Beyond Weisfeiler-Lehman: A Quantitative Framework for GNN Expressiveness

TL;DR

This paper introduces a new quantitative framework based on homomorphism counting to evaluate and compare the expressiveness of GNNs, addressing limitations of the traditional Weisfeiler-Lehman hierarchy.

Contribution

It proposes a unified, practical measure called homomorphism expressivity for assessing GNNs, enabling direct comparisons and understanding of their capabilities.

Findings

Homomorphism expressivity effectively measures GNNs' ability to count substructures.

The framework unifies previous GNN expressiveness assessments.

Experimental results confirm the theory aligns with real-world GNN performance.

Abstract

Designing expressive Graph Neural Networks (GNNs) is a fundamental topic in the graph learning community. So far, GNN expressiveness has been primarily assessed via the Weisfeiler-Lehman (WL) hierarchy. However, such an expressivity measure has notable limitations: it is inherently coarse, qualitative, and may not well reflect practical requirements (e.g., the ability to encode substructures). In this paper, we introduce a unified framework for quantitatively studying the expressiveness of GNN architectures, addressing all the above limitations. Specifically, we identify a fundamental expressivity measure termed homomorphism expressivity, which quantifies the ability of GNN models to count graphs under homomorphism. Homomorphism expressivity offers a complete and practical assessment tool: the completeness enables direct expressivity comparisons between GNN models, while the practicality allows for understanding concrete GNN abilities such as subgraph counting. By examining four classes of prominent GNNs as case studies, we derive simple, unified, and elegant descriptions of their homomorphism expressivity for both invariant and equivariant settings. Our results provide novel insights into a series of previous work, unify the landscape of different subareas in the community, and settle several open questions. Empirically, extensive experiments on both synthetic and real-world tasks verify our theory, showing that the practical performance of GNN models aligns well with the proposed metric.