  A new measure to assess the centrality of vertices in an undirected and
connected graph is proposed. The proposed measure, L1 centrality, can
adequately handle graphs with weights assigned to vertices and edges. The study
provides tools for graphical and multiscale analysis based on the L1
centrality. Specifically, the suggested analysis tools include the target plot,
L1 centrality-based neighborhood, local L1 centrality, multiscale edge
representation, and heterogeneity plot and index. Most importantly, our work is
closely associated with the concept of data depth for multivariate data, which
allows for a wide range of practical applications of the proposed measure.
Throughout the paper, we demonstrate our tools with two interesting examples:
the Marvel Cinematic Universe movie network and the bill cosponsorship network
of the 21st National Assembly of South Korea.
