Domains of Structural Stability for Minimal-Cost Discrete Flows with Reference to the Hierarchical Central-Place Models

Abstract

The purpose of this paper is to construct the domains of structural stability for optimal discrete flows, to compare the discrete case with the case of the Beckmann model of continuous flows, and to find the structurally stable minimal-cost flows for the hierarchical central-place models. The mathematical basis for this construction is the sensitivity analysis of the linear programming problem instead of the generic theory of differential equations and the catastrophe theory in the continuous case. The domain of structural stability is the Cartesian product of the domain (cone) of the existence of flow with a preset topological structure and the domain (wedge) of optimality of the flow with the same structure. A detailed algorithm for the construction of these domains of structural stability is given. This algorithm leads to the vector generalization of method of potentials of Kantorovitch-Dantzig for the transportation problem and also gives a description of structural change for dynamic flows. As an application, the generalized (Christaller-Parr) hierarchical three-level central-place models are investigated, for which structurally stable optimal flows exist.