  Recently, techniques for applying convolutional neural networks to
graph-structured data have emerged. Graph convolutional neural networks (GCNNs)
have been used to address node and graph classification and matrix completion.
Although the performance has been impressive, the current implementations have
limited capability to incorporate uncertainty in the graph structure. Almost
all GCNNs process a graph as though it is a ground-truth depiction of the
relationship between nodes, but often the graphs employed in applications are
themselves derived from noisy data or modelling assumptions. Spurious edges may
be included; other edges may be missing between nodes that have very strong
relationships. In this paper we adopt a Bayesian approach, viewing the observed
graph as a realization from a parametric family of random graphs. We then
target inference of the joint posterior of the random graph parameters and the
node (or graph) labels. We present the Bayesian GCNN framework and develop an
iterative learning procedure for the case of assortative mixed-membership
stochastic block models. We present the results of experiments that demonstrate
that the Bayesian formulation can provide better performance when there are
very few labels available during the training process.
