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Graph Neural Networks (GNNs) have emerged in recent years as a powerful toolto learn tasks across a wide range of graph domains in a data-driven fashion;based on a message passing mechanism, GNNs have gained increasing popularitydue to their intuitive formulation, closely linked with the Weisfeiler-Lehman(WL) test for graph isomorphism, to which they have proven equivalent. From atheoretical 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 piecewisepolynomial activation functions. The aim of our work is to extend this analysison the VC dimension of GNNs to other commonly used activation functions, suchas sigmoid and hyperbolic tangent, using the framework of Pfaffian functiontheory. Bounds are provided with respect to architecture parameters (depth,number of neurons, input size) as well as with respect to the number of colorsresulting from the 1-WL test applied on the graph domain. The theoreticalanalysis is supported by a preliminary experimental study.

VC dimension of Graph Neural Networks with Pfaffian activation functions