A Characterization of Closeness Centrality Indices in Graphs and Its Applications

Location problems on networks involve the decision problem of determining the location of one or more facilities e.g. communication stations to provide service to several clients located at known points in the service area. Such networks are often modeled as graphs, and this paper uses this model as well. The evaluation index for determining the location differs depending on the system. Since these evaluation indices represent the importance or centrality of a point on a network, they are called centrality indices. As centrality indices, degree centrality, neighborhood centrality, eccentricity centrality, closeness centrality, betweenness centrality, and eigenvector centrality etc. are known. In recent years, this problem has expanded to include complex networks such as social networks. In this paper, we study closeness centrality, which is often used in communication networks and social networks. A number of indices have been proposed to mathematically express closeness centrality. However, the common characteristics of these indices have not been clarified or studied. We characterize the common characteristics of these indices as a generalized closeness centrality function, which internally possesses the properties of discrete decrease and discrete convexity. As an application, we clarify the similarities and differences between closeness centrality indices and neighborhood centrality indices., which is relevant for networks like e.g. cellular mobile communication systems, multidimensional networks, etc.