Graph Neural Networks (GNNs) have emerged in recent years as a powerful tool to learn tasks across a wide range of graph domains in a data-driven fashion; based on a message passing mechanism, GNNs have gained increasing popularity due to their intuitive formulation, closely linked with the Weisfeiler-Lehman (WL) test for graph isomorphism, to which they have proven equivalent. From a theoretical point of view, GNNs have been shown to be universal approximators, and their generalization capability (namely, bounds on the Vapnik Chervonekis (VC) dimension) has recently been investigated for GNNs with piecewise polynomial activation functions. The aim of our work is to extend this analysis on the VC dimension of GNNs to other commonly used activation functions, such as sigmoid and hyperbolic tangent, using the framework of Pfaffian function theory. Bounds are provided with respect to architecture parameters (depth, number of neurons, input size) as well as with respect to the number of colors resulting from the 1-WL test applied on the graph domain. The theoretical analysis is supported by a preliminary experimental study.

翻译：近年来，图神经网络（GNNs）作为一种强大的数据驱动工具，能够在广泛的图领域上学习任务；基于消息传递机制，GNNs因其直观的表述而日益流行，该机制与用于图同构的Weisfeiler-Lehman（WL）测试紧密相关，已被证明等价于该测试。从理论角度看，GNNs已被证明具有通用逼近能力，其泛化能力（即Vapnik Chervonekis（VC）维数的界）近期已针对具有分段多项式激活函数的GNNs得到研究。本文旨在利用Pfaffian函数理论框架，将这一针对GNNs VC维数的分析推广至其他常用激活函数，如sigmoid和双曲正切函数。我们提供了关于架构参数（深度、神经元数量、输入大小）以及图领域上应用1-WL测试所得颜色数量的界。该理论分析得到了初步实验研究的支持。