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, and furthermore show that, for this application, ReLU activation functions and non-identity weight matrices with non-negative entries do not help in terms of distinguishing power. These results theoretically match empirical observations of several prior works. Finally, we show 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.

One of the main challenges in cluster analysis is estimating the true number of clusters in a dataset. This paper quantifies a notion of persistence of a clustering solution over a range of resolution scales, which is used to characterize the natural clusters and estimate the true number of clusters in a dataset. We show that this quantification of persistence is associated with evaluating the largest eigenvalue of the underlying cluster covariance matrix. Detailed experiments on a variety of standard and synthetic datasets demonstrate that the proposed persistence-based indicator outperforms the existing approaches, such as, gap-statistic method, $X$-means, $G$-means, $PG$-means, dip-means algorithms and information-theoretic method, in accurately predicting the true number of clusters. Interestingly, our method can be explained in terms of the phase-transition phenomenon in the deterministic annealing algorithm where the number of cluster centers changes (bifurcates) with respect to an annealing parameter. However, the approach suggested in this paper is independent of the choice of clustering algorithm; and can be used in conjunction with any suitable clustering algorithm.