Seeking universal approximation for continuous limits of graph neural networks on large random graphs

We propose a notion of universality for graph neural networks (GNNs) in the large random graphs limit, tailored for node-level tasks. When graphs are drawn from a latent space model, or from a graphon, GNNs on growing graph sequences are known to converge to limit objects called "continuous GNNs", or cGNNs. A cGNN inputs and outputs functions defined on a latent space, and as such, is a non-linear operator between functional spaces. Therefore, we propose to evaluate the expressivity of a cGNN through its ability to approximate arbitrary non-linear operators between functional spaces. This is reminiscent of Operator Learning, a branch of machine learning dedicated to learning non-linear operators between functional spaces. In this field, several architectures known as Neural Operators (NOs) are indeed proven to be universal. The justification for the universality of these architectures relies on a constructive method based on an encoder-decoder strategy into finite dimensional spaces, which enables invoking the universality of usual MLP. In this paper, we adapt this method to cGNNs. This is however far from straightforward: cGNNs have crucial limitations, as they do not have access to the latent space, but observe it only indirectly through the graph built on it. Our efforts will be directed toward circumventing this difficulty, which we will succeed, at this stage, only with strong hypotheses.