Flows, Hierarchies, Potentials

This work deals with the analysis of interregional flows. The central idea is the decomposition of the origin–destination flow matrix into the additive weighted sum of basic matrices. Each basic matrix represents an extremal flow which optimizes some objective function corresponding to some extremal tendency acting in the real flow. The weight of the extremal flow can be treated as the frequency of the choice of the fixed extremal tendency. The geometrical and algebraic algorithm of the decomposition of the flow matrix is based on the Minkovsky theorem about the center of gravity of convex polyhedra. Its spatial content is Weber's well-known principle of industrial location (in inverted form), and Stouffer's principle of intervening opportunities. Each extremal flow defines the spatially represented hierarchy of the regions of origin and destination. The successive selection of the ‘most probable’ tendencies defines the set of changing hierarchy relationships between the regions. The construction of the successive hierarchies is an elaboration of Nystuen and Dacey's ideas about the use of graph theory in flow theory. The most interesting and important fact is the nonuniqueness of the expansion of the real flow by the shares of extremal tendencies, that is, nonuniqueness of results of analysis. This ensues from the fundamental statement that the analysis depends on the investigator's point of view.