In Graph Neural Networks (GNNs), hierarchical pooling operators generate a coarser representation of the input data by creating local summaries of the graph structure and its vertex features. Considerable attention has been devoted to studying the expressive power of message-passing (MP) layers in GNNs, while a study on how pooling operators affect the expressivity of a GNN is still lacking. Additionally, despite the recent advances in the design of effective pooling operators, there is not a principled criterion to compare them. Our work aims to fill this gap by providing sufficient conditions for a pooling operator to fully preserve the expressive power of the MP layers before it. These conditions serve as a universal and theoretically-grounded criterion for choosing among existing pooling operators or designing new ones. Based on our theoretical findings, we reviewed several existing pooling operators and identified those that fail to satisfy the expressiveness assumptions. Finally, we introduced an experimental setup to empirically measure the expressive power of a GNN equipped with pooling layers, in terms of its capability to perform a graph isomorphism test.

翻译：在图神经网络（GNN）中，层次化池化算子通过对图结构及其顶点特征生成局部摘要，从而产生输入数据的粗粒度表示。现有研究已大量关注消息传递（MP）层在GNN中的表达能力，但针对池化算子如何影响GNN表达性的系统性研究仍然缺失。此外，尽管近年来在设计高效池化算子方面取得进展，但尚缺乏用于比较它们的理论准则。本研究旨在填补这一空白，提出确保池化算子完全保留其前置MP层表达能力的充分条件。这些条件可作为选择现有池化算子或设计新型算子的普适性理论基准。基于理论发现，我们系统评估了多种现有池化算子，并识别出不符合表达能力假设的算子。最后，我们设计了一种实验方案，通过图同构测试来实证测量配备池化层的GNN的表达能力。