A graph theory interpretation of nodal regions

Abstract

THE PURPOSE OF THIS PAPER is to describe a procedure for ordering and grouping cities by the magnitude and direction of the flows of goods, people, and communications between them. Current theories of nodal regions and central place hierarchies provide the bases for the recognition of region-wide organization of cities into networks. These two theories were developed by students who recognized that the direction and magnitude of flows associated with social processes are indicators of spatial order in the regional structure of urban society. Whether the flow is local and to the city's hinterland, or regional and to the rank ordering of cities, the notion of central or nodal point is dependent upon the levels of strongest associations within the total flow. 1 The present problem is to develop a method capable of quantifying the degree of association between city pairs in a manner that allows identification of the networks of strongest association. These associations may be in terms of .interactions that occur directly between two cities, or indirectly through one or more intermediary cities. The magnitude of the combined direct and indirect associations is measured by an index that is related to certain concepts of graph theory. This index is used to identify the degree of contact between city pairs and it provides a quantitative basis for grouping cities. The resulting subgroups of cities are analogous to nodal regions. When each city in a study region is assigned to a subgroup, it is possible to specify the rank ordering of cities and to evaluate the functional relations of the nodal hierarchy. In this paper, pertinent geograrphic and graph theoretic concepts are discussed and are then used as a basis for deriving the method of isolating nodal regions. Wihile this method is illustrated by the use of intercity telephone calls in Washington state, the techniques are quite general and may be adapted to