Kolmogorov–Smirnov-Based Edge Centrality Measure for Metric Graphs

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The authors consider the problem of identifying the most central edges in the network. As a result, they propose a novel centrality measure (KS p-value) that is based on the Helmholtz equation and the Kolmogorov-Smirnov test. The performance of the proposed measure is shown on two artificial networks data.

The paper is well-organized with clear descriptions of the problem formulation, while the research problem is clear and relevant to the journal. However, I do not think the manuscript is ready for publication in its current form. My concerns:

1. As mentioned by the reviewers, the concept of centrality is ill-defined, and there is no single, universally accepted definition of a centrality measure. This makes it relatively easy to propose a new centrality metric, which could yield results that differ from other established measures. Therefore, I believe the manuscript should include a more in-depth discussion of the advantages of the proposed approach, along with specific use cases where this metric would be particularly valuable. It would also be helpful to include more detailed examples of real-world cases where this measure is more valuable than other existing centrality measures. Currently, the manuscript only applies the approach to two artificial networks, and there is no clear evidence presented that the results of the KS p-value measure are more accurate for these networks.
2. The manuscript should include a more thorough discussion of the limitations of the model, particularly regarding its computational complexity. Additionally, it would be important to address whether the small number of nodes in the two networks used in the study is a coincidence or if this is reflective of a limitation in the model's applicability. Can this metric be effectively applied to larger networks, and if so, how does its performance scale with network size?
3. The authors compare their metric only to four classical centrality measures, but there are many more centrality metrics available in the field. Have the authors checked if the proposed measure differs significantly from other existing ones? Specifically, the authors mention that the eigenvector centrality seems to overlap with the proposed edge centrality measure. Since there are numerous spectral centrality measures for nodes, it is possible that the new metric is correlated with one of the existing ones. It would be helpful for the authors to investigate and address this potential correlation and discuss how their proposed measure compares to a wider range of centrality metrics.
4. Page 3, lines 112-113: The authors provide the definition of stress centrality instead of betweenness centrality, yet they claim that it is betweenness centrality. This appears to be a mistake and should be corrected.
5. Figures 1-2: It would be beneficial for the reader if the authors also provided images of the original graphs that were used to construct the line graphs. Additionally, it would be helpful if the authors could explain the presence of multiple edges between two nodes in the line graph.
6. The authors use the terms "combinatorial graph" and "combinatorial measures." I believe they should provide clear definitions or descriptions of these notions, as they may not be familiar to all readers.
7. Based on the definition on Page 2 and the examples in Section 5, the authors appear to operate with directed weighted graphs. However, Section 3.1 seems to describe the line graph in the context of undirected, unweighted networks. Could the authors please clarify this discrepancy and explain how the line graph is constructed from directed weighted graphs and how it is utilized in the article?

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

1.The description of the abstract is rather confusing, it is suggested that according to the research background, methods, results, conclusions of four parts to give a concise description.

2.The experimental part should add methods of nearly three years for comparison.

 

Comments on the Quality of English Language

The English could be improved to more clearly express the research.

Author Response

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Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

In this paper, the authors introduce a new measure of edge centrality in a directed weighted metric graph. The measure is based on the energy distribution of eigensolutions of the Helmholtz equation. The authors compare the values ​​of the introduced centrality measure with the values ​​of standard centrality measures (by degree, by eigenvalue, betweenness) on two real graphs.

Overall, the study is promising. However, in my opinion, some aspects of the article require further development and strengthening.

My first remark concerns the motivation for using the new measure. The authors write ‘Unfortunately, the standard combinatorial edge centrality measures are not sufficient for metric graphs as they do not account for the dynamics on the edges governed by the accompanying ordinary differential equations (ODEs) or partial differential equations (PDEs).’ It should be explained why it is so important to take this dynamics into account. It would also be nice if the authors provided an applied meaning of the new measure.

My second remark concerns the complexity of calculating the new measure. The authors ignore this issue, which is nevertheless extremely important when used in applications. The examples considered by the authors contain a small number of nodes (14 and 23). However, in practice, most problems are associated with the study of complex networks containing hundreds of thousands or even more nodes. At first glance, the complexity of calculating the measure does not allow it to be applied to graphs containing even a hundred nodes. Perhaps, I would advise the authors to pay closer attention to the assessment of its computational complexity.

In Section 5, the authors find the value of the centrality measure proposed by them for the edges of two real graphs. Unfortunately, the demonstration of the usefulness of the new measure was not convincing. It is not clear for which practical problem this measure is more preferable than standard centrality measures.

Minor remarks
- Equation (2) should be corrected

Author Response

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Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The manuscript is an improvement over the previous version, and I appreciate the revisions made. However, some of my previous concerns have only been partially addressed.

 

1) The response to my previous concern 1 remains unsatisfactory.

Currently, the authors justify their novel centrality measure by stating that (1) "standard centrality measures ignore edge dynamics, potentially leading to misleading assessments of edge importance and overall network functionality" and (2) the proposed measure's edge rankings correlate more strongly with the drop in energy flow (represented by edge density distribution) compared to four standard measures.

However, this justification does not address the core issue: the comparative analysis is flawed. The authors compare their new model, which is designed for flow networks, against four standard centrality metrics, most of which are NOT designed for flow networks. This is not a fair comparison. The literature already contains numerous centrality measures that explicitly account for energy drop after node/edge removal (e.g., Laplacian centrality, X-degree and X-non-backtracking, entropy variation, efficiency centrality, etc.) or assume the existence of flow in the network (e.g., current-flow closeness, flow betweenness, current-flow betweenness, routing betweenness, bridging capital, physarum centrality, path-transfer, etc.). Many of these metrics have been tested in experiments and arrive at similar conclusions: "the proposed measure is an effective tool for identifying influential nodes whose removal significantly changes the energy distribution". Since the manuscript does not compare the proposed measure to these more relevant metrics, its advantages remain unclear. Additionally, the current analysis is limited to two artificial networks, with no compelling evidence that the KS p-value measure is more accurate in these settings.

As I previously mentioned, the manuscript should provide a more in-depth discussion of the advantages of the proposed approach, including specific use cases where this metric would be particularly valuable. Furthermore, a comparative analysis against relevant flow-based or energy-based centrality measures, as well as real-world network examples, would strengthen the manuscript and clarify its contributions.

 

2) The second response has been only partially addressed. I appreciate the authors for adding a discussion on the computational complexity of the model in the appendix. However, a thorough discussion of the model’s limitations is still missing from the main part of the manuscript. Specifically, it remains unclear for which types of networks and underlying processes the proposed metric is most or least appropriate. Additionally, there is no explicit mention of the network size for which computing the metric remains feasible. These are critical aspects that should be included in the main text, as they are essential for readers assessing the applicability of the model.

Regarding computational complexity, while the authors provide runtime analysis for two separate phases, it would be informative to also include a figure showing the total runtime of the model. Furthermore, the manuscript should present results for graphs with up to approximately 10,000 edges. The authors’ response states that the complexity scales linearly with the number of edges for graphs of this size, but this is not clearly demonstrated in Figure A2, where the experiments and curve fitting appear to be performed on graphs with at most 100 edges.

Lastly, I recommend replacing the experiments on graphs with three vertices and multiple edges between nodes with experiments on artificial graphs, such as ER graphs (e.g. different size n and link probability p). Given that the study focuses on simple graphs rather than multigraphs, this adjustment would ensure greater consistency with the main part of the paper and provide more meaningful insights.

 

3) The authors did not address my comment 3. I strongly believe that introducing a novel measure and comparing it only to four standard centrality measures (simply because they are the “most commonly used measures”) is insufficient.

As previously noted, and as the authors acknowledge, these classical measures are not designed for flow networks. Therefore, comparing a metric that is specifically designed for a particular network process only to metrics, which do not account for this process (they interpret centrality in fundamentally different ways), cannot serve as a valid justification for the proposed new model. Instead, the comparison should focus on metrics that are more relevant to the problem. This is particularly important if the proposed metric has already been shown to correlate well with certain existing measures. Without a relevant comparison to existing measures, the study may struggle to demonstrate the true significance of the proposed metric and its contribution to the field. I believe the authors should make greater efforts to address this issue.

 

4) The description of betweenness centrality needs further revision. While betweenness centrality evaluates the fraction of shortest paths between a given pair of nodes that pass through a particular node, the cumulative betweenness centrality score of a node (i.e., the sum of these fractions) does not necessarily reflect the fraction of shortest paths between all pairs of nodes that pass through that node.

 

5-7) Response to my concerns 5-7: No further comments, thank you for the clarification.

 

Author Response

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Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

All my concerns were answered. 

Comments on the Quality of English Language

English seems to be correct

Author Response

Thank you.

Round 3

Reviewer 1 Report

Comments and Suggestions for Authors

I would like to thank the authors for addressing my previous comments. However, some of my concerns have only been partially addressed.

 

1) My previous concerns #1,3:

 

a) Regarding my comment on the experiments with artificial graphs: while the initial graphs (G14 and the road network in Poland) are real, undirected, and weighted networks, the authors then transform the weighted graph into a weighted line graph using an arbitrary transformation (“each edge in the line graph is weighted by the mean of the normalized average edge densities of the corresponding metric graph”). This transformation causes the initial graph G to lose its original interpretation. As a result, vertex centrality measures are applied to artificial line graphs, which weakens the validity and interpretability of the experiments.

 

b) Regarding the choice of centrality measures, I appreciate the authors’ effort to include additional metrics. However, I still believe further improvements are necessary.

First, I believe the experiments and the comparison of centrality measures are not conducted fairly. As I understand it, the proposed metric is applied to the original metric graph Γ, while classical vertex metrics are applied to the artificial line graph after undergoing an arbitrary transformation (the choice of the transformation directly affects the results). As a result, the authors are comparing outcomes from two fundamentally different networks, which introduces an inherent imbalance in the evaluation.

Second, I believe the choice of centrality metrics is not appropriate. The authors apply standard vertex centrality measures to the line graph, arguing that these measures are defined for vertices. However, is it truly necessary to transform the initial graph into a line graph and apply an additional, somewhat arbitrary transformation to compute standard centralities? In many cases, it is not. Many standard vertex centrality measures have been extended to edges, such as edge betweenness and edge current-flow betweenness, which can be applied directly to the original graph without transformation, yielding different results than those obtained on the line graph. The same holds for many energy-drop-based measures: while originally defined for vertices, they can be easily adapted for edges (by removing an edge instead of a node) and applied directly to the initial graph rather than the artificial line graph.

Third, while the study focuses on edge centrality, it overlooks well-established edge centrality metrics in the field (no edge centralities in the paper).

Therefore, I strongly believe that Section 5 should be significantly rewritten and revised.

Finally, I urge the reviewers to verify the correctness of the applied centrality measures. For example, the authors mention the centiserve package. However, the ‘laplacian’ function in this package is defined for unweighted graphs, whereas the experiments consider weighted graphs.

 

2) My concern #2 (computational complexity).

Thank you for the clarification and for revising the typos. It is now clear that ∣E∣≈10,400, which represents the intersection between the two phases.

I would like to ask the authors about the approximating equations for phases 1 and 2: were these equations constructed experimentally based on the number of edges ∣E∣, which varies from 65 to 100 (a short observation window), or did the authors consider a wider range? For instance, while the equation 2.9∣E∣+109 seems to approximate the data points in Fig. A2 quite well (with the data points almost perfectly aligning with the line), does this pattern hold if you consider ∣E∣=5 edges (≈123 seconds), ∣E∣=10 edges (≈138 seconds), or ∣E∣=200 edges (≈689 seconds) in the graph? Additionally, it is not clear why there is a sudden increase in runtime after around ∣E∣=75 (with a 20-second increase in runtime). Phase II does not exhibit such a large increase.

Regarding my comment on multigraphs, I was primarily concerned about symmetry in the graph: there are 3 unique centrality scores, with ∣E∣/3 of the edges having the same centrality score. Does this symmetry simplify any of the performed calculations (e.g., SVD, the simplex algorithm, or other techniques that might require less time due to this symmetry)?

As a minor comment, the authors mention tol = 1E – 10, but the use of E seems confusing, as it is already used to denote the edge set.

4) In the original paper, the authors mentioned that “Based on the results, the eigenvector centrality seems to overlap with the proposed edge centrality measure.” However, this statement is no longer present in the current version of the manuscript, which is also reflected in the experiments. Was the initial observation incorrect after performing the experiments?

Author Response

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Author Response File: Author Response.pdf

Round 4

Reviewer 1 Report

Comments and Suggestions for Authors

While the authors have addressed some of my concerns, a number of important comments remain unaddressed.

 

1) Response to my concern 1:

a) First, I would like to clarify my earlier comments (concern 1a and the first part of concern 1b). I agree with the authors that applying vertex centrality measures to the line graph L(G) is a well-established and commonly used technique. In that regard, there is no disagreement that this approach is appropriate for the manuscript.

However, in this work, the authors transform a weighted graph into a weighted line graph using a specific transformation (“each edge in the line graph is weighted by the mean of the normalized average edge densities of the corresponding metric graph”). This particular transformation appears to be arbitrary and, to the best of my knowledge, is not commonly found in the existing literature. As such, the choice of transformation introduces a potential bias that could significantly affect the performance and interpretation of the vertex centrality measures. This, in turn, weakens the validity and fairness of the experimental comparisons. While the manuscript cites references [1–3], it is important to note that these works focus on unweighted graphs and therefore do not involve such transformation steps. Reference [4], on the other hand, does consider weighted line graphs, but employs a different approach from the one used here.

Given these points, I strongly encourage the authors to include a discussion and, if possible, a comparison of alternative transformation strategies. Additionally, it would be valuable to explicitly acknowledge this limitation in the manuscript, as it may affect the fairness of the comparisons between the vertex centrality measures and the proposed model.

 

b) I remain concerned about the choice and balance of centrality metrics included in the manuscript. Specifically, there appears to be a disproportionate emphasis on vertex centrality measures applied to the transformed (and potentially biased) weighted line graph, while edge centrality measures are underrepresented. Including only a single edge centrality measure is, in my view, insufficient to support the manuscript’s claims.

I believe the initial selection of centrality measures was not appropriate, and, as evidenced by successive revisions, the inclusion of more relevant metrics (such as CFCC, EVC, and EBC) has led to improved performance in terms of KS test statistics. Since the authors justify their proposed centrality measure by arguing that the proposed measure's edge rankings correlate more strongly with the drop in energy flow (represented by edge density distribution), it is essential to consider additional edge centrality measures, especially those explicitly designed to capture changes in energy flow.

For example, as mentioned in my previous reviews, a broad class of edge centrality measures (e.g. see a class of vitality/induced measures) is discussed in [1]. These measures are well-suited to multigraphs and directly align with the core objective of evaluating edge importance through energy flow changes.

I also find the authors’ justification for their current selection of measures (they include only metrics that “are both well-defined and implemented in widely-used libraries, ensuring the reliability and accessibility of our results”) unsatisfactory. The authors introduce a novel centrality measure designed for a specific type of network (flow networks). Therefore, to convincingly demonstrate the value of their model, it is important to compare it with relevant centrality measures, rather than limiting the evaluation to those that are simply available in standard libraries. Comparing a model specifically designed for flow networks with others that conceptualize centrality in fundamentally different ways does not provide a meaningful assessment. The number of centrality measures included does not necessarily reflect the quality of those measures.

To further support “the reliability and accessibility of the results” (the authors’ point), I recommend that the authors make their experimental setup and datasets publicly available (e.g., via GitHub). This practice is well-established in the field, especially among researchers introducing new centrality measures, who compare their proposed methods to the most relevant existing alternatives.

 

c) The authors state that edge current-flow betweenness centrality was not included in their analysis because it is not compatible with multigraphs. However, it was not clear from the original manuscript that all experiments were conducted exclusively on multigraphs—this is only briefly mentioned in footnote 1 and Appendix A. I recommend that the authors explicitly state this in the main text, as it represents an important limitation of the study. Specifically, the decision not to evaluate the proposed centrality measure on simple graphs should be clearly acknowledged, as it restricts the generalizability of the results and may confuse the reader.

 

d) The authors analyze an approximate road network in Poland, citing reference [27], and report using a graph with 17 vertices and 23 edges. However, the original paper presents a network with 18 vertices and 24 edges (see Fig. 1b and Table 2). Furthermore. the original network is not a multigraph (unlike the one used in the current manuscript). The authors do not clarify why their version of the network differs from the original, nor do they justify the modifications made. Moreover, there are significant discrepancies in edge weights. For example, in the original paper, the edge (v2, v7) has a weight of 12.56, which is the largest in the network. In contrast, the current manuscript assigns a weight of 1 to this same edge, making it one of the smallest. These changes are not explained and raise concerns about the validity, the interpretability and comparability of the experiments. I recommend that the authors clearly explain the reasons for these differences and how they may impact the results.

 

 

2) Response to my concern 2: Thank you for the clarification. As a minor suggestion, I recommend that the authors provide an explanation for the sudden increase in runtime after around ∣E∣=75 (with a 20-second increase in runtime), as they have done in their response letter.

 

3) Response to my concern 3: No further comments, thank you for the clarification.

 

[1] Koschützki, D., Lehmann, K.A., Peeters, L., Richter, S., Tenfelde-Podehl, D., Zlotowski, O.: Centrality indices. In: Brandes, U., Erlebach, T. (eds.) Network Analysis. LNCS, vol. 3418, pp. 16–61. Springer, Heidelberg (2005). doi:10.1007/978-3-540-31955-9_3

Author Response

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