On clustering network-valued data
Community detection, which focuses on clustering nodes or detecting communities in (mostly) a single network, is a problem of considerable practical interest and has received a great deal of attention in the research community. While being able to cluster within a network is important, there are emerging needs to be able to cluster multiple networks. This is largely motivated by the routine collection of network data that are generated from potentially different populations. These networks may or may not have node correspondence. When node correspondence is present, we cluster networks by summarizing a network by its graphon estimate, whereas when node correspondence is not present, we propose a novel solution for clustering such networks by associating a computationally feasible feature vector to each network based on trace of powers of the adjacency matrix. We illustrate our methods using both simulated and real data sets, and theoretical justifications are provided in terms of consistency.
Review Summary: This paper accomplishes a lot in just a few pages. This paper is generally well-written, but the first three sections in particular are very strong. The paper does a great job of balancing discussing previous work, offering theoretical results for their approach, and presenting experimental results on both generated and real data. The paper would benefit from just a touch of restructuring to add a conclusion and future work section as well as more details to some of their experimental methods and results. 
Summary of paper: This paper presents two versions of their framework for clustering networks: NCGE and NCLM. Each version tackles a slightly different variation of the problem of clustering networks. NCGE is applied when there is node correspondence between the networks being clustered, while NCLM is applied when there isn’t. It is impressive that this paper tackles not one but both of these situations, presenting strong experimental results and connecting their work to the theory of networks. 
This paper is superbly written in some sections, which makes the weaker sections almost jarring. For example, sections 2 and 3 do an excellent job of demonstrating how the current work builds on previous work. But then in sharp contrast lines 197-200 rely heavily the reader knowing the cited outside sources to grasp their point. Understandably (due to page lengths) one can not include detailed explanations for everything, but the authors may want to consider pruning in some places to allow themselves more consistency in their explanations. 
The last subsection (page 8, lines 299-315) is the most interesting and strongest part of that final section. It would be excellent if this part could be expanded and include more details about these real networks. Also the paper ends abruptly without a conclusion or discussion of potential future work. Perhaps with a bit of restructuring, both could be included. 
Section 3.2 was another strong moment of the paper, peppering both interesting results (lines 140-142) and forecasting future parts of the paper (lines 149-150). However, this section might be one to either condense or prune just a bit to gain back a few lines as to add details in other sections. 

A few formatting issues, questions, and/or recommendations are below:
Page 1, lines 23,24,26, 35: Citations are out of order. 
Page 2, line 74: The presentation of the USVT acronym is inconsistent with the presentation of the NCGE and NCLM acronyms. 
Page 3, line 97: Citations are out of order. 
Page 4, line 128: Based on the earlier lines, it seems that g(A) = \hat{P} should be g(A_i) = \hat{P_i} 
Lines 159-160: In what sense does the parameter J count the effective number of eigenvalues? 
Page 5, lines 180-183: It appears that $dn$ is a number, but then in line 182, it seems that $dn()$ is a matrix function. Which is it?
Lines 180-183: The use of three different D’s in this area make it a challenge to sort through what all the D’s mean. Perhaps consider using different letters here. 
Line 196: Citations are out of order. 
Page 6, line 247: How is clustering accuracy measured? 
Page 7, line 259: There appears to be an extra space (or two) before the citation [26] 
Lines 266-277: There appears to be a bit of inconsistency in the use of K. For example, in line 266, K is a similarity matrix and in line 268, K is the number of clusters. 
Lines 269-270, table 2, and line 284: What are GK3, GK4, and GK5? How are they different?
Line 285: There is a space after eigenvectors before the comma.