  We perform theoretical and algorithmic studies for the problem of clustering
and semi-supervised classification on graphs with both pairwise relational
information and single-point feature information, upon a joint stochastic block
model for generating synthetic graphs with both edges and node features.
Asymptotically exact analysis based on the Bayesian inference of the underlying
model are conducted, using the cavity method in statistical physics.
Theoretically, we identify a phase transition of the generative model, which
puts fundamental limits on the ability of all possible algorithms in the
clustering task of the underlying model. Algorithmically, we propose a belief
propagation algorithm that is asymptotically optimal on the generative model,
and can be further extended to a belief propagation graph convolution neural
network (BPGCN) for semi-supervised classification on graphs. For the first
time, well-controlled benchmark datasets with asymptotially exact properties
and optimal solutions could be produced for the evaluation of graph convolution
neural networks, and for the theoretical understanding of their strengths and
weaknesses. In particular, on these synthetic benchmark networks we observe
that existing graph convolution neural networks are subject to an sparsity
issue and an ovefitting issue in practice, both of which are successfully
overcome by our BPGCN. Moreover, when combined with classic neural network
methods, BPGCN yields extraordinary classification performances on some
real-world datasets that have never been achieved before.
