Paper:
While existing graph embedding models can generate useful embedding vectors for graph-related tasks, what valuable information can be jointly learned from a graph embedding model is less discussed. In this paper, we consider the possibility of detecting critical structures by a graph embedding model. We propose Ego-CNN to embed graphs, which works in a local-to-global manner to take advantages of CNNs that gradually expands the detectable local regions on the graph as the network depth increases. Critical structures can be detected if Ego-CNN is combined with a supervised task model. We show that Ego-CNN is (1) competitive to state-of-the-art graph embeddings models, (2) can work nicely with CNNs visualization techniques to show the detected structures, and (3) is efficient and can incorporate with scale-free priors, which commonly occurs in social network datasets, to further improve the training efficiency.

Review:
Dear authors,
Thank you for your contribution to ICLR. The problem you are addressing with your work is important. Your paper is well-motivated. Detecting and exploiting "critical structures" in graphs for graph classification is indeed something that is missing in previous work. 
After the introduction you discuss some related work. While I really appreciate the effort you put into this section (including the figures etc.) there are several inaccuracies in the portrayal of existing methods. Especially the comparison to Patchy-san is somewhat vague. Please make sure that you clearly state the differences between patchy-san and Ego-CNNs. What exactly is it that Patchy cannot achieve that you can. I believe I understood what the advantages of the proposed method are but it took a while to get there. Just one example to show you what I mean; you write:
"The reason why the idea of Patchy-San fails to generalize into multiple layers is that its definition of
neighborhood, which is based on adjacency matrix, is not static and may not corresponding to local
regions in the graph. "
It is very difficult to understand what it is that you want to express with the above sentence. Its definition of neighborhood is based on adjacency matrix - what does that mean? A neighborhood is a set of nodes, no? Why is it that their definition of neighborhood might not correspond to local regions? In general, you should try to be more precise and concise when discussing related work. 
Section 3, the most important section in the paper that describes the proposed Ego-CNN approach, should also be written more clearly. For instance, it would be good if you could define the notion of an "Ego-Convolution layer." You use that term without properly defining it and it is difficult to make sense of the approach without understanding it. Also, you contrast your approach with patchy and write that "Our main idea is to use the egocentric design, i.e. the neighborhood at next
layer is defined on the same node." Unfortunately, I find it difficult to understand what this means. In general, this section is very verbose and needs a lot more work. This is at the moment also the crucial shortcoming of the paper. You should spent more time on section 3 and formally and more didactically introduce your approach. In my opinion, without a substantial improvement of this section, the paper should not be accepted. 
The experiments are standard and compare to numerous existing state of the art methods. The data sets are also rather standard. The one thing I would add to the results are the standard deviations. It is common to report those. Also, in the learning for graph structured data, the variance can be quite high and providing the stddev would at least indicate how significant the improvements are. 
I also like the visualizations and the discussion of the critical structures found in some of the graphs.
Overall, I think this is an interesting paper that has a lot of potential. The problem, however, is that the presentation of the proposed approach is verbose and partially incomprehensible. What exactly is different to existing approaches? What exactly is the formal definition of the method? All of this is not well presented and, in my opinion, requires another round of editing and reviews.