Graph Neural Networks (GNNs) have demonstrated superior performance on various graph learning tasks, including recommendation, where they leverage user-item collaborative filtering signals in graphs. However, theoretical formulations of their capability are scarce, despite their empirical effectiveness in state-of-the-art recommender models. Recently, research has explored the expressiveness of GNNs in general, demonstrating that message passing GNNs are at most as powerful as the Weisfeiler-Lehman test, and that GNNs combined with random node initialization are universal. Nevertheless, the concept of "expressiveness" for GNNs remains vaguely defined. Most existing works adopt the graph isomorphism test as the metric of expressiveness, but this graph-level task may not effectively assess a model's ability in recommendation, where the objective is to distinguish nodes of different closeness. In this paper, we provide a comprehensive theoretical analysis of the expressiveness of GNNs in recommendation, considering three levels of expressiveness metrics: graph isomorphism (graph-level), node automorphism (node-level), and topological closeness (link-level). We propose the topological closeness metric to evaluate GNNs' ability to capture the structural distance between nodes, which aligns closely with the objective of recommendation. To validate the effectiveness of this new metric in evaluating recommendation performance, we introduce a learning-less GNN algorithm that is optimal on the new metric and can be optimal on the node-level metric with suitable modification. We conduct extensive experiments comparing the proposed algorithm against various types of state-of-the-art GNN models to explore the explainability of the new metric in the recommendation task. For reproducibility, implementation codes are available at https://github.com/HKUDS/GTE.

翻译：图神经网络（GNN）在各种图学习任务（包括推荐）中已展现出卓越性能，其通过图结构利用用户-物品协同过滤信号。尽管这些方法在先进推荐模型中具有实证有效性，但其理论能力阐述却十分匮乏。近期研究从通用角度探索了GNN的表达能力，证明消息传递型GNN最多与Weisfeiler-Lehman测试等价，而结合随机节点初始化的GNN具有通用逼近性。然而，GNN"表达能力"这一概念仍缺乏明确定义。现有研究大多采用图同构测试作为表达能力度量标准，但这种图级任务可能无法有效评估模型在推荐中的能力——推荐任务的目标是区分不同邻近程度的节点。本文从三个层面对推荐中GNN的表达能力进行了全面理论分析：图同构（图级）、节点自同构（节点级）和拓扑接近度（边级）。我们提出拓扑接近度度量标准，用于评估GNN捕获节点间结构距离的能力，这与推荐目标高度契合。为验证新度量在评估推荐性能中的有效性，我们引入了一种免学习的GNN算法，该算法在新度量下达到最优，且经适当修改后可在节点级度量下达到最优。我们开展了大量实验，将所提算法与各类先进GNN模型进行对比，以探究新度量在推荐任务中的可解释性。为保障可复现性，实现代码已发布于https://github.com/HKUDS/GTE。