Fine-Grained Expressive Power of Weisfeiler-Leman: A Homomorphism Counting Perspective

TL;DR

This paper introduces a unified framework using generalized folklore Weisfeiler-Leman algorithms to analyze and determine the homomorphism counting power of various graph neural networks, enhancing understanding of their expressive capabilities.

Contribution

It proposes a flexible GFWL-based framework for analyzing GNN expressive power and provides a method to algorithmically determine their homomorphism counting capabilities.

Findings

Extends existing analyses to a broad class of GNNs.

Provides a theoretical basis for automating GNN design.

Unifies previous work under a comprehensive framework.

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.