sequence of graphs. In particular, they show that these characterizations are still equivalent in the graphon case. The k-dimensional Weisfeiler-Leman algorithm (k-WL) is a more powerful variant of color refinement that colors k-tuples instead of single vertices, where the terms 1-WL and color refinement are often used interchangeably since they compute equivalent colorings. We show how to adapt the result of Grebík and Rocha to k-WL or, in other words, how k-WL and its characterizations generalize to graphons. In particular, we obtain characterizations in terms of homomorphism densities from multigraphs of bounded treewidth and linear equations. We give a simple example that parallel edges make a difference in the graphon case, which means that the equivalence between 1-WL and color refinement is lost. We also show how to define a variant of k-WL that corresponds to homomorphism densities from simple graphs of bounded treewidth.