Kolmogorov–Smirnov-Based Edge Centrality Measure for Metric Graphs

Abstract

In this work, we introduce an edge centrality measure for the Helmholtz equation on metric graphs, a particular flow network, based on spectral edge energy density. This measure identifies influential edges whose removal significantly changes the energy flow on the network, as indicated by statistically significant p-values from the two-sample Kolmogorov–Smirnov test comparing edge energy densities in the original network to those with a single edge removed. We compare the proposed measure with eight vertex centrality measures applied to a line graph representation of each metric graph, as well as with two edge centrality measures applied directly to each metric graph. Both methods are evaluated on two undirected and weighted metric graphs—a power grid network adapted from the IEEE 14-bus system and an approximation of Poland’s road network—both of which are multigraphs. Two experiments evaluate how each measure’s edge ranking impacts the energy flow on the network. The results demonstrate that the proposed measure effectively identifies influential edges in metric graphs that significantly change the energy distribution.

1. Introduction

Network theory serves as a powerful tool for investigating the structure, dynamics, and function of complex systems [1,2,3] by providing a natural framework to model the system units as vertices and the interactions among the units as edges within a network [4]. In networks, not all vertices have the same level of importance, as some vertices are more influential than others. The problem of identifying the most influential vertices in a network has been extensively studied, as the importance of the vertices within the network can be characterized according to various notions of centralities. By definition, a centrality measure is a function that assigns a numerical value (“importance”) to each vertex within the network, and measures of centrality are often defined on some assumption of how information propagates throughout the network. As there is no exact meaning of “centrality”, there is an abundance of definitions for a centrality measure of individual vertices (the most well-known measures include the degree [5], eigenvector [6], closeness [7], and betweenness [8] centralities). Using centrality measures on networks offers several advantages (e.g., identification of super-spreaders [9,10], efficient resource allocation [11,12], and robustness assessment [13,14]). Consequently, these measures serve as effective tools for understanding, analyzing, and managing the dynamics within networks across various domains (e.g., physics, sociology, biology, and the neurosciences [3,15,16,17,18]).

In this work, we focus on measuring the influence of an edge within a particular flow network, where understanding the significance of the connections can provide additional insights into the network dynamics and functionality (e.g., identification of important pathways [19,20] and fragility assessments [20,21]). While there exist specific centrality measures defined solely for edges within a network (for example, the betweenness, closeness, and eigenvector edge centralities are defined in an analogous manner to their vertex counterparts [19]), there are far fewer metrics [22] and, to the best of our knowledge, none defined specifically for metric graphs, a particular flow network in which each edge is given a positive length, allowing various geometric and analytical methods to be applied. In particular, we examine the effect of removing a single edge or a small group of edges whose removal leads to significant changes in the energy flow on the network. Examples of flow networks include traffic systems, water supply networks, electric circuits and power grids, and ecological systems involving the flow of nutrients and energy between organisms. In these cases, violating capacity constraints can result in congestion, blackouts, or the failure of ecosystems, respectively [23,24].

Consequently, we introduce an edge centrality measure for metric graphs based on the energy distribution of eigensolutions of the Helmholtz equation. The measure identifies influential edges whose removal significantly changes the energy flow (i.e., the edge energy density distribution) on the network, as indicated by statistically significant p-values from the two-sample Kolmogorov–Smirnov test [25]. Specifically, the test compares the cumulative distribution functions of normalized average edge densities for eigenvalues $k\le M$ (for some real number M) between the original metric graph and the metric graph with the edge removed. We compare the performance of the proposed edge measure with that of eight vertex centrality measures applied to a line graph, whose vertices denote the edges of the metric graph, as well as with two edge centrality measures applied directly to the metric graph. Numerical examples are performed on two metric graphs, G14 (a power grid network adapted from an Institute of Electrical and Electronics Engineers (IEEE) benchmark test [26]) and a simplified road network of Poland [27], which are displayed in Figure 1 and Figure 2, respectively. Lastly, we evaluate our proposed measure by comparing edge rankings from different centrality measures for each metric graph through two experiments that identify edges whose removal significantly changes the network’s energy flow.

The structure of the paper is as follows. The definition of a metric graph is provided in Section 2, along with an overview of the Helmholtz equation. Vertex centrality measures are highlighted in Section 3, along with a discussion on edge centrality measures. Section 4 details the proposed edge centrality measure for metric graphs. Section 5 describes the two metric graphs used in the numerical examples and compares the proposed edge centrality measure with eight vertex centrality measures applied to a line graph representation of each metric graph, as well as with two edge centrality measures applied directly to each metric graph; it also presents two experiments that assess the rankings of influential edges and their impact on the energy flow on the networks. A brief discussion and conclusion follow in Section 6, along with directions for future work. Lastly, the computational complexity of the proposed edge centrality measure is detailed in Appendix A.

2. Metric Graphs

Mathematically, a network can be represented by a graph $G=(V,E)$, where $V={\left\{v_j\right\}}_{j=1}^m$ is the set of m vertices and $E={\left\{e_j\right\}}_{j=1}^n$ is the set of n edges connecting vertex pairs. We consider a finite undirected graph $\Gamma$ with the set of vertices $\{v_1,v_2,\dots ,v_m\}\in V$ and arbitrarily oriented edges $\{e_1,e_2,\dots ,e_n\}\in E$. For any edge $e\in E$, we define its terminal vertex as $t\left(e\right)\in V$ and the source vertex as $b\left(e\right)\in V$. We assume the edges $e\in E$ are smooth curves that are equipped with the natural length coordinate x. Each edge is given a length l. Thus, the 1D coordinates on each edge range from $x=0$ at $b\left(e\right)$ to $x=l$ at $t\left(e\right)$. The vertices and edges together define a metric graph $\Gamma$, a particular flow network in which a metric (often edge length) is defined on the edges [28].

The Helmholtz eigenvalue problem on the metric graph is given by

$$-\tilde{\Delta }U=k^{2}U$$

(1)

where $-\tilde{\Delta }$ is the generalized Laplacian, i.e., the standard Laplacian on each edge together with Kirchhoff junction conditions

$$\left\{\begin{array}{c}U\mathrm{is}\mathrm{continuous}\mathrm{on}\Gamma \hfill \\ \mathrm{At}\mathrm{each}\mathrm{vertex}v,\sum _{j=1}^{deg\left(v\right)}{u_j}_x=0\hfill \end{array}\right.$$

where ${u_j}_x$ represents the outgoing flux for edge $e_j$ consistent with the arbitrarily-defined edge orientation. A solution in terms of Fourier harmonics on each edge $e_j$ is

$$u_j\left(x\right)=A_{j}sin\left(kx\right)+B_{j}cos\left(kx\right)$$

An energy measure based on the $L_2$ norm ratio was developed in [29]. For a given k (which is real, non-negative, and countably infinite), the energy density on edge $e_j$ of length $l_j$ is given by

$$E_k\left(e_j\right)={\displaystyle \frac{1}{l_j}}{\int }_0^l{\left(A_{j}sin\left(kx\right)+B_{j}cos\left(kx\right)\right)}^{2}dx$$

(2)

Consequently, the Helmholtz equation is a good choice for the subsequent analysis because its solutions form an orthonormal basis for the metric graph.

3. Vertex Centrality Measures

In the study of complex networks, vertex centrality measures are used to identify the most influential vertices within the network. Specifically, a centrality measure, $C:V\to \mathbb{R}$, is a function that assigns a numerical value to each vertex within the network, where a vertex with a higher centrality value is usually considered more influential than the other vertices. Although many centrality measures have been proposed to rank vertices within a network according to their level of influence, the well-known measures for characterizing the centrality of a vertex include the degree [5], eigenvector [6], closeness [7], and betweenness [8] centralities.

The degree centrality (DC) of a vertex quantifies the number of connections a vertex has to other vertices within the network. In an unweighted network, the degree centrality measures the number of edges each vertex has. For weighted networks, the measure considers the sum of the edge weights incident on the vertex. In both cases, vertices with higher degree centrality values have more or stronger connections with other vertices.

The eigenvector centrality (EC) measures the importance of vertices based on the quality of their connections rather than on the quantity. In an unweighted network, the eigenvector centrality quantifies the influence of a vertex based on both its degree and the degree of the vertices of its connections. In weighted networks, importance is measured by the strength of the vertex as well as the strength of its connections. In both cases, higher eigenvector centrality values indicate vertices are connected to other influential vertices within the network.

The closeness centrality (CC) measures how “near” a vertex is to all other vertices within a network. In unweighted networks, the closeness centrality measures the shortest distance from a vertex to all other vertices within the network (where distance is the number of edges in the shortest path between a vertex pair). In a weighted network, the measure is defined similarly but where distance is the sum of the weights of edges in the shortest path. In both cases, higher closeness centrality values indicate the ability of vertices to reach other vertices efficiently within the network.

The betweenness centrality (BC) quantifies the importance of a vertex based on the extent to which it lies on the shortest paths between other vertices [30]. For unweighted networks, the betweenness centrality calculates the fraction of shortest paths between a pair of vertices that pass through a particular vertex and sums these fractions over all vertex pairs [31]. In weighted networks, the measure is defined similarly but takes the weights of the edges along the shortest paths into account. In both cases, vertices with higher betweenness centrality values have a critical role in facilitating interactions within the network.

While the aforementioned measures identify the centrality of a vertex based upon structural features of the network, there are centralities defined that explicitly account for the drop in energy after the removal of a vertex (e.g., Laplacian centrality [32], X-degree and X-non-backtracking [33], entropy variation [34], and efficiency centrality [35]) or assume the existence of flow within the network (e.g., current-flow closeness and current-flow betweenness [36], flow betweenness [37], and path-transfer [38]). Although not an exhaustive list, a few of these measures are discussed in more detail below.

The Laplacian centrality (LC) measures the importance of a vertex by the drop in the network’s Laplacian energy when it is removed from the network (where the Laplacian energy of a network is calculated as the sum of the squared eigenvalues of its Laplacian matrix). Although defined for weighted networks, Qi et al. [32] demonstrates the utility of this centrality on an unweighted graph.

The entropy variation centrality (EVC) defines the importance of a vertex as the difference in entropy before and after its removal from the network (where entropy is often utilized to characterize the amount of information encoded in the network structure [34]). While defined for directed networks, Ai [34] demonstrates the calculation of the entropy variation using in-degree, out-degree, all-degree, and betweenness as information functions.

The current-flow closeness centrality (CFCC) of a vertex is defined as the reciprocal of the mean distance of the vertex from the other vertices within the network [39]. In this context, the distance between two vertices is defined as the potential difference of vertices when a unit of current is injected into one vertex and removed from another. Brandes and Fleischer [36] present this measure for undirected networks with positive edge weights.

The flow betweenness centrality (FBC) of a vertex is defined as the amount of flow through it when the maximum flow is transmitted from a source vertex to a target vertex, averaged over all source and target vertices [30]. Freeman et al. [37] define the measure for both unweighted and weighted networks and consider all possible paths, not just the shortest paths, in its calculation.

Edge Centrality Measures

A related problem is that of measuring the influence of an edge within a network. While some vertex centrality measures extend to edges in a straightforward manner (e.g., betweenness), many measures do not [40]. Consequently, an edge centrality measure is often defined by applying the corresponding vertex centrality to the line graph of the network [41]. Given an undirected graph $G=(V,E)$, the line graph $L\left(G\right)=(V^{\prime },E^{\prime })$ is a dual representation of G where each vertex $e_k\in V^{\prime }$ corresponds to an edge $(v_i,v_j)=e_k\in E$, and there exists an edge between two vertices of $L\left(G\right)$ if and only if the corresponding edges of G share a vertex [41]. In an undirected line graph, multiple edges between two vertices can occur if the corresponding edges in the original undirected graph are parallel and they are incident on the same vertex. Additionally if G is weighted, then the edges of $L\left(G\right)$ can inherit weights based on some function of the edge weights of G. As an illustration, the line graphs of two metric graphs are provided in Figure 3 and Figure 4.

By applying the vertex centrality on $L\left(G\right)$, the value assigned to each vertex can be mapped to the corresponding edge of G to obtain an edge centrality measure. However, this approach produces a different ranking of influential edges than that produced through a direct definition of the edge centrality on G [42]. Despite this discrepancy, the approach is used in practice—for example, Lockhart et al. [41] compared their proposed edge centrality measure to the degree centrality applied to a line graph, and Ortiz Gaona et al. [43] used vertex centrality measures on line graphs to investigate edge importance in social networks.

While there are a few metrics defined for edge centrality (e.g., Bröhl and Lehnertz [44] extended the closeness and eigenvector vertex centralities to edges), only a handful of edge measures are codified (i.e., explicitly implemented in libraries). For example, the edge betweenness centrality (EBC) [45], defined as the sum of the fraction of all-pairs shortest paths that pass through the edge, is defined in both R’s igraph package [46] and Python’s NetworkX package [47], while the edge current-flow betweenness centrality (ECFBC) [30,36], a measure that uses an electrical current model for information spreading, is defined only in the NetworkX package. Currently, the function edge_current_flow_betweenness_centrality does not support multigraphs (i.e., a graph in which multiple edges are allowed between the same pair of vertices). Consequently, we translated the source code from Python to R to take advantage of R’s support for multigraphs. To verify equivalency, we tested the centrality results on a weighted simple graph.

4. Edge Centrality Measure for Metric Graphs

Unfortunately, most traditional edge centrality measures (e.g., edge betweenness) are not sufficient for flow networks such as metric graphs since they do not account for the dynamics on the edges governed by the accompanying ordinary differential equations (ODEs) or partial differential equations (PDEs). Since these flow dynamics describe quantities such as the propagation of wave signals, electrical currents, and traffic patterns through the network, ignoring them can result in misleading assessments of edge importance and overall network functionality.

Consequently, we propose an edge centrality measure for metric graphs that identifies influential edges whose removal significantly changes the energy flow on the network. The measure uses statistically significant p-values from the two-sample Kolmogorov–Smirnov (KS) test applied to the cumulative distribution functions (CDFs) of normalized average edge densities (i.e., the energy flow) for all eigenvalues $k\le M$ (for some real number M). Specifically, it compares the CDF of the original metric graph $\Gamma$ to that of $\Gamma -e_j$, the metric graph obtained by removing edge $e_j$. Although we apply the KS test in this work, any statistical test that compares the CDF of two samples (e.g., the Cramér–von Mises [48] or Anderson–Darling [49] test) may be used in its place where the selection of the appropriate test depends upon the research question at hand.

Fundamentally, the two-sample KS test answers the question “What is the likelihood that the two samples come from the same (but unknown) probability distribution?” by reporting the absolute maximum difference between the two empirical CDFs (i.e., the test statistic) and corresponding p-value. The set of p-values obtained from the KS test applied to each edge ${\left\{e_j\right\}}_{j=1}^n$ quantifies how significantly the removal of edge $e_j$ alters the overall distribution of normalized averaged edge densities in $\Gamma$. A low p-value (e.g., $p<0.05$) indicates that removing $e_j$ significantly changes the distribution of edge densities, suggesting that the edge plays an influential role in maintaining the energy flow (represented in this context by the edge energy density distribution) on $\Gamma$. Conversely, a high p-value (e.g., $p>0.05$) suggests the removal of $e_j$ has little impact and, thus, lesser centrality.

Given a metric graph $\Gamma$, the procedure to quantify the significance of edge $e_j$ on the distribution of normalized averaged edge densities is outlined below.

• Create a second metric graph, $\Gamma -e_j$, by removing edge $e_j$ from $\Gamma$.
• Calculate the eigenvalues of the Helmholtz equation for $k\le M$ (see Equation (1)) for both $\Gamma$ and $\Gamma -e_j$ in increasing order and their corresponding eigenvectors using the algorithm provided in [26].
• Calculate the energy density (see Equation (2)), ${\displaystyle E_k\left(e_j\right)=\frac{1}{l_j}\int u_j{(k,x)}^{2}dx}$, for each edge in each metric graph.
• Compute the average edge density, ${\displaystyle \overline{E}\left(e_j\right)=\frac{1}{N}\sum _k{E}_k\left(e_j\right)}$, for each edge for each metric graph.
• Normalize the average edge density for each edge for each metric graph by the total energy, ${\displaystyle \sum _j\overline{E}\left(e_j\right)}$.
• 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 $\alpha$; if $p<\alpha$, then edge $e_j$ is concluded to have an influential role in the energy density distribution of $\Gamma$.

To generate a set of p-values, one corresponding to each edge in $\Gamma$, repeat Steps 1–7. Refer to Appendix A for a discussion of the computational complexity of this procedure.

5. Results

The proposed edge centrality measure is applied to two undirected and weighted metric graphs, the G14 network and the approximate road network in Poland. Each metric graph is a multigraph, meaning that multiple edges may exist between the same pair of vertices; this structure distinguishes our setting from most comparative analyses that assume simple graphs.

The results of our proposed edge measure are compared against those of the well-known vertex centrality measures (degree, eigenvector, closeness, and betweenness), vertex removal-based centrality measures (Laplacian and entropy variation), and flow-based vertex centrality measures (current-flow closeness and flow betweenness), all applied to their corresponding undirected and weighted line graphs. In addition, we include two edge centrality measures (edge betweenness and edge current-flow betweenness) applied directly to the metric graphs. While more relevant centrality measures exist in the literature, the alternative centrality measures in this work are defined as degree, eigenvector, closeness, betweenness, edge betweenness, Laplacian, entropy variation, current-flow closeness, flow betweenness, and edge current-flow betweenness. The alternative centrality measures considered in this work serve as baseline comparisons, as they are codified in standard libraries (e.g., R’s igraph and centiserve, Python’s NetworkX) and widely used in practice. Their inclusion, therefore, provides a practical benchmark against which the performance of our proposed edge measure can be evaluated.

In both examples, each edge in the line graph is weighted by the mean of the normalized average edge densities of the corresponding metric graph. For example, the edge connecting vertices e12 and e13 in the line graph in Figure 3 has a weight defined as the mean of the densities on edges e12 and e13 in the metric graph in Figure 1. While there is no single standardized method for assigning weights to edges in line graphs derived from weighted graphs, our approach aims to preserve flow-related characteristics from the original graph within its line graph representation. Since edges in a line graph represent adjacency (i.e., shared vertices) between edges in the original graph, assigning weights based on the average density of the connected edges allows the line graph to encode information about flow in a manner consistent with the original graph. Alternative weighting schemes, such as using the minimum, maximum, or harmonic mean of adjacent edge densities, or defining weights based on spectral similarity or correlated modal energy, could emphasize different flow dynamics or local edge interactions. We note, however, that our choice of the arithmetic mean may introduce bias in how influence is distributed across the line graph compared to the original metric graph.

We evaluated our proposed measure by comparing its edge rankings for each metric graph to those of the alternative centrality measures through two experiments identifying edges whose removal significantly changed the energy flow on the network. In the first experiment, we examined the effect of removing individual ranked edges. In the second experiment, we examined the cumulative effect of removing ranked edges. We measured the effect in each experiment with the two-sample KS test by comparing the normalized average edge density distribution of the original metric graph against the modified metric graph whose ranked edges were removed. The removal of edges with higher KS test statistic values indicates a greater difference between distributions and, consequently, a larger impact on the metric graph’s energy flow.

The code for the proposed edge centrality measure for metric graphs, along with the calculation of the alternative centrality measures, were performed in R using version 2.0.3 of the igraph package, version 1.0.0 of the centiserve package [50], and version 2.8.0 of the sna package [51]. The laplacian function in R does not support weighted graphs; therefore, we modified the source code accordingly for the analysis presented in this work. The calculation of the eigenvalues and edge densities for each metric graph was performed using version 9.13 of MATLAB [52].

5.1. G14 Network

We first consider a power grid network adapted from the IEEE 14-bus system, a benchmark test for the American Electric Power grid in February 1962 [53] that consists of 14 vertices (buses that are specific locations in the power grid) and 18 edges (transformers connecting two vertices in the power grid). For the purposes of generating a model graph, the network was placed arbitrarily on a Cartesian grid, and the coordinates of the vertices were approximated. The length of each edge was then computed using the Euclidean distance between vertices. Vertices of degree 2 were then dissolved into their corresponding edges to create a simpler but equivalent system [28], and each edge was given an arbitrary orientation. When multiple edges were combined into one, their distances were added together. This modified graph is referred to as the G14 network (Figure 1), and its edge lengths are detailed in

Table 1. The length $l_j$ of each edge $e_j$ in the G14 network. All values are rounded to the nearest two significant figures.

Edge ${\mathit{e}}_{\mathit{j}}$	$e_1$	$e_2$	$e_3$	$e_4$	$e_5$
Adjacent Vertices	$v_1,v_3$	$v_1,v_2$	$v_1,v_2$	$v_1,v_3$	$v_2,v_3$
Length ${\mathit{l}}_{\mathit{j}}$	11.91	7.08	6.00	2.24	4.12
Edge ${\mathit{e}}_{\mathit{j}}$	$e_6$	$e_7$	$e_8$	$e_9$	$e_{10}$
Adjacent Vertices	$v_2,v_5$	$v_2,v_7$	$v_3,v_4$	$v_4,v_7$	$v_4,v_8$
Length ${\mathit{l}}_{\mathit{j}}$	1.41	2.00	1.00	4.72	4.47
Edge ${\mathit{e}}_{\mathit{j}}$	$e_{11}$	$e_{12}$	$e_{13}$	$e_{14}$
Adjacent Vertices	$v_4,v_8$	$v_5,v_6$	$v_5,v_7$	$v_7,v_8$
Length ${\mathit{l}}_{\mathit{j}}$	2.00	2.00	1.41	4.47

.

The results of the proposed edge centrality measure (labeled as KS) applied to the metric graph of G14 for $k\le 40$ and the alternative centrality measures—eight vertex measures applied to its weighted line graph (Figure 3) and two edge measures applied directly to the metric graph—are displayed in

Table 2. Results of the proposed edge centrality measure (KS) on the G14 network, along with those of the alternative centrality measures. All values are rounded to three significant figures.

Edge ${\mathit{e}}_{\mathit{j}}$	$e_1$	$e_2$	$e_3$	$e_4$	$e_5$
KS  $\mathit{p}$-value	$0.049$	$0.146$	$0.040$	$0.146$	$0.146$
Degree	$0.522$	$0.579$	$0.579$	$0.522$	$0.541$
Eigenvector	$0.938$	$1.000$	$1.000$	$0.938$	$0.881$
Closeness	$0.549$	$0.618$	$0.618$	$0.549$	$0.738$
Betweenness	$0.000$	$0.000$	$0.000$	$0.000$	$9.000$
Edge Betweenness	$0.000$	$0.000$	$4.000$	$3.000$	$3.000$
Laplacian	$6.107$	$7.005$	$7.005$	$6.107$	$7.870$
Entropy Variation	0.083	0.066	0.066	0.083	0.001
Current-Flow Closeness	$5.117$	$3.070$	$3.070$	$2.874$	$3.070$
Flow Betweenness	$30.000$	$25.000$	$25.000$	$30.000$	$25.000$
Edge Current-Flow Betweenness	$1.833$	$1.929$	$1.850$	$1.581$	$2.063$
Edge ${\mathit{e}}_{\mathit{j}}$	$e_6$	$e_7$	$e_8$	$e_9$	$e_{10}$
KS  $\mathit{p}$-value	$0.049$	$0.040$	$0.446$	$0.316$	$0.146$
Degree	$0.401$	$0.480$	$0.435$	$0.404$	$0.340$
Eigenvector	$0.534$	$0.587$	$0.517$	$0.256$	$0.184$
Closeness	$0.674$	$0.761$	$0.676$	$0.729$	$0.562$
Betweenness	$8.000$	$10.000$	$12.000$	$5.000$	$1.500$
Edge Betweenness	$6.000$	$4.000$	$5.000$	$5.000$	$0.000$
Laplacian	$5.791$	$7.014$	$5.962$	$5.053$	$3.482$
Entropy Variation	0.016	0.003	0.002	0.003	0.105
Current-Flow Closeness	$2.880$	$3.072$	$2.876$	$2.879$	$2.644$
Flow Betweenness	$40.000$	$42.000$	$60.000$	$48.000$	$47.000$
Edge Current-Flow Betweenness	$3.948$	$3.007$	$3.821$	$2.515$	$1.649$
Edge  ${\mathit{e}}_{\mathit{j}}$	$e_{11}$	$e_{12}$	$e_{13}$	$e_{14}$
KS  $\mathit{p}$-value	$0.146$	$0.049$	$0.049$	$0.316$
Degree	$0.340$	$0.122$	$0.314$	$0.334$
Eigenvector	$0.184$	$0.092$	$0.211$	$0.192$
Closeness	$0.562$	$0.526$	$0.651$	$0.637$
Betweenness	$1.500$	$0.000$	$10.000$	$12.000$
Edge Betweenness	$3.000$	$7.000$	$6.000$	$4.000$
Laplacian	$3.482$	$1.567$	$3.894$	$4.201$
Entropy Variation	0.105	0.001	0.019	0.002
Current-Flow Closeness	$2.644$	$1.526$	$2.647$	$2.644$
Flow Betweenness	$47.000$	$20.000$	$57.000$	$48.000$
Edge Current-Flow Betweenness	$1.602$	$3.500$	$3.852$	$2.724$

. Using a significance level of $\alpha =0.05$, the proposed edge centrality measure identified edges 3, 7, 1, 6, 12, and 13 as significant within the G14 network (see Figure 5, which illustrates the empirical CDF of the normalized average edge densities when a significant edge (edge 3) is removed from the graph compared to the removal of a non-significant edge (edge 8), as identified by the proposed measure). In this work, ties in a ranking are broken numerically. For example, although edges 3 and 7 have the same p-value, edge 3 is ranked ahead of edge 7. The top five ranked edges by the alternative centrality measures are provided in

Table 3. The top five ranked edges by the different centrality measures for the G14 network.

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

.

Figure 6 shows the correlation matrix of the edge rankings between the different centralities. The proposed edge measure (KS) is not strongly correlated with any of the alternative centralities, although strong correlations exist elsewhere (e.g., degree with eigenvector, Laplacian and current-flow closeness centralities, respectively).

In the first experiment, each ranked edge was individually removed from the G14 network. As seen in Figure 7, the proposed edge measure (KS) produced an optimal ranking of edges whose removal significantly changed the energy density distribution of the graph. This result is expected since the construction of the proposed edge measure centers around the two-sample KS test. Although the rankings of the other measures identified high-impact edges on the energy flow, their rankings were more variable.

In the second experiment, the top five ranked edges were cumulatively removed from the G14 network. As seen in Figure 8, the proposed edge measure (KS) consistently yielded high KS test statistics as top-ranked edges were cumulatively removed. Thus, the proposed measure was effective at identifying edges whose cumulative removal significantly altered the edge density distribution in the G14 network. Among the traditional vertex centralities applied to the weighted line graph, the degree centrality (DC) and eigenvector centrality (EC) demonstrated strong performances. DC, in particular, led to sharp increases in the KS test statistic by the fourth and fifth cumulative edge removals. For the vertex-removal-based measures applied to the weighted line graph, both the entropy variation centrality (EVC) and Laplacian centrality (LC) reached high KS test statistics after all five top-ranked edges were removed. Among the flow-based measures applied to the weighted line graph, the current-flow closeness centrality (CFCC) achieved the highest KS test statistic after the top four ranked edges were removed. Lastly, among the edge-based measures applied directly to G14, the edge betweenness centrality (EBC) showed a steady increase in the KS test statistic as more ranked edges were cumulatively removed, though it still underperformed relative to the proposed measure.

5.2. Road Network

Now we consider a more complicated network—an approximation to the road network in Poland (Figure 2)—with edge lengths given in

Table 4. The length $l_j$ of each edge $e_j$ in the road network. All values are rounded to the nearest two significant figures.

Edge ${\mathit{e}}_{\mathit{j}}$	$e_1$	$e_2$	$e_3$	$e_4$	$e_5$
Adjacent Vertices	$v_1,v_2$	$v_1,v_7$	$v_2,v_3$	$v_3,v_7$	$v_3,v_4$
Length  ${\mathit{l}}_{\mathit{j}}$	1.00	1.00	1.69	1.71	0.69
Edge  ${\mathit{e}}_{\mathit{j}}$	$e_6$	$e_7$	$e_8$	$e_9$	$e_{10}$
Adjacent Vertices	$v_4,v_6$	$v_4,v_5$	$v_5,v_6$	$v_6,v_8$	$v_8,v_9$
Length  ${\mathit{l}}_{\mathit{j}}$	1.72	2.25	2.11	0.93	2.86
Edge  ${\mathit{e}}_{\mathit{j}}$	$e_{11}$	$e_{12}$	$e_{13}$	$e_{14}$	$e_{15}$
Adjacent Vertices	$v_2,v_7$	$v_7,v_8$	$v_7,v_9$	$v_2,v_{10}$	$v_2,v_{11}$
Length  ${\mathit{l}}_{\mathit{j}}$	1.00	1.18	2.95	1.40	0.96
Edge  ${\mathit{e}}_{\mathit{j}}$	$e_{16}$	$e_{17}$	$e_{18}$	$e_{19}$	$e_{20}$
Adjacent Vertices	$v_3,v_{12}$	$v_4,v_{13}$	$v_5,v_{14}$	$v_8,v_{16}$	$v_1,v_9$
Length  ${\mathit{l}}_{\mathit{j}}$	0.56	0.75	1.16	1.75	5.34
Edge  ${\mathit{e}}_{\mathit{j}}$	$e_{21}$	$e_{22}$	$e_{23}$
Adjacent Vertices	$v_1,v_9$	$v_8,v_{15}$	$v_8,v_{17}$
Length  ${\mathit{l}}_{\mathit{j}}$	2.46	2.82	2.70

. The edge length varies for each edge, roughly proportional to the distance between the cities they represent, with a few exceptions: an equilateral triangle with edges $e_1,e_2,$ and $e_{11}$ was assigned edge lengths of 1 (the metric graph edge lengths need not measure geographic distance). This network is a simplification of the network developed in [27] with vertices of degree 2 absorbed into their adjacent edges.

The results of the proposed edge centrality measure (KS) applied to the metric graph of the road network and the alternative centrality measures—eight vertex measures applied to its weighted line graph (Figure 4) and two edge measures applied directly to the metric graph—are displayed in

Table 5. Results of the proposed edge centrality measure (KS) on the road network, along with those of the alternative centrality measures. All values are rounded to three significant figures.

Edge ${\mathit{e}}_{\mathit{j}}$	$e_1$	$e_2$	$e_3$	$e_4$	$e_5$	$e_6$
KS  $\mathit{p}$-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  ${\mathit{e}}_{\mathit{j}}$	$e_7$	$e_8$	$e_9$	$e_{10}$	$e_{11}$	$e_{12}$
KS  $\mathit{p}$-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  ${\mathit{e}}_{\mathit{j}}$	$e_{13}$	$e_{14}$	$e_{15}$	$e_{16}$	$e_{17}$	$e_{18}$
KS  $\mathit{p}$-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  ${\mathit{e}}_{\mathit{j}}$	$e_{19}$	$e_{20}$	$e_{21}$	$e_{22}$	$e_{23}$
KS  $\mathit{p}$-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

. Using a significance level of $\alpha =0.05$, the proposed edge centrality measure identified edges 2, 11, 15, 20, 21, 5, 16, and 22 as significant within the road network (see Figure 9, which illustrates the empirical CDF of the normalized average edge densities when a significant edge (edge 2) is removed from the graph compared to the removal of a non-significant edge (edge 9), as identified by the proposed measure). The top three ranked edges by the alternative centrality measures are provided in

Table 6. The top three ranked edges by the different centrality measures for the road network.

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

.

Figure 10 shows the correlation matrix of the edge rankings between the different centralities. As with the rankings for the G14 network, the proposed edge measure (KS) is not strongly correlated with any of the alternative centralities, although strong correlations exist elsewhere.

In the first experiment, each ranked edge was individually removed from the road network. As expected (see Figure 11), the proposed edge measure (KS) produced an optimal ranking of edges whose removal significantly changed the energy density distribution of the graph. Although the rankings of the other measures identified high-impact edges on the energy flow, their rankings were more variable.

In the second experiment, the top three ranked edges were cumulatively removed from the road network. As shown in Figure 12, the proposed edge measure (KS) consistently yielded high KS test statistics as top-ranked edges were cumulatively removed. Thus, the proposed measure was effective at identifying important edges in the road network whose cumulative removal produced a measurable disruption to the edge density distribution. Among the traditional vertex centralities applied to the weighted line graph, the degree centrality (DC) and eigenvector centrality (EC) mirrored the KS test statistics of the proposed measure when the first two ranked edges were removed, yet dropped upon removing the top three ranked edges. For the vertex-removal-based measures applied to the weighted line graph, the entropy variation centrality (EVC) achieved high KS test statistics as each ranked edge was cumulatively removed. Among the flow-based measures applied to the weighted line graph, neither the current-flow closeness centrality (CFCC) nor the flow betweenness centrality (FBC) performed as well as the other methods. Finally, among the edge-based measures applied directly to the road network, both the edge betweenness centrality (EBC) and the edge current-flow betweenness centrality (ECFBC) performed strongly, exhibiting steadily increasing KS test statistics as each of the three ranked edges were cumulatively removed.

6. Conclusions

We introduced a new edge centrality measure for metric graphs based on the energy density of the eigenvectors of the Helmholtz equation. The utility of this proposed measure is in the identification of edges whose removal significantly change the energy flow (represented in this context by the distribution of edge densities) on the network. A natural application of such a measure, for example, is the identification of such edges in an electrical network to avoid a power outage or to maintain the particular lasing patterns of a random network laser [54].

The proposed edge centrality measure offers several advantages. First, the measure is defined specifically for edges; this avoids the necessity of applying a vertex centrality to a line graph, which produces a different ranking of influential edges than that produced through a direct definition of an edge centrality measure. Second, the measure considers the dynamics on the edges governed by the accompanying ODEs or PDEs which can provide a more comprehensive assessment of edge importance and overall network functionality. Third, the measure uses p-values from the KS test to determine whether the removal of an edge and subsequent change in energy flow is statistically significant rather than being due to random variations. Fourth, the measure is scale invariant and, thus, less sensitive to differences in network size or scaling effects since the KS test is based on empirical distributions. Lastly, the measure provides a global perspective, which is crucial for identifying edges that are influential but may not necessarily have high centrality in terms of degree or other measure.

We demonstrated the proposed edge centrality measure on two metric graphs, a power grid network and road network, and compared the results to eight vertex centrality measures applied to a line graph representation of each metric graph, as well as with two edge centrality measures applied directly to each metric graph. The edge rankings from the vertex and edge centrality measures differ markedly from those of the proposed edge measure due to fundamental differences in their underlying principles. Unlike the vertex and edge centrality measures, which assess importance based on network topology or flow dynamics, the proposed edge centrality measure directly captures how energy propagates along edges using the spectral properties of the Helmholtz equation. This distinction allows it to identify critical edges based on their role in sustaining network-wide energy distribution, rather than relying on structural proxies derived from a line graph transformation.

We assessed our proposed measure by comparing the ranking of the edges in each metric graph against those determined by the vertex and edge centrality measures (referred to here as the alternative measures) through two experiments. The first experiment, which removed individual ranked edges from the metric graph, demonstrated that the proposed measure identified edges that cause a significant change in the energy distribution, while the alternative measures showed variability in their rankings. The second experiment, where multiple top-ranked edges were cumulatively removed, also showed that the proposed measure identified edges that significantly impacted the energy distribution. Overall, the experiments indicated that the proposed edge centrality measure is an effective tool for identifying influential edges in metric graphs, a particular flow network, whose removal significantly changes the energy distribution.

Future work will investigate the relationship between the proposed edge measure and other centrality measures that are not currently implemented in standard libraries, including spectral-based centralities, vertex and edge removal-based centralities, flow-based centralities, and vitality-based edge centralities. We also plan to explore alternative line graph weighting strategies to assess how different weighting choices influence centrality rankings and their sensitivity to flow dynamics.