PCSY-D-24-00032

Statistical Complexity of Heterogeneous Geometric Networks

PLOS Complex Systems

Dear Dr. Keith Smith,

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.

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Carlo Vittorio Cannistraci

Section Editor

PLOS Complex Systems

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Additional Editor Comments (if provided):

I want to apologize with the Authors for the prolonged time of the review process. The article is quite technical, and we encountered difficulty to secure the adequate number of reviewers. At the end of the Review process we received suggestions that include acceptance, major revision and rejection. For this reason, I had to read myself the article and provide my review to help the Authors to improve the manuscript.

The reason that I decided to provide a review is that I really like the topic of the proposed study. I think that it is of high interest for the journal and I want to help the Authors to improve the quality with my own comments. You will notice that in my list of comments below I will append different references many of which are mine. I apologize in advance for this excessive self-referencing, This will happen because it is easier for me to refer to study that I wrote, however I encourage the authors to consider other articles that are related to the topic and to include them in the study.

Point by point review

+ The author should include and offer quantitative comparisons to previous measures of statistical complexity (for instance: network diversity score or entropy based) explaining extensively the similarity and differences and the pro- and cons-. This means I expect

+ The fact that lower density correlates with higher NHC in artificial models is an important result and should be stressed more in the discussion, because it indicates that sparsity triggers or is associated to complexity in network structures. The fact that the result is not reproducible in real networks is an interesting point and can be further investigated by reducing the density with random removal.

+ the use of the term “attachment mechanisms” for links growth is very confusing. As a matter of fact, it seems to me that the Authors perform neighbored-based topological link prediction that is a terminology more frequently adopted in the field to refer to predicting new edges to add to the network. See for instance these articles of mine:

From link-prediction in brain connectomes and protein interactomes to the local-community-paradigm in complex networks

CV Cannistraci, G Alanis-Lobato, T Ravasi

Scientific reports 3 (1), 1613

Local-community network automata modelling based on length-three-paths for prediction of complex network structures in protein interactomes, food webs and more

A Muscoloni, I Abdelhamid, CV Cannistraci

BioRxiv, 346916

+ Attachment mechanisms line 395: The Combined hierarchical and similarity attachment: isn’t simply a common neighbor mechanism for link prediction? Refer always to the same articles of the point before this one.

+ Data 425 and Discsussion 740: Co-expression networks are not proper physical protein interaction networks and should not be confused with protein networks. See references above and this one:

Network-based prediction of protein interactions

IA Kovács, K Luck, K Spirohn, Y Wang, C Pollis, S Schlabach, W Bian, ...

Nature communications 10 (1), 1240

+ Consider the comments of Reviewer 1 and 2 I suggest to the Authors should include in the study tests on nonuniform popularity-similarity hyperbolic model with tailored community structure and to comment how the proposed normalized hierarchical complexity varies with:

# power law exponent: 2, 2.5, 3. This controls network heterogeneity in a scale free network.

# network size N growing from 10, 100, 1000, etc.

# density by modifying the parameter m which is half of the average node degree

# temperature that controls clustering: T= 0, 0.1, 0.3, 0.5, 0.7, 0.9. The higher the temperature the lower the clustering and the more the network tends to take an ER model structure.

# community structure: pay attention C=0 does not mean absence of community but it means absence of constraint on community structure. Hence, I suggest, C=2,4,6,8,10.

This last point, to study how the NHC (and the other previous proposed measures of statistical complexity) reacts to changes in the mesoscale community structure, is of particular relevance for your study that aims to quantify emergence of complexity.

Reference studies: DOI 10.1088/1367-2630/aac06f and DOI 10.1088/1367-2630/aac6f9

+ Add the letters to panels in the figures to simplify their discussion in the text.

+ Pay attention sometimes you write HC when instead you refer to NHC. Use consistently in the text NHC.

+ Fig. 2 In the plot with x-axis network node size, do you fix the density to a value or do you plot points considering any density value considered in the analysis? A similar question applies to the plot with x-axis density, what values are considered for the size?

+ Fig.2 and Table II, why the values of the normalized hierarchical complexity (NHC) are so low? They are below 0.1. As a matter of fact, NHC measure still presents some interpretability problems. I would expect that you show some examples in artificial and real networks that converge to the upper-bound.

+ add to the discussion a careful part on the limitations of the study and the proposed measure. For instance, that the NHC measure is not really well scaled and bounded between 0 and 1. In many experiments it is lower than 0.1. It is still missing the ability to quantify a condition of maximal complexity.

+ I am not convinced by the results and interpretation in Fig. 4 (bottom panel). I would expect that the Normalized Hierarchical complexity either grow or form a plateau with the growth of the degree heterogeneity in RHGG. The authors comment that: << HC is generally strongest in models with σh between 0.2 and 0.4 and decreases towards low and high heterogeneity. When σh is high the network becomes dominated by the hierarchical relationships, which should make the network more ordered. >> Looking at the plot I have the doubt that the results are not complete enough to provide a strong statement such as this one. It seems that between 0.2 and 0.4 there are more samples, but between 0.6 and 1 there are points that attains NHC close to 0.1. There might be some problems with the sampling or other inconsistency in the test.

+ Discussion line 698 the Authors write: “assess the advantages conferred by the combination of hierarchy and geometry which real-world networks appear to incorporate almost universally”. I disagree on this universality, see the discussion on the fact that networks can have LCP or not LCP organization associated with Fig. 5 of this study of mine.

From link-prediction in brain connectomes and protein interactomes to the local-community-paradigm in complex networks

CV Cannistraci, G Alanis-Lobato, T Ravasi

Scientific reports 3 (1), 1613

I invite the author to adjust the sentence accordingly. They are free to refer at any article discuss the topic.

+ Discussion, line 711: “On the other hand, online social networks of twitch are largely free from geometrical constraints”. It is not clear to me this statement about the absence of geometrical constraints in social networks. To the best of my knowledge, social networks have a similarity constraint associated with many factors such as common neighbor and local community mechanisms. See for reference:

Network geometry

Marián Boguñá, Ivan Bonamassa, Manlio De Domenico, Shlomo Havlin, Dmitri Krioukov & M. Ángeles Serrano

Nature Reviews Physics volume 3, pages114–135 (2021)

Machine learning meets complex networks via coalescent embedding in the hyperbolic space. A Muscoloni, JM Thomas, S Ciucci, G Bianconi, CV Cannistraci

Nature Communications 8 (1), 1 [In this study refer to Fig.7]

From link-prediction in brain connectomes and protein interactomes to the local-community-paradigm in complex networks

CV Cannistraci, G Alanis-Lobato, T Ravasi

Scientific reports 3 (1), 1613

Local-community network automata modelling based on length-three-paths for prediction of complex network structures in protein interactomes, food webs and more

A Muscoloni, I Abdelhamid, CV Cannistraci

BioRxiv, 346916

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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: Yes

Reviewer #2: Yes

Reviewer #3: No

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

Reviewer #1: Yes

Reviewer #2: Yes

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: Yes

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: Yes

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: The manuscript proposes a new complexity measure for heterogeneous geometric networks.

I did not understand why "statistical" was used in the title. Is there any statistical interpretation of this proposal? The authors only calculated a mean and a variance.

$d$ is the density for what? I suggest explicitly describing $d$ at its first appearance. I only figured out what density is when I analyzed Table I.

Could you tell me why the hierarchical complexity is computed using the neighborhood? And why not also compute the neighbor of the neighbors (or the neighbors or the neighbors of the neighbors), and so on? Or another definition of neighborhood?

I suggest showing many simulation results using some well-known random graph models (e.g., Erdös-Rényi, geometric, regular, Watts-Strogatz, Barabási-Albert, stochastic block model, popularity-similarity, etc.) to gain an intuition of the complexity measure behavior. For example, the authors could fix the size n, vary the parameter of the random graph model, and plot how the complexity changes over the parameter.

It would also be interesting to show how robust the proposed measure is when we compare graphs of different sizes. For example, the authors could apply it to two random graphs generated by the same model and parameter but with different sizes. I expect the same complexity since the graphs are identical except for the size.

I would like to know the advantages and limitations of their proposal when compared to other complexity/entropy measures.

A curiosity: is it possible to derive the complexity analytically for at least the Erdös-Rényi and k-regular random graphs? For example, what is the complexity given $n$ and $p$?

Reviewer #2: Heterogeneity and geometry are important issues in the study of complex networks. The author introduces a simple and effective measure and provides interesting conclusions. The paper is very intriguing, with a well-structured organization. I recommend acceptance.

Reviewer #3: The manuscript PCSY-D-24-00032 by Smith and Smith introduces a measure of statistical complexity for complex networks. Such measure has the property to be zero in the limiting cases of regular and Erdos-Renyi random graphs. Later the authors illustrate that the combination of hierarchy and geometry leads to larger statistical complexity, similar to real networks. Thus concluding that real-world networks evolve to increase this measure of statistical complexity. To be precise, the main novelty here is how hierarchical complexity is normalized. Indeed, the work builds on previous work of some of the authors on hierarchical complexity, as it is described in the introduction. In this sense, the new measure can be considered an incremental piece of work with respect to the state of the art. Several values of the normalized hierarchical complexity (nHC) are reported including ER random graphs. Interestingly, for random geometric graphs, the authors report the conjecture that the value tends to 0 but no formal proof is given. Also, no mention of random graphs with arbitrary degree distributions is made which is of more value when interpreting real networks. The final figure points out the relevance of the edge density on nHC. To elucidate the role of the density, I suggest performing a subsampling of the real networks (e.g., removing links while keeping the network connected, or removing the links of the node contributing most to the hierarchy) in an attempt to clarify the role of the density and hierarchies.

In summary, although I find the paper interesting for the specialized reader, the advance in the field is enough to meet the publication criteria of the journal. The introduced measure contributes by normalizing HC, which solves some issues but introduces the question of the role played by the density which is only addressed partially here. Also, the implications to real networks are not very convincing and a deeper analysis is needed.

Minor comments:

Fig.2: To strengthen the result that the value of the nHC tends to 0, either changing the x-axis to log scale and/or finding the scaling towards zero will be helpful.

Fig. 3: y-axis should say “Normalized Hierar…”. Here also an x-axis in log scale could help.

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Reviewer #1: No

Reviewer #2: No

Reviewer #3: No

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Statistical Complexity of Heterogeneous Geometric Networks

PCSY-D-24-00032R1

Dear Dr Smith,

We are pleased to inform you that your manuscript 'Statistical Complexity of Heterogeneous Geometric Networks' has been provisionally accepted for publication in PLOS Complex Systems.

Before your manuscript can be formally accepted you will need to complete some formatting changes, which you will receive in a follow-up email from a member of our team. 

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Réka Albert

Section Editor

PLOS Complex Systems

Hocine Cherifi

Editor-in-Chief

PLOS Complex Systems

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Reviewer Comments (if any, and for reference):

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: All comments have been addressed

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: Yes

Reviewer #3: No

**********

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

Reviewer #1: Yes

Reviewer #3: Yes

**********

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: Yes

Reviewer #3: Yes

**********

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 #3: Yes

**********

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: (No Response)

Reviewer #3: (No Response)

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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: None

Reviewer #3: No

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