  Recently there has been increased interest in semi-supervised classification
in the presence of graphical information. A new class of learning models has
emerged that relies, at its most basic level, on classifying the data after
first applying a graph convolution. To understand the merits of this approach,
we study the classification of a mixture of Gaussians, where the data
corresponds to the node attributes of a stochastic block model. We show that
graph convolution extends the regime in which the data is linearly separable by
a factor of roughly $1/\sqrt{D}$, where $D$ is the expected degree of a node,
as compared to the mixture model data on its own. Furthermore, we find that the
linear classifier obtained by minimizing the cross-entropy loss after the graph
convolution generalizes to out-of-distribution data where the unseen data can
have different intra- and inter-class edge probabilities from the training
data.
