Numerous recent works have analyzed the expressive power of message-passing graph neural networks (MPNNs), primarily utilizing combinatorial techniques such as the $1$-dimensional Weisfeiler-Leman test ($1$-WL) for the graph isomorphism problem. However, the graph isomorphism objective is inherently binary, not giving insights into the degree of similarity between two given graphs. This work resolves this issue by considering continuous extensions of both $1$-WL and MPNNs to graphons. Concretely, we show that the continuous variant of $1$-WL delivers an accurate topological characterization of the expressive power of MPNNs on graphons, revealing which graphs these networks can distinguish and the level of difficulty in separating them. We identify the finest topology where MPNNs separate points and prove a universal approximation theorem. Consequently, we provide a theoretical framework for graph and graphon similarity combining various topological variants of classical characterizations of the $1$-WL. In particular, we characterize the expressive power of MPNNs in terms of the tree distance, which is a graph distance based on the concepts of fractional isomorphisms, and substructure counts via tree homomorphisms, showing that these concepts have the same expressive power as the $1$-WL and MPNNs on graphons. Empirically, we validate our theoretical findings by showing that randomly initialized MPNNs, without training, exhibit competitive performance compared to their trained counterparts. Moreover, we evaluate different MPNN architectures based on their ability to preserve graph distances, highlighting the significance of our continuous $1$-WL test in understanding MPNNs' expressivity.

翻译：近期众多工作分析了消息传递图神经网络（MPNNs）的表达能力，主要利用图同构问题中的一维Weisfeiler-Leman测试（1-WL）等组合技术。然而，图同构目标本质上是二元的，无法揭示两个给定图之间的相似程度。本文通过将1-WL和MPNNs连续扩展到图极限解决该问题。具体地，我们证明1-WL的连续变体能够精确刻画MPNNs在图极限上的表达能力的拓扑特征，揭示这些网络可区分的图及其分离难度。我们识别出MPNNs可分离点的最细拓扑，并证明通用逼近定理。据此，我们融合经典1-WL表征的各种拓扑变体，构建了图与图极限相似性的理论框架。特别地，我们以树距离（基于分数同构概念的图距离）及通过树同态的子结构计数来刻画MPNNs的表达能力，证明这些概念与图极限上的1-WL和MPNNs具有相同表达能力。实验上，我们验证了理论发现：未经训练的随机初始化MPNNs与训练后的对应版本相比，展现出竞争性性能。此外，我们基于不同MPNN架构保持图距离的能力对其评估，凸显连续1-WL测试在理解MPNNs表达性中的重要性。