Gain and Pain in Graph Partitioning: Finding Accurate Communities in Complex Networks^†

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Please see the attachment

Comments for author File: Comments.pdf

Comments on the Quality of English Language

No

Author Response

Dear Reviewer,
First, I would like to express my appreciation for your valuable feedback, which led to significant improvements in the paper. Your reviews were instrumental in enhancing some overlooked aspects of the work. I have incorporated most of your comments into the revised paper, and below, I try to briefly explain what has been done.

Your technical feedback significantly improved the theoretical aspects of the paper. Based on your suggestions, we have revised the proofs for Lemma 1, Theorem 1, and Theorem 2. Additionally, we streamlined Section 2.3 by combining the lemma and the theorem into a single Theorem 4, which now effectively supports our sparse model IPs-MM.

We also addressed the typographical errors and other details you mentioned. We reviewed the entire paper once more to ensure nothing was overlooked or incorrect.

Once again, thank you very much for your valuable feedback.
With warm regards,
Arman Ferdowsi

Reviewer 2 Report

Comments and Suggestions for Authors

This paper proposes a novel method that combines connectivity metrics and max-min modularity for community detection in complex networks. By leveraging both community quality functions, the authors developed a robust algorithm and demonstrated its effectiveness and time efficiency through extensive experiments. The method, which merges connectivity metrics and max-min modularity, shows certain innovativeness and compensates for the limitations of using a single approach. Detailed theoretical proofs provided by the authors ensure the validity of the proposed method, which is also empirically validated through a wide range of experiments, including comparisons with several existing advanced methods. However, the following modifications are suggested:

1. Most importantly, Figure 4 appears to be generated by ChatGPT, which significantly undermines the credibility of this study. It is strongly recommended that the authors make modifications.

2. The paper is very detailed in its descriptions, but due to the lack of a clear framework, it only broadly informs the reader about the integration of multiple algorithms. It is suggested that the authors provide a figure to introduce an overarching research framework for the paper.

3. The authors compare their work with recent studies in the experimental section, using classic algorithms like simulated annealing for comparison. If time permits, it is recommended that the authors further refine the comparisons of these algorithms.

Author Response

Dear Reviewer,

First, I would like to express my appreciation for your valuable feedback, which led to significant improvements in the paper. Your reviews were instrumental in enhancing some overlooked aspects of the work. I have incorporated most of your comments into the revised paper, and below, I address each of your questions individually.

1. Most importantly, Figure 4 appears to be generated by ChatGPT, which significantly undermines the credibility of this study. It is strongly recommended that the authors make modifications.
We had the adjacency matrix for this graph, and our script generated the community data. However, as you correctly pointed out, visualizing such a large and complex graph was challenging, so we sought assistance from an AI for visualization. We agree that including this AI-generated visualization might not have been ideal for readers, so we decided to remove it from the paper.

2. The paper is very detailed in its descriptions, but due to the lack of a clear framework, it only broadly informs the reader about the integration of multiple algorithms. It is suggested that the authors provide a figure to introduce an overarching research framework for the paper.
This comment was constructive in revising our introduction, particularly the Main Contribution section, as well as the conclusion. It also helped clarify many points in Section 2, where we introduce the algorithms. We believe these changes have resulted in a more coherent message and a clearer presentation of the research framework.

3. The authors compare their work with recent studies in the experimental section, using classic algorithms like simulated annealing for comparison. If time permits, it is recommended that the authors further refine the comparisons of these algorithms.
You are correct.
It is worth mentioning that we also used another metric, the Adjusted Rand Index (ARI), in our research, which yielded results similar to those of the NMI. Our algorithm proved superior to others, with consistent performance when using the ARI metric. To avoid making the paper too lengthy and tedious, we decided to omit these results. However, we have now added a sentence to clarify this issue.
Additionally, we have already compared our algorithm to the baseline models in references [15,16], which have demonstrated superiority over many other algorithms. Therefore, we felt it sufficient to state that our model outperforms these baseline models, and this conclusion is clearly supported by the evidence presented in the paper.

Once again, thank you very much for your valuable feedback.
With warm regards,
Arman Ferdowsi

Reviewer 3 Report

Comments and Suggestions for Authors

The paper is well-organized, and the authors present a novel approach to community detection in complex networks by combining connectivity-based metrics with Max-Min Modularity. The methodology is clear, and the hybrid approach provides an interesting approach. However, I have some specific concerns/questions that I would like to address.

1. How does the combination of connectivity-based metrics with Max-Min Modularity contribute to our novel approach to community detection in complex networks?
2. The authors should explain the role the heuristic algorithm plays in shaping the complementary graph discussed in Section 2.1.
3. Can authors outline the specific improvements that the proposed sparse integer programming model (IPs-MM) offers over the previous (IP-RMM) model, as explained in Section 2.2?
4. How were the parameters 𝑘1k1​ and 𝑘2k2​ determined for each network in the evaluation of real-world networks depicted in Figure 1? I think it is better to provide a table for that.
5. Given the significant drop in NMI for the proposed approach at higher noise levels (𝜇=0.9μ=0.9) in the LFR experiments shown in Figure 2, what strategies could be employed to maintain performance at these noise levels?
6. What is the purpose of the visual representation of the LFR artificial network and communities discovered by their algorithm in Figure 4?

The manuscript generally does not have any significant challenges, and it can be accepted after minor revision.

Author Response

Dear Reviewer,
First, I would like to express my appreciation for your valuable feedback, which led to significant improvements in the paper. Your reviews were instrumental in enhancing some overlooked aspects of the work. I have incorporated most of your comments into the revised paper, and below, I address each of your questions individually.

1) How does the combination of connectivity-based metrics with Max-Min Modularity contribute to our novel approach to community detection in complex networks?
First, I would like to mention that we have clarified this issue in the paper, particularly in the Introduction. The connectivity-based metric we used was beneficial in two ways. Specifically, emphasizing this metric and using a heuristic algorithm to optimize it allowed us to systematically improve the complemented graph corresponding to the Max-Min Modularity metric and develop a high-quality initial solution for the integer formulation of our revised Max-Min Modularity maximization problem. Thus, the impact of this metric is both inevitable and significant.

2)The authors should explain the role the heuristic algorithm plays in shaping the complementary graph discussed in Section 2.1.
Thank you for this relevant comment. We have added a paragraph at the beginning of Section 2.1 to explain this.

3) Can authors outline the specific improvements that the proposed sparse integer programming model (IPs-MM) offers over the previous (IP-RMM) model, as explained in Section 2.2?
Yes, this is indeed very important, and Fig. 3 illustrates this by comparing the computational time required for solving IPs-MM and IP-RMM. We have proven that the optimal solution for both models is the same, with IPs-MM being a sparse counterpart of IP-RMM. Therefore, the critical factor in comparing them is their time complexity.

4) How were the parameters ?1k1​ and ?2k2​ determined for each network in the evaluation of real-world networks depicted in Figure 1? I think it is better to provide a table for that.
Table 3 on page 14 addresses this issue. Specifically, k_1 represents the number of times the second phase of the heuristic algorithm is repeated. A higher k_1 value means more refinements within the initial communication obtained from the first phase. Since we have already demonstrated that the first phase alone can achieve reasonably high-quality communities, we can safely reduce k_2 manually to decrease computational complexity. This is crucial when dealing with large networks, where a good initial solution may be more important than a precise result. The same applies to k_2, which is the number of constraints that must be satisfied for optimally solving IPs-MM.

5) Given the significant drop in NMI for the proposed approach at higher noise levels (?=0.9μ=0.9) in the LFR experiments shown in Figure 2, what strategies could be employed to maintain performance at these noise levels?
We should note two things here. First, despite the NMI decreasing significantly when the noise level is high, our proposed algorithm still maintains a higher NMI value, demonstrating its superiority over other methods. On the other hand, it is natural to see this decrease in NMI with high noise levels because the community structures of a network with ?=0.9 are very chaotic and unrealistic.
Nevertheless, as you correctly pointed out, several strategies to improve this inefficiency might be possible, and this can be part of future work. For instance, using a different quality metric (so far, we have not found one that works better) or a revised heuristic approach for forming initial communities could help us form better communities.

6) What is the purpose of the visual representation of the LFR artificial network and communities discovered by their algorithm in Figure 4?
We had the adjacency matrix for this graph, and our script generated the community data. However, as you correctly pointed out, visualizing such a large and complex graph was challenging, so we sought assistance from an AI for visualization. We agree that including this AI-generated visualization might not have been ideal for readers, so we decided to remove it from the paper.

Once again, thank you very much for your valuable feedback.
With warm regards,
Arman Ferdowsi

Reviewer 4 Report

Comments and Suggestions for Authors

The authors present a new approach for community detection in complex networks, which combines some previous methods and proves to be relatively effective. Along with the theoretical derivation, the authors also test the algorithm on empirical data and compare the method with existing approaches. The article is written in a comprehensible manner. Although the method does not represent a major breakthrough in this field, it is novel and improves on existing methods in some aspects. This study is likely to attract the interest of researchers in this field. For these reasons, I support the acceptance of this article, but I suggest that the authors improve several aspects before final publication:

The NMI value for larger networks is rather low, around 2/3. This seems surprisingly low to me, and I am curious as to why this is the case. I believe it would be beneficial to evaluate the similarity of the algorithm's performance using an additional independent metric.

It would also improve the paper if the figure and table captions were made more self-contained. The current legends for most figures are rather brief, and it would be helpful to add more details, such as the main message and what exactly is being shown.

There are still some typographical errors and misplaced commas in the text (e.g., around in-text citations).

The authors could more explicitly highlight in either the introduction or the concluding chapter the significance of communities in real (and diverse) complex networks and why their identification is important.

Author Response

Dear Reviewer,
First, I would like to express my appreciation for your valuable feedback, which led to significant improvements in the paper. Your reviews were instrumental in enhancing some overlooked aspects of the work. I have incorporated most of your comments into the revised paper, and below, I address each of your questions individually.

1) The NMI value for larger networks is rather low, around 2/3. This seems surprisingly low to me, and I am curious as to why this is the case. I believe it would be beneficial to evaluate the similarity of the algorithm's performance using an additional independent metric.
Thanks for your notice.
It is worth mentioning that we also used another metric, the Adjusted Rand Index (ARI), in our research, which yielded results similar to those of the NMI. Our algorithm proved superior to others, with consistent performance when using the ARI metric. To avoid making the paper too lengthy and tedious, we decided to omit these results. However, we have now added a sentence to clarify this issue.
More importantly, I would like to make two comments here. First, despite the NMI decreasing significantly when the noise level is high, our proposed algorithm still maintains a higher NMI value, demonstrating its superiority over other methods. On the other hand, it is natural to see this decrease in NMI in large networks or when we encounter high noise levels because the community structures of such networks are very chaotic.
Nevertheless, as you correctly pointed out, several strategies to improve this inefficiency might be possible, and this can be part of future work. For instance, using a different quality metric (so far, we have not found one that works better) or a revised heuristic approach for forming initial communities could help us form better communities.

2) It would also improve the paper if the figure and table captions were made more self-contained. The current legends for most figures are rather brief, and it would be helpful to add more details, such as the main message and what exactly is being shown.
I very much appreciate the comment. We have already tried to use more elaborate legends and captions.

3) There are still some typographical errors and misplaced commas in the text (e.g., around in-text citations).
Once again, thank you. We reviewed the paper once again and tried to address these errors as much as possible.

4) The authors could more explicitly highlight in either the introduction or the concluding chapter the significance of communities in real (and diverse) complex networks and why their identification is important.
This was indeed very helpful. We have updated both the Introduction and Conclusion and tried to make them clearer and more coherent in explaining the importance of community detection.

Once again, thank you very much for your valuable feedback.
With warm regards,
Arman Ferdowsi