• # 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.

Comparison between the growth of Berlin and our strongly correlated percolation model.

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

Strongly correlated percolation model

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''.

Map of UK used in our studies of scaling laws of urban centers

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.

• # URBAN PLANNING

Throughout this century, the dominant planning policy in many western nations has been the containment of urban growth. This has been effected using several instruments, particularly through the siting of new settlements or new towns at locations around the growing city which are considered to be beyond commuting distance, but also through the imposition of local controls on urban growth, often coordinated regionally as green belts''. One of the key elements in the growth models we have proposed here is the characteristic length scale over which growth takes place. In the case of the gradient percolation model, correlations occur over all length scales, and the resulting distributions are fractal, at least up to the percolation threshold.

In examining the changing development of Berlin in Fig. \ref{dy}a, it appears that the fractal distribution remains quite stable over a period of 70 years and this implies that any controls on growth that there might have been do not show up in terms of the changing settlement pattern, implying that the growth dynamics of the city are not influenced by such control. A rather different test of such policies is provided in the case of London where a green belt policy was first established in the 1930s and rigorously enforced since the 1950s. The question is whether this has been effective in changing the form of the settlement pattern. First, it is not clear that the siting of new towns beyond London's commuting field was ever beyond the percolation field and thus it is entirely possible that the planned new settlements in the 1950s and 1960s based on existing village and town cores simply reinforced the existing fractal pattern.

In the same manner, the imposition of local controls on growth in terms of preserving green field land from development seems to have been based on reinforcing the kind of spatial disorder consistent with morphologies generated through correlated percolation. The regional green belt policy was based on policies being defined locally and then aggregated into the green belt itself, and this seems to suggest that the morphology of nondevelopment that resulted was fractal. This is borne out in a fractal analysis of development in the London region which suggests that the policy has little impact on the overall morphology of the area. Moreover, we note that the coincidence between the settlement area distribution for different cities and different years (Berlin 1920 and 1945, and London 1981) suggests that local planning policies such as the green belt may have a relatively low impact on the distribution of towns. Our model suggests that the area distribution is determined by the degree of interactions among development units, and that its scaling properties are independent of time. Current debates on urban growth have now shifted to the development of brownfield sites in cities, and it would be interesting to quantify the extent to which such future developments might reinforce or counter the natural'' growth of the city as implied in these kinds of models.

To develop more detailed and conclusive insights into the impact of urban policies on growth, it is necessary to develop the model further. This model implies that the area and size distributions, the degree of interaction amongst dependent units of development, and fractal dimension are independent of time. The only time dependent parameter is the gradient lambda and it appears that we might predict future urban forms simply by extrapolating the value of lambda in time. However, we have yet to investigate the influence of topography and other physical constraints on development, the influence of transport routes and the presence of several independent'' central cores or CBDs in the urban region.

These models can also be further adapted to predict bond as well as site percolation and in future work we will explore the extent to which such interactions between sites and cities might be modeled explicitly. Our interest in such examples is in the universality of the exponents that we have demonstrated here, and which we wish to relate to the impact of urban planning policies.

• # More info in the WEB:

For cities in the world,
for cities in USA,
for Swiss cities,
for Africa population

• # Links in the press

Our research has been featured in the press:

• Nature Editorial by M. Batty, Nature 377, 574 (1995).
• Science News , Cover story, The Shapes of Cities: Mapping out fractal models of urban growth'', Science News, 149, 8-9 (6 Jan 1996)
• Daily Telegraph (London) , Roger Highfield, Sept. 25, 1996.
• Diario El Pais , Suplemento Futuro, November 1995.
• Physics World, May 1997, p. 29
• Scienza & Vita 4, 70 (April 1996).
• M & P Computer, La vita artificiale'', A. Vespignani, May 1996, p. 48.
• Physics World, H. A. Makse, H. E. Stanley, and S. Havlin, " Power Laws for Cities", Physics World, 10, 22-23 (October 1997).
• CSIRO, Division of Wildlife and Ecology.

• # Collaborators

S. Havlin (Bar-Ilan), M. Batty (University College London), J. S. Andrade Jr. (UFC, Brazil), and H. E. Stanley (BU).

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