Scaling laws in urban supply networks

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Abstract

In previous work, it has been proposed that urban structures may be understood as a result of self-organization principles. In particular, researchers have identified fractal structures of public transportation networks and land use patterns. Here, we will study spatial distribution systems for energy, fuel, medical, and food supply. It is found that these systems show power-law scaling as well, when the number of “supply stations” is plotted over the population size. Surprisingly, only some supply systems display a linear scaling with population size. Others show sublinear or superlinear scaling. We suggest an interpretation regarding the kind of scaling law that is expected in dependence of the function and constraints of the respective supply system.

Introduction

The classical view of the spatio-temporal evolution of cities in developed countries is that urban spaces are the result of (centralized) urban planning. After the advent of complex systems’ theory, however, people have started to interpret city structures as a result of self-organization processes. In fact, although the dynamics of urban agglomerations is a consequence of many human decisions, these are often guided by optimization goals, requirements, or boundary conditions (such as topographic ones). Therefore, it appears promising to view urban planning decisions as results of the existing structures and upcoming ones (e.g. when a new freeway will lead closeby in the near future). Within such an approach, it would not be surprising anymore if urban evolution could be understood as a result of self-organization.

In fact, already in the 19th century, it was proposed that the migration streams between cities can be understood by a “gravity law” [1], [2], according to which the relevant variables for migration activities are the population sizes of the origin and destination town and the distance between them. According to this, city growth could be solely understood as a result of the birth and death rates and migration activities [3].

Maybe even more intriguing is the existence of Zipf's law [4], according to which the population sizes of cities are inversely proportional to their rank. It also implies that the number of cities of a given size is approximately inversely proportional to their size N, implying a power lawp(N)N-1of the size distribution. Among the many different approaches trying to explain Zipf's law (e.g. Refs. [5], [6], [7]), the one by Gabaix [8] surprises by its simplicity. According to him, the simplest stochastic model with multiplicative noise ξi(t), namelydNidt=[A+ξi(t)]Ni(t)is able to generate Zipf's distribution. In agreement with “Gibrat's law” [9], [10], it assumes that the growth rates Ai(t)=A+ξi(t) are stochastically distributed and varying around a characteristic value A independent of the (population) size Ni(t) of a city i. These days, scientists attempt to explain the different attractiveness of cities, i.e., the time-dependent values Ai(t) through other variables such as productivity, natural resources, quality of life, etc.

Another interesting point is the discovery of self-similar structures in urban systems. For example, Christaller suggested a theory of central places [11], according to which cities are (self-)organized in a hierarchical way: metropoles are surrounded by a hexagonal “ring” of medium-sized cities, each of these by a ring of smaller towns, etc. Later on, the Collaborative Research Project “Natural Constructions”1 studied urban patterns from the perspective of self-organization theory. For example, street systems were compared with the water transport systems within leaves of plants and analyzed for quantities such as the reachability and average detours. Moreover, public transportation systems were found to show self-similar, fractal features [12]. The same applies to urban boundaries and urban sprawl [7], [12], [13], [14], [15]. Usage patterns of public transport networks [16] and road networks [17] show power-law distributions as well.

The EU project ISCOM involving biological physicists, geographers, and scientists of other disciplines continues this line of research and goes beyond previous results by trying to identify the mechanisms behind self-similar urban structures and their scaling laws. This is encouraged by the successful derivation of empirically observed scaling laws for quantities like metabolic rates, maximal population growth, life-spans, gross photosynthetic rates, trunk diameters, etc. in biological systems [18], [19], [20]. The underlying theory is based on the determination of self-similar, tree-like arterial and other supply systems, which minimize energy dissipation under the constraint of space-covering supply.

In this contribution, we will focus on the question whether scaling laws exist for urban supply systems. If yes, what are their exponents? In Section 2, we start with describing our data sources. Section 3 discusses our data evaluation procedure and compares different statistical evaluation procedures such as linear and logarithmic binning. Section 4 presents the scaling laws found for different supply systems. Finally, Section 5 summarizes our results and tries to give an interpretation of the findings.

Section snippets

Data sources

We have studied supply systems of cities in different European countries and analyzed, on the one hand, variables related to the electric energy supply in Germany and, on the other hand, data about so-called “Points of interest” of several European countries.

Our investigation of energy data is based on information by the German Electricity Association [21] for the time period of January to December 2002. The electric energy relates to the power plant of the respective commune, while the

Data evaluation procedure

In our data evaluation, we started with an ordinary least-squares method to a double-logarithmic representation of the data. The data used to scatter in a cone-like manner (see Fig. 1a), which raised questions whether other fit functions could be used as well. However, when we applied a logarithmic binning method [23], which appears to be more appropriate for empirical power-law distributions, the double-logarithmic representation of the number Yi of supply units over population size Ni (i.e.,

Scaling laws of urban supply systems

In the following, we present the evaluation results of our supply system data sets. Due to their greater clarity, we have plotted the data using the logarithmic binning method. The plots for electrical energy supply in Germany are shown in Fig. 3. A summary of the scaling exponents and the 95% confidence intervals (both determined for the original data) is presented in Table 1.

Fig. 4, Fig. 5, Fig. 6 show the plots for the number of petrol stations, post offices, and restaurants. Note that all

Summary and discussion

In this paper, we have analyzed empirical data of urban supply systems. When the number of “supply stations” is fitted as a function of the population size, one can find power-law distributions for quantities as different as electrical energy, petrol stations, car dealers, hospitals, hospital beds, post offices, pharmacies, doctors, and restaurants. For several European countries, the corresponding power-law exponents and their 95% confidence intervals are summarized in Fig. 2. No data were

Acknowledgments

This study has been partially supported by the EU project ISCOM and the DFG project He2789/5-1. The authors would like to thank Luis Bettencourt, Jose Lobo, and Denise Pumain for inspiring discussions. They are also grateful to the VDEW for the data of the German electricity suppliers. Moreover, they acknowledge the data evaluation efforts of Markus Winkelmann in a previous stage of this study.

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