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（GNN）作为一种数据驱动型工具，在广泛图域的学习任务中展现出强大性能；基于消息传递机制，GNN因其与图同构的Weisfeiler-Lehman（WL）测试紧密相连的直观公式而日益普及，并被证明与该测试等价。理论上，GNN已被证明具有通用逼近能力，且近期针对具有分段多项式激活函数的GNN，其泛化能力（即Vapnik-Chervonekis（VC）维数的界）已得到研究。本研究旨在利用Pfaffian函数理论框架，将GNN的VC维数分析拓展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、输入规模）以及图域中应用1-WL测试所得颜色数量的理论界。该理论分析得到了初步实验研究的支持。