A note on the VC dimension of 1-dimensional GNNs

• URL: http://arxiv.org/abs/2410.07829v1
• Date: Thu, 10 Oct 2024 11:33:15 GMT
• Title: A note on the VC dimension of 1-dimensional GNNs
• Authors: Noah Daniëls, Floris Geerts,
• Abstract summary: Graph Neural Networks (GNNs) have become an essential tool for analyzing graph-structured data. This paper focuses on the generalization of GNNs by investigating their Vapnik-Chervonenkis (VC) dimension.
• Score: 6.0757501646966965
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Graph Neural Networks (GNNs) have become an essential tool for analyzing graph-structured data, leveraging their ability to capture complex relational information. While the expressivity of GNNs, particularly their equivalence to the Weisfeiler-Leman (1-WL) isomorphism test, has been well-documented, understanding their generalization capabilities remains critical. This paper focuses on the generalization of GNNs by investigating their Vapnik-Chervonenkis (VC) dimension. We extend previous results to demonstrate that 1-dimensional GNNs with a single parameter have an infinite VC dimension for unbounded graphs. Furthermore, we show that this also holds for GNNs using analytic non-polynomial activation functions, including the 1-dimensional GNNs that were recently shown to be as expressive as the 1-WL test. These results suggest inherent limitations in the generalization ability of even the most simple GNNs, when viewed from the VC dimension perspective.

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