Fine-Grained Expressive Power of Weisfeiler-Leman: A Homomorphism Counting Perspective
Junru Zhou, Muhan Zhang
TL;DR
The paper tackles the problem of quantifying graph neural networks' expressive power by focusing on homomorphism counting. It introduces generalized folklore Weisfeiler-Leman (GFWL) algorithms as a broad, unifying design space for WL-type GNNs and develops a meta-algorithm that determines the exact homomorphism-counting power of any GFWL instance. The core results establish a precise equivalence between WL indistinguishability and differences in homomorphism counts for a finite family of query graphs, using EF and Cops-Robber pebble games to connect color refinement with counting power. The work provides a principled, automatable framework for analyzing and potentially guiding GNN design, while acknowledging current limitations and outlining directions for extending the theory to broader WL variants and substructure counting.
Abstract
The ability of graph neural networks (GNNs) to count homomorphisms has recently been proposed as a practical and fine-grained measure of their expressive power. Although several existing works have investigated the homomorphism counting power of certain GNN families, a simple and unified framework for analyzing the problem is absent. In this paper, we first propose \emph{generalized folklore Weisfeiler-Leman (GFWL)} algorithms as a flexible design basis for expressive GNNs, and then provide a theoretical framework to algorithmically determine the homomorphism counting power of an arbitrary class of GNN within the GFWL design space. As the considered design space is large enough to accommodate almost all known powerful GNNs, our result greatly extends all existing works, and may find its application in the automation of GNN model design.