PCSY-D-25-00046

Persistence Partitions of Real and Synthetic Networks

PLOS Complex Systems

Dear Dr. Webb,

Thank you for submitting your manuscript to PLOS Complex Systems. After careful consideration, we feel that it has merit but does not fully meet PLOS Complex Systems's publication criteria as it currently stands. Therefore, we invite you to submit a revised version of the manuscript that addresses the points raised during the review process.

Please submit your revised manuscript within 60 days Oct 24 2025 11:59PM. If you will need more time than this to complete your revisions, please reply to this message or contact the journal office at complexsystems@plos.org. When you're ready to submit your revision, log on to https://www.editorialmanager.com/pcsy/ and select the 'Submissions Needing Revision' folder to locate your manuscript file.

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We look forward to receiving your revised manuscript.

Kind regards,

Hector Zenil, Ph.D.

Academic Editor

PLOS Complex Systems

Hector Zenil

Academic Editor

PLOS Complex Systems

Hocine Cherifi

Editor-in-Chief

PLOS Complex Systems

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Reviewer 1 Comments:

I found this to be an engaging and ambitious manuscript that brings together persistent homology and network partitioning in a novel way. The introduction of the concept of persistence surfaces and their use to generate node partitions is both interesting and potentially impactful. I particularly appreciated the authors’ clear exposition in Section 2, where they take care to build the construction step by step and motivate the idea of persistence partitions through worked examples.

That said, there are some important issues that I believe need to be addressed before I could recommend this paper for publication.

1. Comparisons with State-of-the-Art Partitioning Methods

My main concern is the set of baseline methods against which the proposed approach is compared. The authors devote considerable effort to comparisons with classical assortative community detection methods and stochastic block models. While this is useful for grounding the results, the choice of comparators is somewhat dated and risks underselling the significance of the work.

In particular, the authors should compare persistence partitions against more modern node-partitioning methods that explicitly account for multiple structural scales and roles. A relevant benchmark is this framework published in Nature Machine Intelligence (doi: 10.1038/s42256-018-0005-0), which provides a principled role-based embedding and partitioning approach. This seems much closer in spirit to what persistence partitions are capturing than traditional modularity maximization or stochastic block models.

Without situating their results relative to such state-of-the-art techniques, it is difficult to assess whether persistence partitions are offering qualitatively new insights, or simply rediscovering structures that modern role-based algorithms already capture.

2. Positioning with Respect to Roles vs. Communities

As the authors themselves note, persistence partitions are not intuitively linked to assortative “communities.” Rather, they seem more naturally aligned with role discovery, core-periphery structure, or functional equivalence. Yet Section 3 places disproportionate emphasis on comparisons with modularity-based community detection. I recommend re-ordering the results so that the most distinctive contributions—such as the symmetry results in Section 3.3---come first, and that role-based or function-oriented comparisons are given more prominence.

In this regard, in addition to the Nature Machine Intelligence paper above, I encourage the authors to engage with the recent social science literature on role detection methods.

3. Technical Clarifications

Uniqueness claim (Section 2): The statement that “the resulting persistence partition is unique” requires qualification. It is only unique conditional on the chosen distance metric. Since multiple node-to-node metrics are possible (beyond shortest-path geodesic distance), the uniqueness claim should be softened or rephrased.

Computational complexity: the complexity of the PSA algorithm is given. The mitigation strategy (restricting ego-graph radius) is mentioned, but not quantified.

Synthetic experiments: Using a non-degree-corrected SBM mixes degree heterogeneity with community structure. A degree-corrected SBM baseline would provide a cleaner point of comparison for what persistence partitions are, and are not, capturing.

In summary, this is a creative and promising manuscript. The technical development is solid, and the exposition is strong. However, to reach the standard of PLOS Complex Systems, the authors should (i) broaden their experimental comparisons to include state-of-the-art role-based partitioning methods (particularly the Nature Machine Intelligence reference above), (ii) clarify technical claims about uniqueness and complexity, and (iii) re-order and reframe their results to emphasize what persistence partitions uniquely contribute.

With these revisions, I believe the paper could make a valuable addition to the literature at the interface of persistent homology and network science.

Please, also take into consideration the valuable suggestions by the other reviewer and address their concerns.

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Reviewer 2 Comments:

Reviewer's Responses to Questions

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1. Does this manuscript meet PLOS Complex Systems’s publication criteria? Is the manuscript technically sound, and do the data support the conclusions? The manuscript must describe methodologically and ethically rigorous research with conclusions that are appropriately drawn based on the data presented.-->

Reviewer #1: Partly

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Reviewer #1: N/A

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Reviewer #1: Yes

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Reviewer #1: Yes

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-->5. Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)-->

Reviewer #1: I greatly enjoyed reading this paper and generally found it to be a very nice bringing together of the previously seemingly disparate threads of community detection in networks and of persistent homology. I particularly appreciated the careful way that the authors describe their approach to persistence partitions throughout Section 2. However, I think there are some important shortcomings in Section 3 that would need to be addressed in a revision before I would recommend this paper for publication.

My primary constructive criticism of the paper is that I find the emphasis on discussing and comparing with traditional assortative community detection methods to be completely misplaced. The method you propose for partitioning nodes wouldn’t seem to intuitively have anything to do with assortative communities. You acknowledge this but seem to devote a lot of text and emphasis in the paper to this, when you also say that there are other ways of partitioning nodes. Indeed, you reference 3 different methods of partitioning nodes according to “roles” which would seem to be much more intuitively related to what persistent partitions are all about, and yet you do not compare with any of those methods for assessing node roles. Why not? I think some kind of comparison is warranted.

You are also missing any reference to the discussion of roles that has been in the social science literature for decades, albeit many of those methods have some key computational issues. In addition to further discussing and comparing with the 3 references about roles you already have, you could access and compare with some of the more computationally efficient methods that have recently appeared in the social science literature by looking at chapter 10 of Rawlings CM, Smith JA, Moody J, McFarland DA. Network Analysis: Integrating Social Network Theory, Method, and Application with R. Cambridge University Press; 2023.

Similarly, when you get to the second paragraph of the Discussion, I think your emphasis is misplaced: I don’t think there’s any reason at all that persistence communities should have anything to do with assortative communities, except that you partition nodes into groups and use the word “communities”. You are instead much more clearly related to the literature about roles, as you allude to in lines 505-506, but by failing to compare with any of the literature about roles, I think you’ve missed an essential opportunity here.

Continuing in this theme, I believe the paper would be stronger if you used chapter 3 to first dive into your main results about persistent partitions that are in section 3.3 and only then later discuss the experiments from 3.1 and 3.2 that largely indicates what persistent partitions are not. You could then also add role calculations for comparison to the later parts of such a re-ordered chapter 3. As it is written now, I found it very confusing as a reader to have you devote so much early space to comparison with assortative communities when there was no intuitive reason why I would expect your method to have anything to match them.

Additionally, I take some issue with the claim that “The resulting persistence partition is unique”, as this is only true after the selection of the distance function used, which is itself not unique. There are different distance metrics between nodes one might use on a network besides geodesic distances. I think you should be more careful with this claim that it only applies after the selection of a distance, but that it may be reasonable to consider a different distance metric.

Given that the whole procedure follows from having a distance metric, I would also recommend that you simply say so in lines 69-72 where you currently mention that the work “can be applied to weighted, directed, and time-dependent networks in similar fashion.” Being straightforward about how it follows from picking a metric would I think make this point much clearer.

Near the end of Section 2, you quote a complexity on line 285 but then seem to say in the next sentence that this very large complexity can be mitigated in large networks as it can be “approximated by limiting the size of each vertex’s ego-graph”. I think it is important for you to explicitly discuss the complexity for the approximate calculation and tell the reader how it scales with the selected radius. In particular, does this only change the factor of d, or does it also change the Ripser cost in some predictable way if you rewrite it in terms of the numbers of nodes and edges in the largest ego-graphs considered? Please be explicit.

As a side thought about your synthetic experiment, comparing with a non-degree-corrected SBM mixes node roles and assortative communities. I believe you will get clearer results about what your method is and is not If you compared with a degree-corrected SBM with a reasonably heterogeneous degree distribution.

Again, I cannot stress enough that I very much appreciated your careful walking through examples in Section 2. I think it’s important in bringing together these different threads to have a carefully constructed section like this that is easily accessible to readers from both backgrounds.

Finally, the following small items appear to be typos, grammatical quirks, or perhaps my minor misunderstandings:

In the author summary, “As it is possibly to equate” seems awkward.

On lines 83-84, I don’t understand why you define E_\delta to be a “set of 1-simplices” appear to do everything or almost everything after that in terms of G_\delta. Does E_\delta serve some special purpose here?

On line 176, “president”.

When you introduce Zachary’s karate club, you should mention you are using the unweighted version.

I also find it confusing to say “forming two communities of maximum modularity” (lines 345-346) and “The network’s two communities of maximum modularity” (Fig. 6 caption) when the maximum modularity partition of this network has four communities. It would be correct to instead reference this partition as that having maximum modularity among all 2-community partitions.

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Reviewer #1: No

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PCSY-D-25-00046R1

Persistence Partitions of Real and Synthetic Networks

PLOS Complex Systems

Dear Dr. Webb,

Thank you for submitting your manuscript to PLOS Complex Systems. After careful consideration, we feel that it has merit but does not fully meet PLOS Complex Systems's publication criteria as it currently stands. Therefore, we invite you to submit a revised version of the manuscript that addresses the points raised during the review process.

Please submit your revised manuscript by May 07 2026 11:59PM. If you will need more time than this to complete your revisions, please reply to this message or contact the journal office at complexsystems@plos.org. When you're ready to submit your revision, log on to https://www.editorialmanager.com/pcsy/ and select the 'Submissions Needing Revision' folder to locate your manuscript file.

Please include the following items when submitting your revised manuscript:

* A letter that responds to each point raised by the editor and reviewer(s). You should upload this letter as a separate file labeled 'Response to Reviewers'. This file does not need to include responses to any formatting updates and technical items listed in the 'Journal Requirements' section below.

* A marked-up copy of your manuscript that highlights changes made to the original version. You should upload this as a separate file labeled 'Revised Manuscript with Track Changes'.

* An unmarked version of your revised paper without tracked changes. You should upload this as a separate file labeled 'Manuscript'.

If you would like to make changes to your financial disclosure, competing interests statement, or data availability statement, please make these updates within the submission form at the time of resubmission. Guidelines for resubmitting your figure files are available below the reviewer comments at the end of this letter.

We look forward to receiving your revised manuscript.

Kind regards,

Hector Zenil

Academic Editor

PLOS Complex Systems

Marc Hütt

Section Editor

PLOS Complex Systems

Hocine Cherifi

Editor-in-Chief

PLOS Complex Systems

Journal Requirements:

If the reviewer comments include a recommendation to cite specific previously published works, please review and evaluate these publications to determine whether they are relevant and should be cited. There is no requirement to cite these works unless the editor has indicated otherwise.

Additional Editor Comments (if provided):

Dear authors,

Please, take care of the latest round of revision and submit a revision. In particular, please re-assess the substantial number of self-citations one of the reviewers has flagged.

Kind regards

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Reviewers' Comments:

Reviewer's Responses to Questions

-->Comments to the Author

1. If the authors have adequately addressed your comments raised in a previous round of review and you feel that this manuscript is now acceptable for publication, you may indicate that here to bypass the “Comments to the Author” section, enter your conflict of interest statement in the “Confidential to Editor” section, and submit your "Accept" recommendation.-->

Reviewer #1: (No Response)

Reviewer #2: All comments have been addressed

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-->2. Does this manuscript meet PLOS Complex Systems's publication criteria? Is the manuscript technically sound, and do the data support the conclusions? The manuscript must describe methodologically and ethically rigorous research with conclusions that are appropriately drawn based on the data presented.-->

Reviewer #1: Yes

Reviewer #2: Yes

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-->3. Has the statistical analysis been performed appropriately and rigorously?-->

Reviewer #1: Yes

Reviewer #2: Yes

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Reviewer #1: Yes

Reviewer #2: Yes

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-->5. Is the manuscript presented in an intelligible fashion and written in standard English?

PLOS Complex Systems does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.-->

Reviewer #1: Yes

Reviewer #2: Yes

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-->6. Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)-->

Reviewer #1:  The authors have addressed all of my concerns and I recommend publication.

One appeal I will make to the authors is to ask them whether they could not possibly find some additional non-self citations to add to the article with which to make their points. Currently, it appears that (at least) 16 of the references are to their own work.

I also note there are a few typographical issues that I presume will be adjusted during the publication process. I do not promise I caught them all, but I noticed the following:

On line 313, I believe it would be more correct to say "core-periphery".

On line 398, there is a missing reference.

In Figure 7, the labels should be "DCSBM" and "SBM", and the last word of the caption before the parentheses should be "network".

On line 531, I believe it should be "recovers".

Reviewer #2:  I have reviewed the revised manuscript "Persistence Partitions of Real and

Synthetic Networks" by Jenkins, Callor, Boyd, Gledhill, Wonnacott, and Webb.

This is a very interesting work that combines persistent homology and node

partitioning by introducing the concept of persistence surfaces and the

associated persistence partition of a network. Through this construction, the

authors develop a non-parametric homology-based framework for grouping nodes

according to how they see the network's cycles and higher-dimensional

features, and they examine how the resulting partitions relate to network

roles, communities, and core-periphery organization.

I have also noted that this manuscript has already been through a first round

of revision, and I have thus studied the comments and responses from the other

reviewers. In my view, the authors have effectively responded to the main

concerns raised in the previous round, and these revisions have significantly

strengthened the paper and improved both its positioning and presentation.

I found the manuscript scientifically sound, conceptually original, and highly

relevant to readers interested in complex systems, topology-informed data

analysis, and network structure. In particular, I believe the paper makes a

valuable contribution by showing how persistent homology can induce

informative node-level equivalence classes that are not reducible to more

conventional partitioning approaches.

Overall, I greatly enjoyed reading this manuscript. I believe it is a fine

piece of work, carefully developed and clearly written, and I am happy to

recommend publication in its present form. I congratulate the authors on this

elegant and thoughtful contribution.

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Reviewer #1: No

Reviewer #2: No

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Persistence Partitions of Real and Synthetic Networks

PCSY-D-25-00046R2

Dear Dr. Webb,

We're pleased to inform you that your manuscript has been judged scientifically suitable for publication and will be formally accepted for publication once it meets all outstanding technical requirements.

Within one week, you'll receive an e-mail detailing the required amendments. When these have been addressed, you'll receive a formal acceptance letter and your manuscript will be scheduled for publication.

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Kind regards,

Marc Hütt

Section Editor

PLOS Complex Systems

Additional Editor Comments (optional):

Reviewers' comments: