  Graph convolutional networks (GCNs) are \emph{discriminative models} that
directly model the class posterior $p(y|\mathbf{x})$ for semi-supervised
classification of graph data. While being effective, as a representation
learning approach, the node representations extracted from a GCN often miss
useful information for effective clustering, because the objectives are
different. In this work, we design normalizing flows that replace GCN layers,
leading to a \emph{generative model} that models both the class conditional
likelihood $p(\mathbf{x}|y)$ and the class prior $p(y)$. The resulting neural
network, GC-Flow, retains the graph convolution operations while being equipped
with a Gaussian mixture representation space. It enjoys two benefits: it not
only maintains the predictive power of GCN, but also produces well-separated
clusters, due to the structuring of the representation space. We demonstrate
these benefits on a variety of benchmark data sets. Moreover, we show that
additional parameterization, such as that on the adjacency matrix used for
graph convolutions, yields additional improvement in clustering.
