We introduce novel measures, L1 prestige and L1 centrality, for quantifying the prominence of each vertex in a strongly connected and directed graph by utilizing the concept of L1 data depth (Vardi and Zhang, Proc. Natl. Acad. Sci. U.S.A.\ 97(4):1423--1426, 2000). The former measure quantifies the degree of prominence of each vertex in receiving choices, whereas the latter measure evaluates the degree of importance in giving choices. The proposed measures can handle graphs with both edge and vertex weights, as well as undirected graphs. However, examining a graph using a measure defined over a single `scale' inevitably leads to a loss of information, as each vertex may exhibit distinct structural characteristics at different levels of locality. To this end, we further develop local versions of the proposed measures with a tunable locality parameter. Using these tools, we present a multiscale network analysis framework that provides much richer structural information about each vertex than a single-scale inspection. By applying the proposed measures to the networks constructed from the Seoul Mobility Flow Data, it is demonstrated that these measures accurately depict and uncover the inherent characteristics of individual city regions.

翻译：本文基于 L1 数据深度（Vardi and Zhang, Proc. Natl. Acad. Sci. U.S.A. 97(4):1423–1426, 2000）的概念，提出了两种新的度量指标——L1 声望与 L1 中心度，用于量化强连通有向图中各顶点的显著程度。前者度量每个顶点在接收选择方面的显著程度，而后者评估其在给出选择方面的重要性程度。所提出的度量能够处理同时包含边权重与顶点权重的图，也适用于无向图。然而，使用在单一“尺度”上定义的度量来考察图，不可避免地会导致信息损失，因为每个顶点在不同局部性水平上可能展现出不同的结构特征。为此，我们进一步开发了所提出度量的局部版本，并引入可调节的局部性参数。利用这些工具，我们提出了一个多尺度网络分析框架，该框架能够提供比单尺度考察更为丰富的各顶点结构信息。通过将所提出的度量应用于基于首尔人流移动数据构建的网络，证明了这些度量能够准确描绘并揭示各个城市区域的内在特征。