Computational Capabilities of Graph Neural Networks

In this paper, we will consider the approximation properties of a recently introduced neural network model called graph neural network (GNN), which can be used to process-structured data inputs, e.g., acyclic graphs, cyclic graphs, and directed or undirected graphs. This class of neural networks implements a function tau(G, n) isin R ^m that maps a graph G and one of its nodes n onto an m-dimensional Euclidean space. We characterize the functions that can be approximated by GNNs, in probability, up to any prescribed degree of precision. This set contains the maps that satisfy a property called preservation of the unfolding equivalence, and includes most of the practically useful functions on graphs; the only known exception is when the input graph contains particular patterns of symmetries when unfolding equivalence may not be preserved. The result can be considered an extension of the universal approximation property established for the classic feedforward neural networks (FNNs). Some experimental examples are used to show the computational capabilities of the proposed model.

I. Introduction

In a large number of practical and engineering applications, the underlying data are often more conveniently represented in terms of graphs. In fact, a graph naturally represents a set of objects (nodes) and their relationships (edges). For example, in an image, it is natural to represent as nodes regions of the image that have similar intensity or color, and to represent the relationship among these regions by edges. This is often known as a region adjacency graph. As another example, it is convenient to model the individual web pages as nodes of a graph, and the hyperlink connections among the web pages as edges of the graph.