  Recent work on graph generative models has made remarkable progress towards
generating increasingly realistic graphs, as measured by global graph features
such as degree distribution, density, and clustering coefficients. Deep
generative models have also made significant advances through better modelling
of the local correlations in the graph topology, which have been very useful
for predicting unobserved graph components, such as the existence of a link or
the class of a node, from nearby observed graph components. A complete
scientific understanding of graph data should address both global and local
structure. In this paper, we propose a joint model for both as complementary
objectives in a graph VAE framework. Global structure is captured by
incorporating graph kernels in a probabilistic model whose loss function is
closely related to the maximum mean discrepancy(MMD) between the global
structures of the reconstructed and the input graphs. The ELBO objective
derived from the model regularizes a standard local link reconstruction term
with an MMD term. Our experiments demonstrate a significant improvement in the
realism of the generated graph structures, typically by 1-2 orders of magnitude
of graph structure metrics, compared to leading graph VAEand GAN models. Local
link reconstruction improves as well in many cases.
