Centrality functions defined by minimum cuts in a network

Abstract

It is of interest in graph theory to discuss the degree of centrality of a vertex in a graph or to locate the most central point in the graph representation of systems with network structures such as communication and traffic networks. Various expressions for the centrality of a point in a graph or a network have been considered in association with the above problem and are called centrality functions in general. Most of them are based on the distance between two points, while an expression for the centrality by the gauge between two points in a network is also proposed. Most of the previous centrality functions were defined by distance, whereas in this paper we consider the centrality function defined by the minimum cut separating two points in a network (or the maximum flow between two points). According to this centrality function, the centrality of the point under consideration may happen to be the sum of the maximum flows from the point to every other point in the special case and hence it seems to be significant in estimating traffic networks and reliability with respect to circuit faults in communication networks.

Centrality functions of a network with capacities on vertices and edges

“…Thus, a weak centrality function resulting from a minimum mixed cut matches the centrality function resulting from a minimum cut in Ref. 9. Therefore, the centrality function obtained in this research includes the centrality function resulting from a minimum cut as a special case.…”

“…A centrality function for a network with capacity in its edges, as in Ref. 9, will then be shown as a special case.…”