Spatial distributions: Density-equalizing map projections, facility location, and two-dimensional networks.

Abstract

Geography is a strong force behind data of statistical, economical and technological interest. The distribution of human population and the location of industries, for example, follow non-trivial geographic patterns. Service facilities such as department stores and hospitals are constrained by geography since they must be in geographic proximity to consumers and patients. And in transportation networks, geography is important since nearby places can be more easily connected than those far apart. We use tools from statistical physics to analyze the influence of geography on several social and technological phenomena. Variations in population density often go a long way in explaining geographic data. This can be visualized with cartograms, maps in which the sizes of geographic regions such as countries or provinces appear in proportion to their population. The challenge in creating cartograms is to scale regions and still have them fit together. Here we present a new technique based on linear diffusion that is conceptually simple and produces useful and easily readable maps. A number of applications, including the results of the 2004 US presidential election, illustrate the technique. Many geographic problems of practical interest involve the optimization of point locations. Here we treat in detail a problem related to the distribution of service facilities, the <italic>p</italic>-median problem. It consists of finding the positions of <italic>p</italic> facilities in geographic space such that the mean distance between a member of the population and the nearest facility is minimal. An algorithm for approximate numerical solutions and analytical results are presented. The remainder of this dissertation concentrates on networks whose vertices have definite geographic positions. This includes transportation systems, utility networks, or the Internet. We focus on the cost of a network, as represented by the total length of all its links, and its efficiency in terms of the directness of routes from point to point. Although minimizing the cost and maximizing the efficiency are often conflicting goals, real distribution networks achieve remarkably good compromises. Models and numerical simulations are presented to explain this observation. We close with an analysis of structure and flow in optimally designed spatial networks.