CONDITIONAL NETWORK EMBEDDINGS
Network Embeddings (NEs) map the nodes of a given network into d-dimensional Euclidean space R d . Ideally, this mapping is such that 'similar' nodes are mapped onto nearby points, such that the NE can be used for purposes such as link prediction (if 'similar' means being 'more likely to be connected' or 'having similar neighborhoods') or classification (if 'similar' means 'being more likely to have the same label'). In recent years various methods for NE have been introduced, all following a similar strategy: defining a notion of similarity between nodes, a distance measure in the embedding space, and a loss function that penalizes large distances for similar nodes and small distances for dissimilar nodes. A difficulty faced by existing methods is that certain networks are fundamentally hard to embed due to their structural properties: (approximate) multipartiteness, certain degree distributions, assortativity, etc. To overcome this, we introduce a conceptual innovation to the NE literature and propose to create Conditional Network Embeddings (CNEs); embeddings that maximally add information with respect to given structural properties (e.g. node degrees, block densities, etc.). We use a simple Bayesian approach to achieve this, and propose a block stochastic gradient descent algorithm for fitting it efficiently. We demonstrate that CNEs are superior for link prediction and multi-label classification when compared to state-ofthe-art methods, and this without adding significant mathematical or computational complexity. Finally, we illustrate the potential of CNE for network visualization.
The authors propose a generative model of networks by learning embeddings and pairing the embeddings with a prior distribution over networks. The idea is that the prior distribution may explain structure that the embeddings would not have to capture.
The motivation for doing this is that this structure is typically hard to model for network embeddings.
The authors propose a clean -if improper- prior on networks and proceed to perform maximum likelihood inference on it.
The experiments show that the approach works fine for link porediction and can be used for visualization.
Two points: 
a) Why not try to do this with Variational inference? It should conceptually still work and be fast and potentially more robust.
b) The prior seems to be picked according to properties of the observed data and expressed in a product of constraints. This seems clunky, I would have been more impressed with a prior structure that ties in closer with the embeddings and requires less hand-engineering.
A key point of interest is the following: very exciting recent work (GraphRNN: Generating Realistic Graphs with Deep Auto-regressive Models by You et al ICML2018) has proposed neural generative models of networks with a high degree of fidelity and much less hand-picked features.  The work here tries to not learn a lot of these structures but impose them. Do the authors think that ultimately learning priors with models like GraphRNN might be more promising for certain applications?
The drawback in this model here is that ultimately networks are embedded, but not really generated during test time. A more predictive generative model that makes less hard assumptions on graph data would be interesting.
Update After rebuttal:
Given the authors' rebuttal to all reviews, I am upgrading my score to a 6. I still feel that more learning (as inGraphRNN) to build a fuller generative model of the graph would be interesting, but the authors make a strong case for the usefulness and practicality of their approach.