The Power of Graph Convolutional Networks to Distinguish Random Graph Models: Short Version

TL;DR

This paper analyzes the ability of graph convolutional networks (GCNs) to distinguish between different random graph models, revealing depth-dependent limitations and conditions under which simple GCNs succeed, supported by theoretical and empirical evidence.

Contribution

It provides a theoretical framework linking GCN depth to graph model distinguishability and identifies conditions for effective differentiation of graphons.

Findings

Deeper GCNs (logarithmic in graph size) cannot distinguish certain well-separated graphons.

Simple GCNs can distinguish graphons with degree profile separation.

Theoretical results align with empirical observations from prior studies.

Abstract

Graph convolutional networks (GCNs) are a widely used method for graph representation learning. We investigate the power of GCNs, as a function of their number of layers, to distinguish between different random graph models 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 exhibit an infinite class of graphons that are well-separated in terms of cut distance and 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. These results theoretically match empirical observations of several prior works. Finally, we show a converse result that for pairs of graphons satisfying a degree profile separation property, a very simple GCN architecture suffices for distinguishability. To prove our results, we exploit a connection to random walks on graphs.