A Set-Theoretic Approach to Modeling Network Structure

Round 1

Reviewer 1 Report

In this paper, the authors have introduced an interesting set-theoretic approach for modeling a complex network. Instead of representing graphs using popular matrix notations, the authors have used set notations and to represent and manipulate graphs. Following are some of my specific comments about the paper.   Since the authors have used a different approach for analysing a complex network, it would be helpful if the author can clearly state the advantages of the proposed framework over the matrix-based approaches in the abstract and the introduction section.   Section 2 of the paper is very heavy and I found it very difficult to understand some notations. For example, what does Y.\tau represent? Is it the image of Y under \tau? In the second paragraph, the author has used the symbol {\cal S}. Is it the same as S in the first paragraph? Similarly, in the fourth paragraph, it seems that the author has used two different symbols, {\cal N} and N, to represent the same set.   Can the author explain different operators with the help of a very simple graph with few nodes/edges.   In the paper the author has made few claims about the properties of the network. I wonder are these hypotheses or known facts? For example the statement, "Such a node, z whose “horizon” is contained in that of y, contributes very little to the information content of the network so that its removal from {y}.n will result in little information loss". Can the author clarify this in the manuscript?   Since the authors have used a different approach to analyze networks, I would suggest to include a dedicated section about the computational cost of different operations discussed in the paper and how their cost compares with traditional approaches. The authors have only discussed the computation cost of the code given in Figure 7.   The properties reported in Table 1 might not be a good indicator of network similarities. It might be a good idea to include some more properties in the table. Clustering coefficient might be a better alternative for the number of triangles.   The author might find it useful to report the values of the Fiedler eigenvectors (that corresponds to the second smallest eigenvalue of the network) for the three generated networks. This eigenvector plays a very important role in community detection in a network.

Author Response

Thank you for your comments.

I believe I have clarified the introduction rather considerably.  You ask me to clarify the "advantages of the proposed framework over the matrix-based approaches".  They are not in "competition"; for the most part the yield different results.

\CALN is meant to represent the network as a whole, N is the set of its nodes.

You were right. The sections of pseudocode should not be displayed as "figures", I have changed that.

Finally, thank you for pointing out the role of Fiedler eigenvectors; I had not known of them.  I felt that actually displaying the Fiedler eigenvector for even the small network for which I gave the principal eigenvector would have been overkill.  But, I did add a mention to them in the "community" section for the interested reader.

I very much appreciate your careful reading. Thank you

 

 

Reviewer 2 Report

In this paper, the author provide 3 algorithms by using set operations and relational comparator. Based on the concept of "interior" to discuss some properties and apply in social questions. Here are some comments.

Line4 of Sec1, why is it "most" understandable. As I know, using graphs is more simple than using networks.

Line-9 of p1, "In it" is not suitable.

The author should provide some reason to consider the 3 models. I do not see some relations with old ones.

The introduction is too weak for a algorithm-based paper.

Line 3 of sec2, "An operator is said to be ".

Line-3, "Readily" is not proper. Provide details. 

Figure 7 is not a figure, it is a code list.

How do you expand from figure 4 to figure 11?

 

In this work, the idea is good, and the author want to post well three methods. But the presentation is needed to improve largely. Also some details should be added. So I give it as a major revision.

Author Response

Thank your for your suggested revisions.  I believe I have incorporated all of them in addition to others from the 2nd reviewer.

Networks and graphs are synonyms.  I prefer the term network because it suggests applications realms such as social networks.

I have completely rewritten the Introduction; it was too weak.  I think you will find it better now.

The pseudocode lists are no longer figures.  They should not have been treated that way.

The displayed Pseudocode III describes the method of expansion from figure 4 to figure 11, except they are figures 3 and 8 in the new version.

I appreciate your reading it all the way through and making perceptive comments. Thank you 

Round 2

Reviewer 2 Report

This version seems to be publishable.