Kishi G. (1981) On centrality functions of a graph. In: Saito N., Nishizeki T. (eds) Graph Theory and Algorithms. Lecture Notes in Computer Science, vol 108. Springer, Berlin, Heidelberg
For a connected nondirected graph, a centrality function is a real valued function of the vertices defined as a linear combination of the numbers of the vertices classified according to the distance from a given vertex. Some fundamental properties of the centrality functions and the set of central vertices are summarized. Inserting an edge between a center and a vertex, the stability of the set of central vertices are investigated.
For a weakly connected directed graph, we can prove similar theorems with respect to a generalized centrality function based on a new definition of the modified distance from a vertex to another vertex.