Graph Neural Networks: Architectures, Stability and Transferability

TL;DR

This paper explores the architectures, stability, and transferability of Graph Neural Networks (GNNs), demonstrating their permutation equivariance, stability to graph deformations, and convergence properties, which explain their empirical success across various applications.

Contribution

It introduces the theoretical foundations of GNNs' stability and transferability, including convergence to graphon neural networks, and illustrates their practical applications.

Findings

GNNs are permutation equivariant and stable to graph deformations.

GNNs converge to graphon neural networks as graphs grow large.

Transferability of GNNs across different graph sizes is justified.

Abstract

Graph Neural Networks (GNNs) are information processing architectures for signals supported on graphs. They are presented here as generalizations of convolutional neural networks (CNNs) in which individual layers contain banks of graph convolutional filters instead of banks of classical convolutional filters. Otherwise, GNNs operate as CNNs. Filters are composed with pointwise nonlinearities and stacked in layers. It is shown that GNN architectures exhibit equivariance to permutation and stability to graph deformations. These properties help explain the good performance of GNNs that can be observed empirically. It is also shown that if graphs converge to a limit object, a graphon, GNNs converge to a corresponding limit object, a graphon neural network. This convergence justifies the transferability of GNNs across networks with different number of nodes. Concepts are illustrated by the application of GNNs to recommendation systems, decentralized collaborative control, and wireless communication networks.