Weisfeiler-Leman Indistinguishability of Graphons

TL;DR

This paper extends the Weisfeiler-Leman algorithm and its characterizations from finite graphs to graphons, the limit objects in dense graph theory, revealing new insights into their indistinguishability and homomorphism densities.

Contribution

It generalizes the Weisfeiler-Leman algorithm and its characterizations to graphons, including new variants that account for simple graphs and multigraphs with parallel edges.

Findings

$k$-WL characterizations extend to graphons with homomorphism densities.

Parallel edges affect the equivalence between $1$-WL and color refinement in graphons.

A variant of $k$-WL matches homomorphism densities from simple graphs of bounded treewidth.

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. Greb\'ik and Rocha (2022) have recently shown how color refinement and notions that characterize it generalize to graphons, which emerged as limit objects in the theory of dense graph limits. 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 Greb\'ik 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 more general case of graphons, which means that, there, the equivalence between $1$-WL and color refinement does not hold anymore. We also show how this equivalence can be recovered by defining a variant of $k$-WL that corresponds to homomorphism densities from simple graphs of bounded treewidth.