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.
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