The Power of Graph Convolutional Networks to Distinguish Random Graph Models: Short Version
This work addresses the theoretical limitations of GCNs for graph representation learning, which is important for researchers in graph machine learning, though it is incremental as it builds on prior empirical findings.
The paper investigates the ability of Graph Convolutional Networks (GCNs) to distinguish between random graph models based on graphons, showing that GCNs with sufficient depth (logarithmic in graph size) fail to distinguish certain well-separated graphons, matching prior empirical observations, and conversely, that simple GCNs can distinguish graphons with degree profile separation.
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.