PCSY-D-25-00031

Probability-turbulence divergence: A tunable allotaxonometric instrument for comparing heavy-tailed categorical distributions

PLOS Complex Systems

Dear Dr. Dodds,

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 Jul 14 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,

Aming Li, Ph.D.

Academic Editor

PLOS Complex Systems

Hocine Cherifi

Editor-in-Chief

PLOS Complex Systems

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Additional Editor Comments (if provided):

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Reviewers' Comments:

Reviewer's Responses to Questions

Comments to the Author

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

Reviewer #2: No

Reviewer #3: Yes

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2. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #1: No

Reviewer #2: I don't know

Reviewer #3: Yes

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3. Have the authors made all data underlying the findings in their manuscript fully available (please refer to the Data Availability Statement at the start of the manuscript PDF file)?

The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception. The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #1: No

Reviewer #2: Yes

Reviewer #3: Yes

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4. 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: No

Reviewer #3: 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: This paper introduces a new tool called PTD to compare different sets of data, like word frequencies in books or social media posts. The idea is interesting but needs big improvements. The good parts: PTD lets users adjust how much it focuses on rare vs. common items. The examples from books and Twitter show it can be useful. The graphs make it easier to see differences between datasets.

1. The biggest problems: The paper is hard to understand. Words like "allotaxonometry" need simpler explanations. The authors should add a short summary in plain language at the start.

2. The paper claims PTD is better than old methods but doesn’t prove it. They must compare PTD to at least 2-3 standard tools using real and test data.

3. The method has a tuning number (α) but doesn’t explain how to choose it. Picking 3/4 or 5/12 seems random. They should give clear rules, like "use 0.5 for rare items, 1 for common ones."

4. Some math parts don’t match the text, especially when α gets very small. The authors must check all equations and explain limits better.

5. PTD might be too slow for huge datasets like millions of tweets. They should discuss ways to speed it up, like using samples.

6. Small fixes: Make graph labels clearer (always mark log10 scales). Fix typos (like writing *hapax legomena* in italics).

Reviewer #2: In this manuscript, the authors introduce and explore the probability-turbulence divergence, a tunable, interpretable instrument. As the probability-based analogy of the rand-turbulence divergence, the instrument is able to perform well when comparing heavy-tailed size-rank distributions of type frequencies. After reading this work, I have found the topic of the study interesting. However, I do not see the innovation of the work, although I am not an expert in this field. It is not very clear to me why such instrument is able to perform well for comparing heavy-tailed categorical distributions. In addition, the authors need to format the manuscript using the template of the journal since they choose to submit the work to this journal, which denotes the attitude towards this journal and its editors and/or reviewers. Furthermore, I find that the abstract is very complicated and there are too many key words. From these parts, it is hard to find the important results of the work.

Reviewer #3: This manuscript presents an innovative extension of allotaxonometry through the introduction of probability-turbulence divergence (PTD), a tunable and interpretable tool designed for comparing heavy-tailed categorical distributions. The work is motivated by the need for more adaptable divergence measures that can accommodate the complex, often turbulent structures found in real-world probability distributions, such as those observed in natural language use, online communication, and ecological systems.

The paper is clearly written and well-organized. It carefully builds upon prior work on rank-turbulence divergence (RTD), providing a concise yet sufficient formal introduction to PTD. The authors' decision to anchor the development of PTD in familiar concepts like the Sørensen-Dice coefficient and F1 score is effective, making the technical ideas more accessible to a broad audience. I appreciate the inclusion of concrete examples across diverse domains (n-grams from literature and social media, as well as ecological data), which convincingly illustrate the utility and flexibility of the proposed method.

One of the notable strengths of the paper is its thoughtful treatment of visualization through allotaxonographs. The emphasis on making differences interpretable, rather than merely quantifiable, is an important and often underappreciated contribution, especially for applications in exploratory data analysis. The new addition for types with probability zero in one system is clear and useful.The accompanying code, online resources, and flipbooks further enhance the reproducibility and practical accessibility of the work.

That said, a few areas could benefit from additional clarification:

While the analogy to RTD is helpful, readers unfamiliar with Ref. [1] may find some arguments terse. Providing a slightly expanded, self-contained summary of RTD’s motivation—particularly the key points from Section 2.4 of Ref. [1]—would help readers appreciate the probability-based extension without needing to cross-reference extensively.

The behavior of PTD in the α → 0 and α → ∞ limits is intriguing, and the connections to classical similarity measures, particularly the Sørensen-Dice index and F1 score, are well-motivated. I appreciated these links, which help ground the method within familiar frameworks. However, it would be helpful to contrast PTD more explicitly with traditional Lp divergence measures. For instance, in some n-gram comparisons, α = 1 (or the equivalent Lp-rank) may not meaningfully highlight salient differences, especially when rare but important elements are overwhelmed by frequency differences among common types. An example illustrating how PTD better captures meaningful divergence compared to a standard Lp divergence in such cases would strengthen the practical motivation for the method.

It would further strengthen the impact of the work to explore in more detail how varying α offers practical advantages depending on distributional characteristics (e.g., sparsity, extreme skew). In particular, it would be valuable to discuss whether certain features of the compared distributions could be minimized to suggest an optimal α value—perhaps analogous to heuristic methods like the elbow method in k-means clustering. Such a discussion would also help elucidate the differences between PTD and other divergences, especially since α = 1 need not always be the best choice. While the tunability of α is ultimately a strength, further guidance for practitioners on selecting or interpreting α would be valuable. Without principled defaults or heuristics, there is some risk of subjective tuning, especially in exploratory data analysis settings.

Overall, this paper is a strong and meaningful contribution. It provides both theoretical grounding and practical tools that should be of interest across disciplines that engage with complex systems, language modeling, ecology, and social data analysis. I recommend it for publication after addressing the minor clarifications suggested above.

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Reviewer #1: Yes: Ebrahim E. Elsayed

Reviewer #2: No

Reviewer #3: Yes: Alexander Gates

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Probability-turbulence divergence: A tunable allotaxonometric instrument for comparing heavy-tailed type-probability distributions

PCSY-D-25-00031R1

Dear Dr. Dodds,

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.

An invoice for payment will follow shortly after the formal acceptance. To ensure an efficient process, please log into Editorial Manager at https://www.editorialmanager.com/pcsy/ click the 'Update My Information' link at the top of the page, and double check that your user information is up-to-date. For questions related to billing, please contact billing support at https://plos.my.site.com/s/.

If your institution or institutions have a press office, please notify them about your upcoming paper to help maximize its impact. If they'll be preparing press materials, please inform our press team as soon as possible -- no later than 48 hours after receiving the formal acceptance. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information, please contact complexsystems@plos.org.

Kind regards,

Réka Albert

Section Editor

PLOS Complex Systems

Additional Editor Comments (optional):

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 #3: 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: (No Response)

Reviewer #3: Yes

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

Reviewer #1: (No Response)

Reviewer #3: Yes

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4. Have the authors made all data underlying the findings in their manuscript fully available (please refer to the Data Availability Statement at the start of the manuscript PDF file)?

The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception. The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #1: (No Response)

Reviewer #3: Yes

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5. Is the manuscript presented in an intelligible fashion and written in standard English?<br/><br/>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: (No Response)

Reviewer #3: Yes

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6. Review Comments to the Author<br/><br/>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 paper has been suitably revised. The authors have revised the manuscript accordingly and consequently improved the quality of the work. The revised version may be accepted for publication.

Reviewer #3: The authors have addressed all of my concerns and substantially improved the manuscript. Therefore, I recommend for publication.

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7. PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files.

Do you want your identity to be public for this peer review?  If you choose “no”, your identity will remain anonymous but your review may still be made public.

For information about this choice, including consent withdrawal, please see our Privacy Policy.

Reviewer #1: Yes: Ebrahim E. Elsayed

Reviewer #3: Yes: Alexander Gates

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