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EN
The aim of the study was to present a method for performing neighborhood analyses of regions, based on a set of selected spatial data, whose topological data have the form of a geometric graph. The method proposed in the paper consists of the following: m recording spatial data in the form of model graphs, m assigning the values of functions describing spatial data to graph elements, m determining the objective function as a search for the minimum path values within the graph, m evaluating and interpreting results by comparing the values of vectors describing the neighborhood of selected regions. Spatial data may be represented by drawings, tables or matrices. The matrix form is very useful in the case of a mathematical description of analyses (Lewandowicz, Bałandynowicz 2005). A map drawing may be easily turned into a graph structure. The paper (Lewandowicz 2004) presents methods for transforming a map drawing in its representation f by a graph G: G = [W, K, f] Numerical values describing the attributes of spatial data, in the form of metrics or other measures, may be assigned to graph elements. The description depends on space interpretation. Various descriptions enable its multivariant representations. As a result, we obtain a variety of mathematical models of spatial data: M = [G,{w},{k}]: M . mathematical model, G=[W, K, j] . graph describing selected spatial data, {w} . set of functions w : W r (R+ E 0) determined for a set of nodes. {k} set of functions k : K R (R+ E0) determined for a set of edges. Based on M we can perform multivariate spatial analyses. The study focuses on a case of the application of the matrix form of a graph in the process of estimating commune neighborhood in a region (region is a second level of local government administration in Poland). The relationships between communes within this administrative unit should be analyzed while planning their sustainable development. Results of such analyses shall provide a basis for choosing a center of the region, i.e. a commune whose location can be considered optimal in terms of neighborhood of other communes belonging to this region. The fact of direct neighborhood of areas is a positive economic and organizational factor, especially when the adjacent area is well developed. A graph of commune neighborhood was constructed on the basis of a map of the administrative division of a region. It was assumed that communes are to be presented as graph nodes. Two nodes are connected by an edge if the areas of communes corresponding to them share a common boundary line. In this way we obtain a graph dual to the graph of administrative division, easy to generate on the basis of commune boundaries. Such a graph may be algebraically described by the matrix A of node neighborhood. The value of neighborhood is evaluated for each of the subregions by searching for vectors describing the minimum path values within the graph A between a given region and the other regions. In the case discussed in the paper the neighborhood of a selected region with all other regions was examined. Adequate weighting of graph edges allowed to obtain different values of vectors describing the neighborhood of areas. In this kind of analysis we focus on comparing the values of neighborhood vectors, and not on these values alone. In the geographical space the neighborhood of regions is affected not only by their location in space, but also by infrastructure. One of key factors is communication network. A graph describing a road network provided a basis for determining new values of neighborhood vectors. They supplemented the results of analyses obtained based on the location of regions in space. The above method may be successfully applied while searching for the proper location that is to be chosen from among several or more options. The numerical values obtained as a result of analyses may play an important role in the decision-making process.
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