On a Notion of Graph Centrality Based on $L_{1}$ Data Depth

A fundamental and indispensable tool for analyzing graph data is a graph centrality measure, which evaluates the prominence of vertices in a given graph structure (Sabidussi, 1966; Freeman, 1978). Several measures have been suggested based on different concepts of what constitutes a ‘central’ vertex (see, e.g., Borgatti and Everett, 2006; Rodrigues, 2019). The concept of centrality has been developed primarily in social network analysis and, more broadly, in the field of social sciences (Wasserman and Faust, 1994).

Meanwhile, in statistics, the concept of data depth for multivariate data has been extensively studied since Tukey (1975). Data depth is a multivariate generalization of a univariate rank, but in a center-outward manner, starting from the deepest point(s) and extending towards the outer points. Notable instances of data depth include half-space depth (Tukey, 1975), simplical depth (Liu, 1990), projection depth (Zuo and Serfling, 2000a), and $L_{1}$ data depth (Vardi and Zhang, 2000), among others. See Zuo and Serfling (2000a); Mosler and Mozharovskyi (2022) for an extensive review. The data depth is a robust and nonparametric analytic tool for multivariate data. It is known for its powerful usage in robust estimation (Zuo et al., 2004), regression analysis (Rousseeuw and Hubert, 1999), and functional data analysis (López-Pintado and Romo, 2009; Sun and Genton, 2011).

Therefore, both graph centrality and data depth have a surprisingly comparable idea in that they measure the degree of ‘centralness’ of a point or a vertex w.r.t. a given set of data. Nevertheless, there are a limited number of studies that establish a connection between these two fields of research. Aamari et al. (2021) developed a data depth function by generating a neighborhood graph based on the provided multivariate data and calculating the centrality of that graph. Cerdeira and Silva (2021) introduced a new centrality measure, termed Tukey centrality, which leverages the popular concept of half-space depth. However, as indicated in the latter study, the computation of Tukey centrality is NP-hard, so there is no reason to prefer it over the existing ones. To our knowledge, these are the only works connecting the two fields.

The aim of this paper is to introduce a new centrality measure, called $L_{1}$ centrality, for vertices of an undirected and connected graph, analogous to the $L_{1}$ data depth of Vardi and Zhang (2000). This measure can handle graphs with weights assigned to both vertices and edges in a straightforward manner. Nevertheless, the computation of this measure is simple and does not entail a higher computational cost than the existing centrality measures, unlike the approach proposed by Cerdeira and Silva (2021). Furthermore, due to the connection between our work and the notion of data depth for multivariate data, there are several practical possibilities for expanding the proposed measure based on the extensive literature on data depth. For instance, the $L_{1}$ centrality-based neighborhood and the local $L_{1}$ centrality introduced in this study are closely related to the concept of local depth in Paindaveine and Van Bever (2013). In addition, we provide various graphical and multiscale analysis tools that rely on the $L_{1}$ centrality. The tools are based on an insightful interpretation of the suggested measure (Remark 1), distinguishing our centrality measure from others. Through these tools, we demonstrate that even the basic concept of centrality has a wide range of practical uses.

The rest of this paper is organized as follows. The $L_{1}$ centrality measure is defined in Section 2, and its properties are discussed, along with comparisons to the existing measures. Section 3 presents a visualization tool, the target plot, that utilizes the $L_{1}$ centrality. Section 4 expands upon the $L_{1}$ centrality measure using a multiscale approach. The $L_{1}$ centrality-based neighborhood of a vertex is precisely defined and utilized to derive the local $L_{1}$ centrality. The method for representing edges at multiple scales is also described. In Section 5, we propose a group heterogeneity index to analyze graph data effectively at the group level. In Sections 2–5, the Marvel Cinematic Universe movie network is utilized to aid our explanation. Then, in Section 6, we consider the bill cosponsorship network of 317 members in the 21st National Assembly of South Korea, which effectively demonstrates the practicality of the tools developed in the study. Finally, Section 7 provides concluding remarks with further discussion. All technical proofs are in the Appendix.

All methods and data sets used in this paper are provided via the R package L1centrality (Kang and Oh, 2024), available from the Comprehensive R Archive Network. Also, the codes for reproducing the figures and analysis in this paper are available from https://github.com/seungwoo-stat/L1centrality-paper.

In this section, we propose a novel graph visualization tool that aims to embed a graph onto a two-dimensional plane in a way that effectively represents the structural information of each vertex, specifically the $L_{1}$ centrality. As in the previous section, we examine a graph with $n$ vertices that is undirected, connected, and has weighted edges. However, we impose a constraint that all vertices have the same multiplicity, meaning that we exclusively examine vertex unweighted graphs. The reason for this restriction will be explained shortly.

Various graph drawing approaches are available for representing graphs on a two-dimensional plane, and each approach has a distinct perspective on what constitutes a visually pleasing representation. The algorithms proposed by Kamada and Kawai (1989) and Fruchterman and Reingold (1991) are often used in this domain. However, to our knowledge, none of the existing algorithms take graph centrality into account when deciding where to place points in their plot. Typical methods do not place the most central vertex in the center of the plot, nor do they put the least central vertex in the periphery.

In light of Remark 1, it is intuitive to position the graph median at the central point and the remaining vertices distributed around concentric circles whose radii correspond to their $L_{1}$ centrality. A graph is visually represented by the plot in a way that effectively conveys the structural characteristics of each vertex’s centrality. In pursuit of this objective, we present a method of plotting graphs called target plot.

1. 1.

   We aim to determine a configuration of $n$ points on a two-dimensional plane, with each vertex represented as an individual point. Let $\mathbf{x}_{i}\in\mathbb{R}^{2}$ denote the point corresponding to vertex $v_{i}$ ($i=1,\dots,n$). In circular coordinates, this point is denoted as $(r_{i},\theta_{i})\in[0,\infty)\times[0,2\pi)$. The target plot constraints $\mathbf{x}_{i}$ to lie on a concentric circle with radii $r_{i}=-\log C(v_{i})$, so the graph median is placed at the center of the circles. The log transformation converts $L_{1}$ centralities in the interval $(0,1]$ to a radius in the interval $[0,\infty)$. This transformation also reflects the observation that a vertex distant from all other vertices is likely to have a low $L_{1}$ centrality, according to equation (2.1) and Theorem 1 (P4). However, this observation may not be valid if there are distinct multiplicities on the vertices, i.e., a vertex far from all other vertices may exhibit high $L_{1}$ centrality if it has a high degree of multiplicity due to Theorem 1 (P3). For this reason, in this section, we focus only on unweighted vertex graphs.

2. 2.

   Next, we use the nonmetric multidimensional scaling method (nonmetric MDS; Kruskal, 1964a, b) with the constraint specified in 1. In essence, the goal is to find a configuration that retains the geodesic distances on the graph as much as possible while the constraint is enforced. Specifically, we aim to find a configuration of points that minimizes a metric called stress, adopted from Kruskal (1964a):

   $\displaystyle S$

   $\displaystyle\coloneqq\sqrt{\frac{S^{*}}{T^{*}}},\quad\text{where}\quad S^{*}\coloneqq\sum_{i,j}(d_{ij}-\widehat{d}_{ij})^{2},\quad T^{*}\coloneqq\sum_{i,j}d_{ij}^{2}.$

   Here, $d_{ij}\coloneqq\|\mathbf{x}_{i}-\mathbf{x}_{j}\|$, where $\|\cdot\|$ denotes the usual Euclidean norm, and $\widehat{d}_{ij}$ are chosen to minimize the stress metric given $d_{ij}$, subject to the following monotonicity condition:

   $\displaystyle\begin{cases}\text{whenever }d(v_{i},v_{j})<d(v_{k},v_{l}),&\text{then }\widehat{d}_{ij}\leq\widehat{d}_{kl},\\ \text{whenever }d(v_{i},v_{j})=d(v_{k},v_{l}),&\text{then }\widehat{d}_{ij}=\widehat{d}_{kl}.\end{cases}$

   There is a fast and efficient algorithm for fitting $\widehat{d}_{ij}$. See Kruskal (1964b) for details.

   A starting configuration is set using the classical MDS (Mardia, 1978). If $v_{i}$ is the graph median, $\mathbf{x}_{i}=\mathbf{0}$; otherwise, $\mathbf{x}_{i}$ is set to $r_{i}(\mathbf{y}_{i}-\mathbf{y}_{*})/\|\mathbf{y}_{i}-\mathbf{y}_{*}\|$, where $\mathbf{y}_{i}\in\mathbb{R}^{2}$ is the point representing $v_{i}$ as a result of classical MDS, and $v_{*}$ is the graph median with $\mathbf{y}_{*}$ being the corresponding point derived from classical MDS. Given the configuration, it can be readily shown that

   $\displaystyle\frac{\partial S}{\partial\theta_{i}}$

   $\displaystyle=\sqrt{\frac{T^{*}}{S^{*}}}\frac{1}{T^{*}}\sum_{j\neq i}r_{i}r_{j}\sin(\theta_{i}-\theta_{j})\left(1-\frac{S^{*}}{T^{*}}-\frac{\widehat{d}_{ij}}{d_{ij}}\right).$

   The gradient descent method is employed to iteratively update the value of $\theta_{i}$ until convergence. Specifically, denoting $\boldsymbol{\theta}\coloneqq(\theta_{1},\dots,\theta_{n})^{\top}$ and $\mathbf{g}\coloneqq\partial S/\partial\boldsymbol{\theta}=(\partial S/\partial\theta_{1},\ldots,\partial S/\partial\theta_{n})^{\top}$, the gradient descent method is applied in a similar way to Kruskal (1964b): $\boldsymbol{\theta}_{\text{new}}=\boldsymbol{\theta}_{\text{old}}-\alpha\frac{\mathbf{g}}{\operatorname{mag}(\mathbf{g})},$ where $\operatorname{mag}(\mathbf{g})\coloneqq\|\mathbf{g}\|/\sqrt{\sum_{i=1}^{n}r_{i}^{2}}$ (the relative magnitude of $\mathbf{g}$), and $\alpha$ is the step size with an initial value of 0.2 and is set to a smaller value by $\alpha_{\text{new}}=0.95\cdot\alpha_{\text{old}}$ in each iteration. The above process is repeated until $\operatorname{mag}(\mathbf{g})$ is small enough.

3. 3.

   From 1 and 2, we obtain a configuration $(r_{i},\theta_{i})$ representing the vertex $v_{i}$. After plotting each point on a two-dimensional plane, contours of concentric circles are drawn to assist in recognizing and understanding the graph structure. For example, in Figure 2 (b), the radii of the four concentric circles correspond to the first (25%) to fourth (maximum) quartiles of the $L_{1}$ centralities. Hence, a quarter of the points lie within the smallest circle, and these points correspond to the vertices with the highest 25% $L_{1}$ centrality. From its appearance, it is clear why we call the resulting figure a target plot. This plot can be enhanced by including the color or shape of the points to facilitate more appropriate interpretation. For instance, there might be identifiable patterns in which the least central vertices share, indicating that it is desirable to remove vertices that exhibit those patterns before further analysis. This analysis will be demonstrated in Section 6.

Before closing this section, we make three remarks. First, the concept of the target plot is not applicable to other graph centrality measures. A small problem is that the radii of each point cannot be determined using a straightforward log transformation. A crucial aspect is connected to Remark 1. For example, plotting vertices with high-degree centralities near the center of the concentric circles is utterly inadequate since it conveys the misleading impression that vertices with high-degree centralities are related. Second, the gradient descent method can converge to a local minimum of the stress measure, not an overall minimum. However, as Kruskal (1964b) pointed out, this is not a significant problem. The local minimum configuration is not the final result of the analysis, and it should be of interest if it is meaningful. Refer to the discussion in Kruskal (1964b, Section 5). Finally, the target plot resembles the depth contour, a visualization tool commonly used in data depth literature (e.g., Zuo and Serfling, 2000b). Depth contours, similar to our concentric circles, aid in identifying points with high and low data depths.

The degree centrality is a local measure considering only the vertices directly connected to a particular vertex. On the other hand, the closeness and betweenness centralities are global measures of centrality because they consider all vertices in the graph while calculating these measures. The $L_{1}$ centrality, in the same sense, is a global centrality measure. However, the structural properties of vertices can vary at both local and global scales, and all graph centrality measures, including the $L_{1}$ centrality, can only capture one of the two.

This section introduces a local extension of $L_{1}$ centrality based on the local depth proposed by Paindaveine and Van Bever (2013), where the degree of locality can be selected to a range of levels. This extension provides a multiscale view of a single graph with the centrality values at various locality levels. As described below, the local measure is derived by conditioning the graph to suitable $L_{1}$ centrality-based neighborhoods.

This section provides a visualization tool and an index representing heterogeneity in the proposed centrality measures for groups of vertices or the entire graph. The proposed visualization tool and index are important for many types of analysis. For example, it can be used to evaluate if a group or graph exhibits any peculiar structural characteristics, detect trends across several groups or graphs (e.g., graphs that change over time), or explore the relationship of the heterogeneity of the group’s vertices to other attributes of the group. Furthermore, it can be employed to compare various groups and graphs and categorize them into similar types (Badham, 2013).

Several indices have been developed to quantify the heterogeneity of centrality measures. Some of the prominent measures in this field are the concept of centralization by Freeman (1978), the variance index of Snijders (1981), and the notion of hierarchization by Coleman (1964, Chapter 14). Recently, Badham (2013) discussed using the Gini coefficient to measure heterogeneity; the author argued that it has favorable theoretical characteristics compared to the previously discussed indices.

However, the Gini coefficient has yet to be well studied as a descriptive measure of centrality heterogeneity in network science. Interestingly, the Lorenz curve and Gini coefficient have been studied in data depth. Liu et al. (1999) used the Lorenz curve and Gini coefficient to establish a metric for quantifying kurtosis in multivariate data. Given that the notion of graph centrality in this study is based on a specific data depth, the purpose of this section is to revisit the Lorenz curve and Gini coefficient as a tool for analyzing the heterogeneity of the proposed global and local $L_{1}$ centrality measures. Here is a detailed description of the benefits of using these tools for global and local $L_{1}$ centralities.

The Lorenz curve is defined as follows. Given a univariate cumulative distribution function (cdf) $F$ with a nonnegative support, it is a plot of $(p,L(p))$, with

where $F^{-1}(t)\coloneqq\inf\{v\in\mathbb{R}:F(v)\geq t\}$ and $X\sim F$ (Gastwirth, 1971). By definition, $L(p)$ is nondecreasing in $p$ and $L(p)\leq p$. Furthermore, $L(1)=1$ and $L(0)=0$. Given $m$ centrality measurements, $C_{\alpha}(v_{1}),\dots,C_{\alpha}(v_{m})\geq 0$, $F$ of (5.1) is substituted by the empirical cdf $F_{m}(v)=\frac{1}{m}\sum_{i=1}^{m}I(C_{\alpha}(v_{i})\leq v)$, where $I$ denotes the indicator function. The Lorenz curve of the centrality measurements serves as an intuitive visual indicator of heterogeneity. The greater the deviation of the curve from the diagonal $L(p)=p$, the more it indicates that the group of $m$ vertices has a diverse structural attribute in a particular graph, i.e., their $L_{1}$ centralities show a high level of heterogeneity.

The Gini coefficient, defined as twice the area between the Lorenz curve and the diagonal $L(p)=p$, is a simple and widely used metric for quantifying the level of diversity in income and wealth, making it valuable for comparing multiple distributions. When $F$ is replaced with $F_{m}$ using the measurements $C_{\alpha}(v_{1}),\dots,C_{\alpha}(v_{m})$, it can be readily verified that the Gini coefficient $G$ can be represented as

where $\bar{C}_{\alpha}\coloneqq(1/m)\sum_{i=1}^{m}C_{\alpha}(v_{i})$. It is the expected difference of the centrality measurements with a suitable normalization. This also shows that the Gini coefficient is an appropriate index for quantifying heterogeneity among centrality values. Consistent with the results of the Lorenz curve, the larger the heterogeneity index, the less similar the nodes are in terms of centrality.

There are several advantages to using the Lorenz curve and Gini coefficient to quantify the heterogeneity of the proposed $L_{1}$ centrality. First, the Lorenz curve provides a visual diagnostic tool for comparing several groups or graphs rather than directly summarizing centrality values as a single number. Second, these tools are scale-invariant. As mentioned, local $L_{1}$ centralities generally exhibit higher values than global $L_{1}$ centralities. Consequently, the proposed method can be applied to compare heterogeneity across multiple locality levels, even if the scales of $L_{1}$ centralities across several locality levels differ. Third, the Gini coefficient is inherently standardized to 0 and 1. Some indices, such as centralization and variance, require a normalization procedure that normalizes the index w.r.t. the highest feasible value for a graph with the same number of vertices (Freeman, 1978; Snijders, 1981; Coleman, 1964). Moreover, these procedures only apply to graphs without weights, and it remains uncertain how similar normalization procedures may be implemented for the edge-weighted graphs considered in this study. Thus, the normalization procedure limits the applicability of given indices to the entire graph. In contrast, the Lorenz curve and Gini coefficient can be utilized for a set of vertices, not necessarily the entire graph, with the index inherently standardized.

In this section, we focus on the network of legislators in the 21st National Assembly of South Korea (May 30th, 2020–May 29th, 2024). Our main emphasis is on analyzing the network at multiple scales, demonstrating the utility of the tools presented so far. At the time of this writing, the 21st National Assembly is still in session, so our focus will be limited to the first 40 months of the assembly, from Jun. 2020 to Sep. 2023. In South Korea, a bill (legislative proposal) must be supported by at least 10 assembly members. We call them cosponsors of the bill. In this study, we do not distinguish between a member who presents the bill (as a representative) and those who cosponsor the bill. For simplicity, the term cosponsor is generically used in this section to refer to all members involved in the proposal. Therefore, the number of cosponsored bills by a member denotes the number of bills to which the person has agreed. Similarly, the number of cosponsored bills between two members indicates the number of bills the two members jointly supported.

A commonly used approach in the social sciences to identify assembly members’ social relationships is to build a network of them using bill cosponsorship information (Fowler, 2006)In this study, we constructed a graph of 317 assembly members, each representing a single vertex. An edge is established between two members if they have cosponsored at least one bill together. The weight of this edge is defined as the reciprocal of the number of cosponsored bills between two members, so the higher the number of cosponsored bills between two members, the shorter the path length between the corresponding vertices representing these members. A graph is formed by utilizing all 25164 bills proposed over 40 months, provided by the Bill Information System of the National Assembly of South Korea (https://likms.assembly.go.kr/bill/main.do). The resulting graph is undirected and connected, with 317 vertices and 47657 edges, each edge having a weight. The multiplicities of all vertices are set to 1. We refer to this network as the assembly network for the remainder of this paper.

As of Sep. 30th, 2023, the party composition of the 317 members is as follows (for simplicity, independent members are categorized by their former political party): There are two major parties, the Democratic Party with 184 members and the People Power Party with 123 members. Additionally, one minor party, the Justice Party, has six members. There are also four small parties with only one member each: the Basic Income Party, Hope of Korea, Progressive Party, and Transition Korea. Moreover, the 317 vertices comprise two groups: those who served as 21st National Assembly members for 40 months (281) and those who did not (36). The latter group includes members who started their term through a by-election, resigned, or lost their seat for any reason during the 40 months.

Figure 7 (a) is the target plot of the assembly network, which shows that the assembly network is organized into communities by the two major political parties. Furthermore, individuals absent from the office for 40 months exhibit lower global $L_{1}$ centrality, likely because their time in the assembly is shorter than that of the others, resulting in fewer opportunities to cosponsor bills and build relationships with other members. Therefore, we eliminate these vertices for further analysis, which can be seen as analogous to trimming in traditional statistical analysis. Furthermore, we have removed two chairpersons of the 21st National Assembly with inactive involvement in cosponsoring legislation. As a result, the assembly network is reduced to a subgraph of $n=279$ vertices and 38222 edges. This reduced network will be utilized for the remaining study.

Next, we performed a multiscale analysis of the reduced assembly network. We calculated the global and local $L_{1}$ centrality for each member by setting the locality parameter to $\alpha=15/279$. Figure 7 (b) shows a graph comparing the global and local $L_{1}$ centrality values transformed to have a uniform margin. To facilitate explanation, we assign a random pseudonym to each member with a prefix indicating its political party. A notable observation is that vertices with high $L_{1}$ centralities also tend to have high local $L_{1}$ centralities. Similarly, vertices with low $L_{1}$ centralities also exhibit this phenomenon. Nevertheless, a small number of vertices do not follow this pattern. The six points in the top-left of the plot (J1–J6) and the two vertices in the bottom-right of the plot (P90 and P57) exhibit contrary behavior at the global and local levels. The paradoxical behavior of these eight vertices, which can be compared to the famous Simpson’s paradox, is worth examining.

• •

  All six members of the Justice Party (J1–J6) have low global $L_{1}$ centrality while demonstrating high local $L_{1}$ centrality at $\alpha=15/279$. This can be attributed to the fact that Justice Party members form a tight community with each other: the edges connecting Justice Party members are significantly stronger (i.e., lower edge weights) than those connecting a Justice Party member and a member from another party.

  While examining the edges connected to one end of a particular Justice Party member, we consistently observed that the top five edges with the lowest weight were always connected to the other five Justice Party members. Table 1 also lists the number of cosponsored bills between the Justice Party members (i.e., the reciprocal of the edge weight connecting two Justice Party members). The magnitude of these values is relatively high for the entire population. The minimum number of cosponsored bills among the members of the Justice Party is 416 bills between J3 and J4, which falls in the 99.64% quantile in the distribution of the number of cosponsored bills among each member in the reduced assembly network. Hence, it is evident that the edge among the members of the Justice Party is solid.

  Even so, the Justice Party members do not maintain a strong link with other party members. The highest number of cosponsored bills by each Justice Party member with a member of another party ranges between 221 and 317, much lower than the values in Table 1. This implies that the Justice Party is a cohesive community in which bill cosponsorship occurs primarily within the party. Therefore, inside the 15/279 $L_{1}$ centrality-based neighborhood of each Justice Party member, it is observed that there are four to six other Justice Party members included. As a result, each member of the Justice Party has a high level of centrality within the conditioned graph due to its cohesiveness.

  	J1	J2	J3	J4	J5	J6
  J1	—	439	438	425	438	452
  J2	439	—	421	418	416	417
  J3	438	421	—	416	436	430
  J4	425	418	416	—	431	427
  J5	438	416	436	431	—	431
  J6	452	417	430	427	431	—
  Total cosponsored bills	818	566	619	638	660	683

  However, the total number of bills cosponsored by each member of the Justice Party is too small (last row of Table 1). The median number of cosponsored bills for all network members is 929, and the first quartile is 612. So, while these members are central within the party and their neighbors, the overall strength of those edges is not impressive. As a result, the global $L_{1}$ centrality is relatively low. This may be the behavior and strategy of a ‘niche’ party that is distinct from the legislative activity patterns of the major parties. They work together effectively, but their small party size limits their ability to engage actively in legislative action.

• •

  P57 and P90 exhibit contrasting behaviors to the members of the Justice Party. Their local $L_{1}$ centrality is low, but their global $L_{1}$ centrality is high, falling in the 91.40% (P57) and 97.85% (P90) quantiles of the global $L_{1}$ centrality distribution. This is because these two vertices essentially play an important role between the two major parties. As shown from the target plot in Figure 7 (a), each of the two major parties establishes a distinct community within the network. A few members, including P57 and P90, serve as a ‘bridge’ vertex between these two communities.

  We started by counting cosponsored bills between and within both major political parties. The distribution is shown in Figure 8. There is a significant scarcity of bills cosponsored amongst members between the two parties compared to the bills cosponsored by the members of each party. That is, cosponsors of a bill typically consist of only Democratic Party members or only People Power Party members.

  

  It is important to note that one network member has switched its affiliation from the Democratic Party to the People Power Party, hence being classified as a member of the People Power Party. This individual has cosponsored more bills with members of the Democratic Party than members of the People Power Party. So, except for this member, the top five highest number of cosponsored bills that occurred between the two major parties are as follows: P90–D139 (118 bills), P90–D63 (87 bills), P90–D128 (81 bills), P57–D140 (64 bills), and P8–D140 (60 bills). It is evident that P90 and P57 act as a ‘bridge’ between the two parties. Therefore, P90 and P57, along with vertices D139, D63, D128, D140, and P8, exhibit high global $L_{1}$ centrality. D128 is in the 72.40% quantile of the global $L_{1}$ centrality distribution, and the others are all above the 90% quantile.

  However, among all the vertices with global $L_{1}$ centrality over the 90% quantile, P57 has the fewest cosponsored bills (467), and P90 has the second-fewest cosponsored bills (905). Furthermore, among the vertices with global centrality exceeding the 95% quantile, P90 has the fewest cosponsored bills, which means that the connection between each of the two vertices to its nearby vertices is weak (high edge weight). Therefore, rather than their closest neighbors, vertices with high global $L_{1}$ centrality are incorporated into the set of $L_{1}$ centrality-based neighbors for these vertices. This is a phenomenon contrary to that of the Justice Party members. Hence, a small $L_{1}$ centrality is obtained in the conditioned graph. In summary, the two exceptional vertices, P90 and P57, attain their high global centrality by serving as a ‘bridge’ between the two large communities in the network but have a weak connection to either of the communities.

The analysis presented demonstrates the utility of examining a single graph at multiple locality levels. Indeed, vertices with similar global $L_{1}$ centrality may exhibit contrasting behavior at a local level and vice versa. The proposed global and local $L_{1}$ centrality captures this specific aspect that cannot be accounted for by other centrality measures.

In this analysis, we used two values of $\alpha=15/279$ and 1. However, a more refined multiscale analysis can be performed with densely chosen values of $\alpha$. By adjusting the locality level, we may create a functional data set in which each function corresponds to a member of the reduced assembly network. Figure 9 shows the variation in the local $L_{1}$ centrality of each vertex over $\alpha=5/279,10/279,\dots,275/279$. The local $L_{1}$ centrality values are transformed to a uniform margin for each locality level. For easier comprehension, each curve is then standardized to have a mean of zero, meaning that the average of the functional values is zero for each curve. The curves representing the Justice Party members deviate from the other curves. Collecting structural information of each vertex as a function can be more informative than other single-valued centrality measures. This functional data can be utilized for various analyses. For example, one can classify or cluster vertices or understand the relationship between the structural attribute of each vertex and other relevant characteristics. Extensive research on functional data analysis can be utilized with this data set (see, e.g., Ramsay and Silverman, 2005).

Finally, we present a multiscale edge representation of the reduced assembly network with two locality levels, $\alpha=15/279$ and $1$. In Figure 10, the local medians are represented by larger vertices, each labeled with the pseudonym of its member. Panel (a) reveals the presence of 15 local medians, with D16 being identified as the local median originating from nearly half of the vertices. As shown in panel (b), D16 serves as the graph median of the reduced assembly network. It would be worthwhile to investigate some local medians, such as D139, which serves as the local median for a number of People Power Party members, and D140, which is designated as the local median for all six Justice Party members. Altering the value of $\alpha$ to different values makes a comprehensive multiscale representation of the assembly network possible. Conversely, a plot with all 38222 edges would be incomprehensible.

In this paper, we have introduced the $L_{1}$ centrality for evaluating vertex centrality in graphs and related tools for multiscale and graphical analysis. The proposed method borrows ideas from existing literature on the data depth for multivariate data and particular instruments and methods used in the field. The tools provided in this paper are only a first step. There are many potential applications beyond the current discussion. The literature on data depth suggests various areas where the $L_{1}$ centrality can be used successfully. A few examples from the data depth literature include classification and clustering (Jörnsten, 2004; Li et al., 2012; Paindaveine and Van Bever, 2013), missing value imputation (Mozharovskyi et al., 2020), and nonparametric test (Liu and Singh, 1993).

This study only considered undirected graphs to define measures and tools. We briefly explore the potential extension of the concepts to directed graphs. In the literature, the concept of ‘centralness’ in directed graphs is divided into two distinct notions: prestige and centrality (Wasserman and Faust, 1994, Chapter 5.3). The prestige measure estimates the level of importance of a vertex in terms of receiving a choice. Conversely, the centrality measure in directed graphs assesses the level of importance in terms of giving a choice. It is feasible to extend the $L_{1}$ centrality to encompass directed graphs. Define the $L_{1}$ prestige for directed graphs in the following way. The graph median for prestige measure is redefined as the vertex $v_{i}$ that minimizes $\sum_{j=1}^{n}\eta_{j}d(v_{j},v_{i})$, where $d(v_{j},v_{i})$ denotes the geodesic distance from $v_{j}$ to $v_{i}$. Here, $d(v_{i},v_{j})$ is not necessarily equal to $d(v_{j},v_{i})$, which means it is not a distance function. Assuming that every vertex can be reached from another, the $L_{1}$ prestige can be defined according to equation (2.1) with the indices in the function $d$ swapped. Nevertheless, since the triangle inequality, such as $d(v_{i},v_{k})-d(v_{i},v_{j})\leq d(v_{k},v_{j})$, does not necessarily hold—since $d$ is not a distance function—the resulting prestige measure cannot be ensured to fall within the range of $[0,1]$. Together with the analogous theorem and propositions in this paper that apply to directed graphs, we plan to carry out a separate study on this topic elsewhere.

Next, recall that the concept of $L_{1}$ centrality is derived from the $L_{1}$ data depth. It is of interest to explore the theoretical connection between these two notions. A possible link can be made in the latent space model for graph data (Smith et al., 2019). In this model, edges are assumed to be generated depending on the positions of points in the underlying geometric space that represent the two vertices. With this model, we may ask if the $L_{1}$ centrality is a ‘good’ estimate of the $L_{1}$ depth w.r.t. the underlying probability distribution in the latent space. Then, we can consider the $L_{1}$ centrality as a desirable statistic rather than an ad-hoc measure. This can also construct a comprehensive distribution theory for a centrality statistic. For example, the large sample ($n\to\infty$) behaviors of the centralities. We have established a proposition that may be useful for further study.

The LHS is similar to equation (2.1), except that instead of using geodesic distance in the graph, it uses the Euclidean norm. Thus, if the geodesic distance in the graph approaches the Euclidean distance in the underlying geometric space at a suitable speed, it is possible to establish the convergence of the $L_{1}$ centrality to the $L_{1}$ depth in the latent space. Much literature proves that the geodesic distance in the graph converges to the geodesic distance in various latent space models (Alamgir and von Luxburg, 2012; Hwang et al., 2016). Nevertheless, there is uncertainty about which latent space models may effectively achieve the desired convergence speed and the conditions necessary to establish a concrete connection between the two concepts. This topic can be explored in future research, perhaps with a graph centrality based on the notion of another data depth function.

This research was supported by the National Research Foundation of Korea (NRF) funded by the Korea government (2021R1A2C1091357).

The MCU movie network and the assembly network are available from the R package L1centrality (https://CRAN.R-project.org/package=L1centrality).