  Graph Convolutional Networks (GCNs) are one of the most popular architectures
that are used to solve classification problems accompanied by graphical
information. We present a rigorous theoretical understanding of the effects of
graph convolutions in multi-layer networks. We study these effects through the
node classification problem of a non-linearly separable Gaussian mixture model
coupled with a stochastic block model. First, we show that a single graph
convolution expands the regime of the distance between the means where
multi-layer networks can classify the data by a factor of at least
$1/\sqrt[4]{\mathbb{E}{\rm deg}}$, where $\mathbb{E}{\rm deg}$ denotes the
expected degree of a node. Second, we show that with a slightly stronger graph
density, two graph convolutions improve this factor to at least
$1/\sqrt[4]{n}$, where $n$ is the number of nodes in the graph. Finally, we
provide both theoretical and empirical insights into the performance of graph
convolutions placed in different combinations among the layers of a network,
concluding that the performance is mutually similar for all combinations of the
placement. We present extensive experiments on both synthetic and real-world
data that illustrate our results.
