L1 Prominence Measures for Directed Graphs

We introduce novel measures, $L_1$ prestige and $L_1$ centrality, for quantifying the prominence of each vertex in a strongly connected and directed graph by utilizing the concept of $L_1$ 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.