A note on the VC dimension of 1-dimensional GNNs

TL;DR

This paper investigates the VC dimension of 1-dimensional GNNs, revealing they have infinite VC dimension even with simple architectures and activation functions, indicating limitations in their generalization capabilities.

Contribution

It extends existing results to show that 1D GNNs with a single parameter have infinite VC dimension, highlighting inherent generalization limitations.

Findings

1D GNNs with a single parameter have infinite VC dimension.

Analytic non-polynomial activation functions do not reduce VC dimension.

Results imply fundamental limitations in GNN generalization.

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.