Fundamental Limits of Deep Graph Convolutional Networks

TL;DR

This paper investigates the fundamental limitations of deep graph convolutional networks (GCNs) in distinguishing between different graph models, providing theoretical characterizations and empirical evidence of their capabilities and constraints.

Contribution

It offers a precise theoretical characterization of when GCNs can or cannot distinguish between graphons, revealing fundamental limits based on network depth and graph properties.

Findings

GCNs with depth at least logarithmic in graph size cannot distinguish certain graphon pairs.

A simple GCN architecture can distinguish some graphons outside the indistinguishability class.

Empirical results show indistinguishable graph distributions occur in real datasets.

Abstract

Graph convolutional networks (GCNs) are a widely used method for graph representation learning. To elucidate the capabilities and limitations of GCNs, we investigate their power, as a function of their number of layers, to distinguish between different random graph models (corresponding to different class-conditional distributions in a classification problem) on the basis of the embeddings of their sample graphs. In particular, the graph models that we consider arise from graphons, which are the most general possible parameterizations of infinite exchangeable graph models and which are the central objects of study in the theory of dense graph limits. We give a precise characterization of the set of pairs of graphons that are indistinguishable by a GCN with nonlinear activation functions coming from a certain broad class if its depth is at least logarithmic in the size of the sample graph. This characterization is in terms of a degree profile closeness property. Outside this class, a very simple GCN architecture suffices for distinguishability. We then exhibit a concrete, infinite class of graphons arising from stochastic block models that are well-separated in terms of cut distance and are indistinguishable by a GCN. These results theoretically match empirical observations of several prior works. To prove our results, we exploit a connection to random walks on graphs. Finally, we give empirical results on synthetic and real graph classification datasets, indicating that indistinguishable graph distributions arise in practice.