Structural Models in the Subterranean World

FOR many years geographers have employed various indexes based on the ideas and concepts of graph theory to measure connectivity and other facets of network analysis. They have noted the limits of these indexes and have sought other tools, some, such as those drawing on combinatorial topology, intimately related to graph theory but others farther afield (Garrison 1960; Nystuen and Dacey 1961; Tinkler 1988). One difficulty with applying mathematics to real-world settings is that strategies which serve in laboratory sciences often do not function equally well in the real world. Thus indexes that can be used over and over in carefully controlled laboratories may not be well suited to complexities in applications that lack such controls. Modifications of theoretical tools that yield good results for one project might not be suited when further modified for another project. Thus it is often preferable to return to the original theoretical underpinnings and adjust the mathematics to fit the real-world situation at hand. In such situations tailor-made mathematical suits are superior to those taken from the racks of traditional models. There are geographical examples of this art, and there is room for more (Harary 1969; Arlinghaus 1994; Arlinghaus, Arlinghaus, and Harary forthcoming). In the latter spirit I offer two simple examples that display both the power and the elegance that carefully constructed structural models can bring to geographical analysis. PROBLEM OF LAYERS Any system, such as a transportation system, that is forced to have different physical levels for entry and exit becomes a target for policy difficulties of various kinds. Elevated and subterranean trains are a response to providing efficient commuting in a densely populated environment: to prevent collisions these networks generally have a number of different horizontal layers. Clearly it is inconvenient for passengers to move vertically as well as horizontally during transfers from one route to another. Although the extra dimension removes a collision hazard between trains, it increases the potential for collisions among passengers. Different layers add a host of security problems for security personnel on the lookout for muggers and thieves who prey on a population closely confined underground. Unless elevators or ramps are installed, multiple-level stations exclude transfer possibilities for individuals confined to wheelchairs. In some locales funding is directly tied to the extent of handicap access to be provided in a public project. Different access layers offer a way to overcome congestion and collisions; however, their mere presence can keep an otherwise worthwhile project from being funded, if they are not constructed in a manner that permits barrier-free access. To begin, therefore, I propose a theorem that determines when a set of layers is arranged to achieve the goal of barrier-free access. ELEVATOR THEOREM A persistent issue in any logical approach to a problem is knowing whether or not a solution exists. It is counterproductive to search for solutions that are known not to exist. One standard theorem from the realm of continuous mathematics is the intermediate-value theorem. Stated informally, this theorem ensures that a continuous function on [a, b] assumes all values between f(a) and f(b). That is, choose a value m on the y axis between f(a) and f(b)--there exists a value c in the interval on the x axis between a and b such that f(c) = m. In this context m is the problem, and the value c is its solution. Stated more formally, the intermediate-value theorem is often cast in the following manner: suppose that a function f is continuous throughout a closed interval [a, b] and that m is any number between f(a) and f(b); then there is at least one number c in [a, b] such that f(c) = m. Geometrically it is not difficult to imagine that instead of a continuous function f over a continuous closed interval [a, b], there might be a discrete function f with separated values

Geography · Data Management and Algorithms · Geographic Information Systems Studies · Urban Design and Spatial Analysis

• Connectivity of the Interstate Highway System

  Open Access•William L Garrison•Papers of the Regional Science…•1960

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• A Graph Theory Interpretation of Nodal Regions

  Open Access•John D Nystuen, Michael F Dacey•Papers of the Regional Science…•1961

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