URBAN GROWTH

Predicting urban growth is important for the challenge it presents to theoretical frameworks for cluster dynamics. Recently, the model of diffusion limited aggregation (DLA) has been applied by Mike Batty, (CASA) in the book Fractal Cities (Academic Press, 1994) to describe urban growth, and results in tree-like dendritic structures which have a core or ``central business district'' (CBD). The DLA model predicts that there exists only one large fractal cluster that is almost perfectly screened from incoming ``development units'' (people, capital, resources, etc), so that almost all the cluster growth occurs in the extreme peripheral tips.

Here we propose and develop an alternative model to DLA that describes the morphology and the area distribution of systems of cities, as well as the scaling of the urban perimeter of individual cities. Our results agree both qualitatively and quantitatively with actual urban data. The resulting growth morphology can be understood in terms of the effects of interactions among the constituent units forming a urban region, and can be modeled using the correlated percolation model in the presence of a gradient [H. A. Makse, S. Havlin, and H. E. Stanley, Modelling Urban Growth Patterns , Nature 377, 608-612 (1995). Accompanied by News & Views editorial by M. Batty, p. 574. Cover story, ``The Shapes of Cities: Mapping out fractal models of urban growth'', Science News 149 , 8-9 (6 Jan 1996); H. A. Makse, J. S. de Andrade, M. Batty, S. Havlin, and H. E. Stanley, Modeling Urban Growth Patterns with Correlated Percolation, Phys. Rev. E (1 December, 1998); H. E. Stanley, L. A. N. Amaral, S. V. Buldyrev, A. L. Goldberger, S. Havlin, H. Leschhorn, P. Maass, H. A. Makse, C.-K. Peng, M. A. Salinger, M. H. R. Stanley, and G. M. Viswanathan, Scaling and Universality in Animate and Inanimate Systems, Physica A 231, 20-48 (1996)].

Traditional approaches to urban science as exemplified in the work of Christaller, Zipf, Stewart and Warntz, Beckmann, and Krugman are based on the assumption that cities grow homogeneously in a manner that suggests that their morphology can be described using conventional Euclidean geometry. However, recent studies have proposed that the complex spatial phenomena associated with actual urban systems is rather better described using fractal geometry consistent with growth dynamics in disordered media.

Predicting urban growth dynamics also presents a challenge to theoretical frameworks for cluster dynamics in that different mechanisms clearly drive urban growth from those which have been embodied in existing physical models. In this paper, we develop a mathematical model that relates the physical form of a city and the system within which it exists, to the locational decisions of its population, thus illustrating how paradigms from physical and chemical science can help explain a uniquely different set of natural phenomena - the physical arrangement, configuration, and size distribution of towns and cities. Specifically, we argue that the basic ideas of percolation theory when modified to include the fact that the elements forming clusters are not statistically independent of one another but are correlated, can give rise to morphologies that bear both qualitative and quantitative resemblance to the form of individual cities and systems of cities.

...................Berlin 1875 ....................... Berlin 1920 .................... Berlin 1945.............

We consider the application of statistical physics to urban growth phenomena to be extremely promising, yielding a variety of valuable information concerning the way cities grow and change, and more importantly, the way they might be planned and managed. Such information includes (but is not limited to) the following:
(i) the size distributions of towns, in terms of their populations and areas;
(ii) the factal dimensions associated with individual cities and entire systems of cities;
(iii) interactions or correlations between cities which provide insights into their interdependence;
(iv) the relevance and effectiveness of local planning policies, particularly those which aim to manage and contain growth.

The size distribution of cities has been a fundamental question in the theory of urban location since its inception in the late 19th century. In the introduction to his pioneering book published over 60 years ago, Christaller posed a key question: ``Are there laws which determine the number, size, and distribution of towns?'' This question has not been properly answered since the publication of Christaller's book, notwithstanding the fact that Christaller's theory of central places and its elaboration through theories such the rank-size rule for cities embody one of the cornerstones of human geography.

Our approach produces scaling laws that quantify such distributions. These laws arise naturally from our model, and they are consistent with the observed morphologies of individual cities and systems of cities which can be characterized by a number of fractal dimensions and percolation exponents. In turn, these dimensions are consistent with the density of location around the core of any city, and thus the theory we propose succeeds in tying together both intra- and inter-urban location theories which have developed in parallel over the last 50 years. Furthermore, the striking fact that cities develop a power law distribution without the tuning of any external parameter might be associated with the ability of systems of cities to ``self-organize''.

So far, we have argued how correlations between occupancy probabilities can account for the irregular morphology of towns in a urban system. The towns surrounding a large city like Berlin are characterized by a wide range of sizes. We are interested in the laws that quantify the town size distribution N(A), where A is the area occupied by a given town or ``mass'' of the agglomeration, so we calculate the actual distribution of the areas of the urban settlements around Berlin and London, and find that for both cities, N(A) follows a power-law.

This new result of a power law area distribution of towns, N(A), can be understood in the context of our model. Insight into this distribution can be developed by first noting that the small clusters surrounding the largest cluster are all situated at distances r from the CBD such that p(r) < p_c or r > r_f. Therefore, we find N(A), the cumulative area distribution of clusters of area A, to be

N(A) ~ A^-(tau+1/d_f nu)

we have plotted the power-law for the area distribution predicted by the above equation along with the real data for Berlin and London. We find that the slopes of the plots for both cities are consistent with the prediction for the case of highly correlated systems. These results quantify the qualitative agreement between the morphology of actual urban areas and the strongly correlated urban systems obtained in our simulations.