Cities are living organisms. They are out of equilibrium, open systems that never stop developing and sometimes die. The local geography can be compared to a shell constraining its development. In brief, a city's current layout is a step in a running morphogenesis process. Thus cities display a huge diversity of shapes and none of the traditional models, from random graphs, complex networks theory, or stochastic geometry, takes into account the geometrical, functional, and dynamical aspects of a city in the same framework. We present here a global mathematical model dedicated to cities that permits describing, manipulating, and explaining cities' overall shape and layout of their street systems. This street-based framework conciliates the topological and geometrical sides of the problem. From the static analysis of several French towns (topology of first and second order, anisotropy, streets scaling) we make the hypothesis that the development of a city follows a logic of division or extension of space. We propose a dynamical model that mimics this logic and that, from simple general rules and a few parameters, succeeds in generating a large diversity of cities and in reproducing the general features the static analysis has pointed out.
We explore real telecommunication data describing the spatial geometrical structure of an urban region and we propose a model fitting procedure, where a given choice of different non-iterated and iterated tessellation models is considered and fitted to real data.This model fitting procedure is based on a comparison of distances between characteristics of sample data sets and characteristics of different tessellation models by utilizing a chosen metric. Examples of such characteristics are the mean length of the edge-set or the mean number of vertices per unit area. In particular, after a short review of a stochastic-geometric telecommunication model and a detailed description of the model fitting algorithm, we verify the algorithm by using simulated test data and subsequently apply the procedure to infrastructure data of Paris.
We study the tessellation obtained in the intersection of two independent planar Poisson-Voronoi tessellations and derive the means of its main geometrical characteristics. We distinguish six types of cell depending on the position of nuclei of the original tessellations. The intensity and the mean area of each type of cell are computed either in closed form or via asymptotic expansions. The model can be used to represent different types of local zones of two competing telecommunication operators where the interconnection of two subscribers induces a specific cost within each type of zone.
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