Football Games Consist of a Self-Similar Sequence of Ball-Keeping Durations

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

1. The title effectively conveys the central theme of the study, highlighting the self-similar nature of ball-keeping durations in football games. The abstract is well-structured but could be more concise. Bit more explanation of the concept of local and global interactions is required.
2. The introduction provides a comprehensive overview of self-organization in football games, referencing previous works effectively. Consider including more recent references to similar studies in team sports to reinforce the relevance of self-similarity in dynamic systems.
3. The literature review is well-developed, establishing a strong theoretical foundation for the study. The authors could consider discussing other statistical models besides power-law distributions to provide a broader analytical perspective.
4. The hypothesis is clearly stated, linking local and global scales in football game dynamics. The hypothesis could be further refined by explicitly stating the expected relationship between local player dynamics and global game flow.
5. The dataset from the J1 League provides a robust sample size, incorporating data from 10 games. Clarify the criteria for game selection to address potential biases in data collection (e.g., team performance, game duration, weather conditions).
6. The authors utilize power-law fitting combined with maximum likelihood estimation, a sound analytical approach for this dataset. It would be beneficial to include a justification for selecting power-law fitting over other statistical models, such as Weibull or exponential distributions.
7. The results are systematically presented, effectively utilizing graphical representations of probability distributions. Incorporate additional visualizations to depict the transition from exponential to power-law distribution more clearly.
8. The statistical analysis is thorough, providing adequate explanation for parameter estimation and fitting procedures. Include a confidence interval for the power-law exponent (α) to assess the robustness of the fitting process.
9. The study’s implications for understanding game dynamics and player decision-making are clearly articulated. Discuss potential applications of the findings in coaching strategies, team performance analysis, or predictive modeling of game outcomes.
10. The manuscript is well-structured, with clear section divisions and logical flow of content. Minor formatting issues were noted in the references section, such as inconsistent citation styles. Standardize to maintain academic rigor.
11. The conclusion succinctly summarizes the main findings, reiterating the presence of hierarchical self-similarity in football dynamics. Highlight the study’s contribution to advancing the understanding of complex systems in sports science.

 

 

 

Author Response

Comments 1: The title effectively conveys the central theme of the study, highlighting the self-similar nature of ball-keeping durations in football games. The abstract is well-structured but could be more concise. Bit more explanation of the concept of local and global interactions is required. 
Response 1: Thank you for your valuable pointing out. We agree with your comment and have revised the abstract to make it more concise including the concept of local and global (p. 1, lines 1-8).

Comments 2: The introduction provides a comprehensive overview of self-organization in football games, referencing previous works effectively. Consider including more recent references to similar studies in team sports to reinforce the relevance of self-similarity in dynamic systems.
Response 2: Thank you for your comments. As you pointed out, we have added a description of the previous works. Please refer to the revised manuscript (p. 1, lines 22-24; p. 2, lines 34-36).

Comments3: The literature review is well-developed, establishing a strong theoretical foundation for the study. The authors could consider discussing other statistical models besides power-law distributions to provide a broader analytical perspective.
Response 3: Thank you for your valuable pointing out. We have added the description about the other Weibull and exponential distributions by making sense for another suggestion.   (p. 2, lines 73-75).

Comments 4: The hypothesis is clearly stated, linking local and global scales in football game dynamics. The hypothesis could be further refined by explicitly stating the expected relationship between local player dynamics and global game flow.
Response 4: Thank you for your valuable pointing out. We agree with your comment and have revised the description about hypothesis. (p. 2, lines 55-62).

Comments 5: The dataset from the J1 League provides a robust sample size, incorporating data from 10 games. Clarify the criteria for game selection to address potential biases in data collection (e.g., team performance, game duration, weather conditions).
Response 5: Thank you for your valuable pointing out. As you pointed out, we have modified and added the description about the criteria for game selection (p. 3, lines 82-84).

Comments 6: The authors utilize power-law fitting combined with maximum likelihood estimation, a sound analytical approach for this dataset. It would be beneficial to include a justification for selecting power-law fitting over other statistical models, such as Weibull or exponential distributions.
Response 6: Thank you for your valuable pointing out. As you pointed out, we have added a description of the reasons for using the power-law (p. 2, lines 73-76).

Comments 7: The results are systematically presented, effectively utilizing graphical representations of probability distributions. Incorporate additional visualizations to depict the transition from exponential to power-law distribution more clearly.
Response 7: Thank you for your comments. As you pointed out, the original manuscript lacked a clear visualization of the transition. We have revised Figure 3 to include a semi-log plot and log-log plot for a single game, in order to clarify the transition and address Reviewer 2’s suggestion. Please refer to the revised manuscript (p. 6, Figure 3).

Comments 8: The statistical analysis is thorough, providing adequate explanation for parameter estimation and fitting procedures. Include a confidence interval for the power-law exponent (α) to assess the robustness of the fitting process.
Response 8: Thank you for your comments. As you pointed out, we added the explanation about confidence interval of the power-law exponent in the main text (p. 7, lines 198-199).

Comments 9: The study’s implications for understanding game dynamics and player decision-making are clearly articulated. Discuss potential applications of the findings in coaching strategies, team performance analysis, or predictive modeling of game outcomes.
Response 9: Thank you for your valuable comments. We agree with your suggestion and have added a description of the implications (p. 8, lines 273-283).

Comments 10: The manuscript is well-structured, with clear section divisions and logical flow of content. Minor formatting issues were noted in the references section, such as inconsistent citation styles. Standardize to maintain academic rigor.
Response 10: Thank you for your valuable comments. As you pointed out, there were some inconsistencies in the format. We have corrected the format of references, so please refer the revised manuscript (p. 9, lines 312-344).

Comments 11: The conclusion succinctly summarizes the main findings, reiterating the presence of hierarchical self-similarity in football dynamics. Highlight the study’s contribution to advancing the understanding of complex systems in sports science.
Response 11: Thank you for your valuable pointing out. We agree with your comment and add the summary of the main findings and the highlight the study’s contribution in the section of ``Conclusions” (p. 9, lines 285-291).

 

 

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The work presents an innovative approach to description of complicated activity of the soccer football players and teams. The paper is interesting both for sport professionals and to lay readers. However, the paper is written in a rather unclear manner, many things could be new for readers, but they are not given in a sufficiently explicit way. More attention should be paid to explanations to make the paper more readable There also are some flaws which should be corrected and explained clearly.

1. In Figure 1, what the red and blue colors mean? Are those for the two team A and B? Then why the length T_g1 and also T_g2 cover both segments with A and B? Below that figure, in the lines 90-91, it is mentioned that “The game variable T_g is the duration for which one team maintains possession of the ball”, so again – why there are both colors, and A and B? The readers should not only guess what is what in such descriptions.
2. About the formula (2), in the integral, is the lower limit infinity? Shouldn’t it be minus infinity as in a CDF? Also, if we are interested in estimation of the parameter alpha, why to skip the term 1/(-alpha +1) in the expression at the right-hand side?
3. How the formula (3) was obtained? What the prime means in alpha’ ? By the way, it could be noticed that this formula defines 1/(alpha’ -1) as the harmonic mean of the logarithm of the values T_i/T_min, so what is the meaning of it?
4. The application of the formula (4) is not clear either, for example: why in the lines 132-133 it is stated that the resulting p value was used, and “If it was greater than 0.1, the power law was plausible” – usually a small p-value below some 0.05 or another confidence level is considered as a reasonable value for a test. It should be clarified as well.
5. Duration times T_g and T_p for a team and for players in it, respectively, seem should be related in a simple way: a total time by all players keeping the ball should be close/equal to the time the team has the ball. There are 11 players in a soccer team, so T_g should be about 11 times bigger than T_p. This kind of relation can be also seen for the number of times n_p and n_g: in the line 146, the average numbers n_p/n_g = 1238/109=11.35, so about 11 times bigger, by the number of players. It means that the estimations should be performed taking into account such a relation.
6. This “11-term” assumption is well supported by the results shown in Table 1, which should be explained in detail and discussed more in the paper.
7. Figure 2 can be simplified. In the panels (a, b), it doesn’t require the first and second halves, one is enough. Then all panels for Game can be put to one row, and all panels for Player can be put to the second row. Not clear why the CDF are shown in the opposite direction as decreasing functions. Both semi-logarithmic and logarithmic plots are redundant, on type of them is sufficient.
8. Figure 3 has 10 panels by games, but they are practically undistinguishable and not very informative, so just one of the games can be enough for illustration.
9. The developed approach can be compared with more standard techniques used in this area of sport research and described, for example, in: Beggs, Soccer Analytics: An Introduction Using R; CRC, 2024.
10. In principle, for the article to Fractals journal, more should be said about the fractal aspect of the data and/or the results.

 

Comments on the Quality of English Language

it can be improved.

Author Response

Comments 1: In Figure 1, what the red and blue colors mean? Are those for the two team A and B? Then why the length Tg1 and also Tg2 cover both segments with A and B? Below that figure, in the lines 90-91, it is mentioned that “The game variable Tg is the duration for which one team maintains possession of the ball”, so again – why there are both colors, and A and B? The readers should not only guess what is what in such descriptions.
Response 1: Thank you for your comments. As you pointed out, we didn’t write the color meaning in Figure 1 and have added this description in caption. Moreover, the description about the “The game variable Tg is the duration for which one team maintains possession of the ball”, is my mistake. We modified this sentence as follows: “The game variable Tg represents the duration for which the game is actively in progress -that is, when the ball is in play and possession is continuously maintained by one or more players.” (p. 3, lines 102-104).

Comments 2: About the formula (2), in the integral, is the lower limit infinity? Shouldn’t it be minus infinity as in a CDF? Also, if we are interested in estimation of the parameter alpha, why to skip the term 1/(-alpha +1) in the expression at the right-hand side?
Response 2: Thank you for your comments. As you correctly pointed out, the use of infinity in the integral was incorrect. I have revised the equation so that the lower limit is T and the upper limit is infinity. In terms of the parameter alpha, as you pointed out, the term 1/(-alpha +1) is necessary for proper normalization. However, we are mainly interested in the scaling behavior of P(T) rather than its exact normalization. Therefore, we used a proportional relation to simplify the expression (p. 4, formula (2)).

Comments 3: How the formula (3) was obtained? What the prime means in alpha’ ? By the way, it could be noticed that this formula defines 1/(alpha’ -1) as the harmonic mean of the logarithm of the values Ti/Tmin, so what is the meaning of it?
Response 3: Thank you for your comments. In response to your suggestions, we have added the new formula (3) to explain how the prime in alpha is obtained. Moreover, we added a corresponding description of its meaning. Please refer to the revised manuscript (p. 4, lines 132-143).

Comments 4: The application of the formula (4) is not clear either, for example: why in the lines 132-133 it is stated that the resulting p value was used, and “If it was greater than 0.1, the power law was plausible” – usually a small p-value below some 0.05 or another confidence level is considered as a reasonable value for a test. It should be clarified as well.
Response 4: Thank you for your comments. As you correctly pointed out, the original manuscript lacked sufficient description. We have now added the necessary explanations. Please refer to the revised manuscript (p. 5, lines 150-157).

Comments 5: Duration times Tg and Tp for a team and for players in it, respectively, seem should be related in a simple way: a total time by all players keeping the ball should be close/equal to the time the team has the ball. There are 11 players in a soccer team, so Tg should be about 11 times bigger than Tp. This kind of relation can be also seen for the number of times np and ng: in the line 146, the average numbers np/ng = 1238/109=11.35, so about 11 times bigger, by the number of players. It means that the estimations should be performed taking into account such a relation.
Response 5: Thank you for your valuable comments. As you pointed out, our original manuscript lacked a discussion on the number of variables, which may carry important implications. We have added a possible explanation regarding this point. Please refer to the revised manuscript (p. 5, lines 171-176).

Comments 6: This “11-term” assumption is well supported by the results shown in Table 1, which should be explained in detail and discussed more in the paper.
Response 6: Thank you for your valuable comments. In response to this and the previous comment, we have added a detailed explanation about the 11-term assumption. Please refer to the revised manuscript (p. 5, lines 171-176).

Comments 7: Figure 3 has 10 panels by games, but they are practically undistinguishable and not very informative, so just one of the games can be enough for illustration.
Response 7: Thank you for your insightful comments. In response, we simplified Figure 3 by displaying only one representative game to improve clarity and readability. The original figure containing all 10 game panels has been moved to the supplementary materials. Additionally, in accordance with Reviewer1’s suggestion, we include both logarithmic and semi-logarithmic plots to clear the transition from exponential to power-law distribution. Please refer to the revised manuscript (p.7, figure 3).

Comments 8: The developed approach can be compared with more standard techniques used in this area of sport research and described, for example, in: Beggs, Soccer Analytics: An Introduction Using R; CRC, 2024.
Response 8: Thank you for your valuable comments. We agree with your suggestion and have added a description of the approach developed in this study, with reference to the book you recommended. Please refer to the revised manuscript (p. 9, lines 279-280).

Comments 9: In principle, for the article to Fractals journal, more should be said about the fractal aspect of the data and/or the results.
Response 9: Thank you for your comments. As you pointed out, we have added the term about fractal in the revised manuscript.

 

Reviewer 3 Report

Comments and Suggestions for Authors

The authors discuss an application of power-law distributions for the real-life example concerning the ball possession times. Generally, the paper should be interesting for the readers and present scientifically sound thinking. Only some minor issues should be solved before its publication:

1. The paper looks unfinished. There are no conclusions or ideas concerning future research. Please, add the respective “Conclusions” section at the end of the paper containing such considerations. It will be helpful for the readers.
2. Please explain if the data, raw data, or code will be available for the readers (e.g., upon request) or if they are strictly confidential.  

Considering the mentioned issues, I advise a minor revision of the paper.

 

Author Response

Comments 1: The paper looks unfinished. There are no conclusions or ideas concerning future research. Please, add the respective “Conclusions” section at the end of the paper containing such considerations. It will be helpful for the readers.
Response 1: Thank you for your valuable pointing out. We agree with your comment and have added the “Conclusions” section. Please refer to the revised manuscript (p. 9, lines 290-296).

 

Comments 2: Please explain if the data, raw data, or code will be available for the readers (e.g., upon request) or if they are strictly confidential. 
Response 2: Thank you for your comment. We modified the data availability statement. Please refer to the revised manuscript (p. 9, lines 310-313).