In Graph Neural Networks (GNNs), hierarchical pooling operators generate local summaries of the data by coarsening the graph structure and the vertex features. Considerable attention has been devoted to analyzing the expressive power of message-passing (MP) layers in GNNs, while a study on how graph pooling affects the expressiveness of a GNN is still lacking. Additionally, despite the recent advances in the design of pooling operators, there is not a principled criterion to compare them. In this work, we derive 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 analyze several existing pooling operators and identify those that fail to satisfy the expressiveness conditions. Finally, we introduce an experimental setup to verify empirically the expressive power of a GNN equipped with pooling layers, in terms of its capability to perform a graph isomorphism test.

翻译：在图神经网络中，分层池化算子通过粗化图结构和顶点特征来生成数据的局部摘要。大量研究致力于分析消息传递层的表达能力，但关于图池化如何影响图神经网络表达能力的系统性研究仍显不足。此外，尽管池化算子设计近期取得了进展，但目前缺乏比较它们的理论准则。本文推导了池化算子完全保留其前序消息传递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算子的通用理论基础。基于理论发现，我们分析了多种现有池化算子，识别出未能满足表达能力条件的算子。最后，我们引入实验框架，通过图同构测试能力对配备池化层的图神经网络进行经验性验证。