Zero-One Laws of Graph Neural Networks

TL;DR

This paper establishes zero-one laws for graph neural networks, showing that as graph size increases, the probability of a GNN output converges to either zero or one, revealing fundamental limits of their capacity.

Contribution

It introduces a novel theoretical framework demonstrating zero-one laws for GNNs on large Erdős-Rényi graphs, highlighting inherent capacity limitations.

Findings

Zero-one laws hold for GNNs on large random graphs.

As graph size grows, GNN outputs tend to deterministic limits.

Empirical results confirm theoretical asymptotic behavior on small graphs.

Abstract

Graph neural networks (GNNs) are the de facto standard deep learning architectures for machine learning on graphs. This has led to a large body of work analyzing the capabilities and limitations of these models, particularly pertaining to their representation and extrapolation capacity. We offer a novel theoretical perspective on the representation and extrapolation capacity of GNNs, by answering the question: how do GNNs behave as the number of graph nodes become very large? Under mild assumptions, we show that when we draw graphs of increasing size from the Erd\H{o}s-R\'enyi model, the probability that such graphs are mapped to a particular output by a class of GNN classifiers tends to either zero or to one. This class includes the popular graph convolutional network architecture. The result establishes 'zero-one laws' for these GNNs, and analogously to other convergence laws, entails theoretical limitations on their capacity. We empirically verify our results, observing that the theoretical asymptotic limits are evident already on relatively small graphs.