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 treelike 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, 608612 (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 , 89 (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, 2048 (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 ranksize 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 interurban 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 ``selforganize''.
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 powerlaw.
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 powerlaw 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, 89 (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, 2223 (October 1997).

CSIRO, Division of Wildlife and Ecology.
Collaborators
S. Havlin (BarIlan), M. Batty
(University College London),
J. S. Andrade Jr. (UFC, Brazil), and
H. E. Stanley (BU).
