Kolmogorov–Smirnov-Based Edge Centrality Measure for Metric Graphs
Abstract
:1. Introduction
2. Metric Graphs
3. Vertex Centrality Measures
Edge Centrality Measures
4. Edge Centrality Measure for Metric Graphs
- Create a second metric graph, , by removing edge from .
- Calculate the energy density (see Equation (2)), , for each edge in each metric graph.
- Compute the average edge density, , for each edge for each metric graph.
- Normalize the average edge density for each edge for each metric graph by the total energy, .
- Apply the two-sample KS test on the two empirical CDFs of the normalized average edge densities to obtain a p-value.
- Compare the p-value to a chosen significance level ; if , then edge is concluded to have an influential role in the energy density distribution of .
5. Results
5.1. G14 Network
5.2. Road Network
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. Computational Complexity
- Phase 1. Given a metric graph , generate the graph ; this part is computationally trivial. Then, find the Helmholtz eigenvalues on and in range .
- Phase 2. Find the corresponding eigenvectors, calculate the average energy density over all eigensolutions, and run a KS test to compare the distributions.The time complexity of this computation depends on several factors:
- –
- Eigenvalue upper bound, M: Although the specific eigenvalues vary between graphs, at some point the energy density distribution stabilizes beyond a certain point. For instance, in the G14 network, the distributions for and are nearly indistinguishable.
- –
- Total length of the graph: The number of eigenvalues satisfying follows Weyl’s asymptotic estimate, growing as
- –
- Choice of algorithm: Optimization strategies may improve computational efficiency, potentially at the cost of accuracy. Possible enhancements include leveraging the sparse structure of the matrix of junction conditions, exploiting known eigenvalue spacings, parallelization, and decreasing the error threshold.
Appendix A.1. Phase I Complexity Analysis
Appendix A.2. Phase II Complexity Analysis
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Edge | |||||
Adjacent Vertices | |||||
Length | 11.91 | 7.08 | 6.00 | 2.24 | 4.12 |
Edge | |||||
Adjacent Vertices | |||||
Length | 1.41 | 2.00 | 1.00 | 4.72 | 4.47 |
Edge | |||||
Adjacent Vertices | |||||
Length | 2.00 | 2.00 | 1.41 | 4.47 |
Edge | |||||
KS -value | |||||
Degree | |||||
Eigenvector | |||||
Closeness | |||||
Betweenness | |||||
Edge Betweenness | |||||
Laplacian | |||||
Entropy Variation | 0.083 | 0.066 | 0.066 | 0.083 | 0.001 |
Current-Flow Closeness | |||||
Flow Betweenness | |||||
Edge Current-Flow Betweenness | |||||
Edge | |||||
KS -value | |||||
Degree | |||||
Eigenvector | |||||
Closeness | |||||
Betweenness | |||||
Edge Betweenness | |||||
Laplacian | |||||
Entropy Variation | 0.016 | 0.003 | 0.002 | 0.003 | 0.105 |
Current-Flow Closeness | |||||
Flow Betweenness | |||||
Edge Current-Flow Betweenness | |||||
Edge | |||||
KS -value | |||||
Degree | |||||
Eigenvector | |||||
Closeness | |||||
Betweenness | |||||
Edge Betweenness | |||||
Laplacian | |||||
Entropy Variation | 0.105 | 0.001 | 0.019 | 0.002 | |
Current-Flow Closeness | |||||
Flow Betweenness | |||||
Edge Current-Flow Betweenness |
Centrality Measure | Top Five Ranked Edges |
---|---|
KS (Proposed) | 3, 7, 1, 6, 12 |
Degree | 2, 3, 5, 1, 4 |
Eigenvector | 2, 3, 1, 4, 5 |
Closeness | 7, 5, 9, 8, 6 |
Betweenness | 8, 14, 7, 13, 5 |
Edge Betweenness | 12, 6, 13, 8, 9 |
Laplacian | 5, 7, 2, 3, 1 |
Entropy Variation | 10, 11, 1, 4, 2 |
Current-Flow Closeness | 1, 7, 2, 3, 5 |
Flow Betweenness | 8, 13, 9, 14, 10 |
Edge Current-Flow Betweenness | 6, 13, 8, 12, 7 |
Edge | |||||
Adjacent Vertices | |||||
Length | 1.00 | 1.00 | 1.69 | 1.71 | 0.69 |
Edge | |||||
Adjacent Vertices | |||||
Length | 1.72 | 2.25 | 2.11 | 0.93 | 2.86 |
Edge | |||||
Adjacent Vertices | |||||
Length | 1.00 | 1.18 | 2.95 | 1.40 | 0.96 |
Edge | |||||
Adjacent Vertices | |||||
Length | 0.56 | 0.75 | 1.16 | 1.75 | 5.34 |
Edge | |||||
Adjacent Vertices | |||||
Length | 2.46 | 2.82 | 2.70 |
Edge | ||||||
KS -value | 0.124 | 0.007 | 0.056 | 0.224 | 0.047 | 0.110 |
Degree | 0.293 | 0.306 | 0.275 | 0.287 | 0.232 | 0.202 |
Eigenvector | 0.549 | 0.766 | 0.308 | 0.523 | 0.154 | 0.129 |
Closeness | 0.510 | 0.559 | 0.560 | 0.619 | 0.559 | 0.515 |
Betweenness | 16.000 | 4.000 | 35.000 | 20.000 | 50.000 | 25.000 |
Edge Betweenness | 9.000 | 6.000 | 27.000 | 18.000 | 35.000 | 15.000 |
Laplacian | 4.055 | 4.943 | 3.458 | 4.347 | 2.683 | 2.294 |
Entropy Variation | 0.006 | 0.001 | 0.012 | 0.008 | 0.014 | 0.012 |
Current-Flow Closeness | 4.896 | 2.940 | 2.941 | 2.941 | 2.758 | 2.535 |
Flow Betweenness | 67.000 | 37.000 | 109.000 | 79.000 | 197.000 | 92.000 |
Edge Current-Flow Betweenness | 11.147 | 5.065 | 14.157 | 12.186 | 18.996 | 9.884 |
Edge | ||||||
KS -value | 0.408 | 0.224 | 0.408 | 0.056 | 0.018 | 0.124 |
Degree | 0.202 | 0.165 | 0.309 | 0.405 | 0.316 | 0.405 |
Eigenvector | 0.061 | 0.110 | 0.589 | 1.000 | 0.585 | 0.923 |
Closeness | 0.456 | 0.442 | 0.561 | 0.506 | 0.584 | 0.624 |
Betweenness | 12.000 | 8.000 | 36.000 | 14.000 | 32.000 | 44.000 |
Edge Betweenness | 18.000 | 12.000 | 25.000 | 7.000 | 18.000 | 30.000 |
Laplacian | 1.840 | 1.781 | 4.216 | 5.396 | 4.845 | 5.950 |
Entropy Variation | 0.023 | 0.026 | 0.009 | 0.003 | 0.007 | 0.002 |
Current-Flow Closeness | 2.537 | 2.262 | 2.942 | 3.098 | 3.095 | 3.229 |
Flow Betweenness | 93.000 | 92.000 | 143.000 | 78.000 | 60.000 | 84.000 |
Edge Current-Flow Betweenness | 11.672 | 11.782 | 20.068 | 14.853 | 9.574 | 13.674 |
Edge | ||||||
KS -value | 0.110 | 0.124 | 0.022 | 0.047 | 0.110 | 0.246 |
Degree | 0.339 | 0.156 | 0.154 | 0.116 | 0.118 | 0.082 |
Eigenvector | 0.898 | 0.213 | 0.211 | 0.127 | 0.045 | 0.023 |
Closeness | 0.533 | 0.476 | 0.479 | 0.493 | 0.444 | 0.367 |
Betweenness | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
Edge Betweenness | 8.000 | 16.000 | 16.000 | 16.000 | 16.000 | 16.000 |
Laplacian | 5.156 | 2.256 | 2.257 | 1.717 | 1.403 | 0.822 |
Entropy Variation | 0.001 | 0.006 | 0.006 | 0.001 | 0.001 | 0.002 |
Current-Flow Closeness | 2.941 | 2.260 | 2.260 | 1.916 | 1.916 | 1.469 |
Flow Betweenness | 33.000 | 35.000 | 35.000 | 57.000 | 31.000 | 35.000 |
Edge Current-Flow Betweenness | 6.705 | 8.000 | 8.000 | 8.000 | 8.000 | 8.000 |
Edge | ||||||
KS -value | 0.117 | 0.022 | 0.022 | 0.047 | 0.056 | |
Degree | 0.237 | 0.316 | 0.316 | 0.248 | 0.244 | |
Eigenvector | 0.585 | 0.871 | 0.871 | 0.608 | 0.599 | |
Closeness | 0.476 | 0.444 | 0.444 | 0.464 | 0.469 | |
Betweenness | 0.000 | 0.500 | 0.500 | 0.000 | 0.000 | |
Edge Betweenness | 16.000 | 0.000 | 1.000 | 16.000 | 16.000 | |
Laplacian | 3.515 | 3.732 | 3.732 | 3.509 | 3.511 | |
Entropy Variation | 0.004 | 0.084 | 0.084 | 0.005 | 0.005 | |
Current-Flow Closeness | 2.535 | 2.756 | 2.756 | 2.535 | 2.535 | |
Flow Betweenness | 39.000 | 47.000 | 47.000 | 39.000 | 39.000 | |
Edge Current-Flow Betweenness | 8.000 | 6.598 | 6.254 | 8.000 | 8.000 |
Centrality Measure | Top Three Ranked Edges |
---|---|
KS (Proposed) | 2, 11, 15 |
Degree | 10, 12, 13 |
Eigenvector | 10, 12, 13 |
Closeness | 12, 4, 11 |
Betweenness | 5, 12, 9 |
Edge Betweenness | 5, 12, 3 |
Laplacian | 12, 10, 13 |
Entropy Variation | 20, 21, 8 |
Current-Flow Closeness | 1, 12, 10 |
Flow Betweenness | 5, 9, 3 |
Edge Current-Flow Betweenness | 9, 5, 10 |
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Durón, C.; Kravitz, H.; Brio, M. Kolmogorov–Smirnov-Based Edge Centrality Measure for Metric Graphs. Dynamics 2025, 5, 16. https://doi.org/10.3390/dynamics5020016
Durón C, Kravitz H, Brio M. Kolmogorov–Smirnov-Based Edge Centrality Measure for Metric Graphs. Dynamics. 2025; 5(2):16. https://doi.org/10.3390/dynamics5020016
Chicago/Turabian StyleDurón, Christina, Hannah Kravitz, and Moysey Brio. 2025. "Kolmogorov–Smirnov-Based Edge Centrality Measure for Metric Graphs" Dynamics 5, no. 2: 16. https://doi.org/10.3390/dynamics5020016
APA StyleDurón, C., Kravitz, H., & Brio, M. (2025). Kolmogorov–Smirnov-Based Edge Centrality Measure for Metric Graphs. Dynamics, 5(2), 16. https://doi.org/10.3390/dynamics5020016