Synthesis of a flow network with a given centrality on each vertex

Abstract

This paper considers the problem of constructing a flow network when the centrality at each vertex is given. Here, attention is given to a centrality function such that the centrality at a vertex under consideration is the sum of maximum flow values between it and all other vertices. It is a representative centrality function among those representing the centrality of each vertex in an undirected flow network in which edges have capacity. First, we introduce a necessary and sufficient condition wherein a given sequence is a centrality sequence for a flow network. Next, we present a procedure for determining a terminal capacity matrix of a flow network with a given centrality sequence. Furthermore, we determine a terminal capacity matrix of a flow network with a given centrality sequence such that the sum of edge capacity is minimum. From these discussions we can see that the problem of constructing a flow network when a centrality is given, can be reduced to the previously known problem of constructing a flow network when a terminal capacity matrix is given.

Synthesis of flow networks minimizing the sum of edge capacities

“…) It is known that the necessary and sufficient condition for the given symmetrical matrix to be a terminal capacity matrix of a flow network is that the matrix should be partitioned by the principal partition [ 1, 51. Using this property, the following theorem can easily be derived [6].…”

“…Proof. Let one of the Characteristic cuts of D, be PI = (V,, V, ) and let one of the characteristic cuts of D, be P, = (V3, V, ) where v,, 6 Vl and vn E V,. Let Vx = V, n V, and V, = V2 u V, .…”