The Origins of Scaling in Cities
The Equations Underlying Cities
Cities are complex systems of which functioning depends upon many social, economic, and environmental factors. Bettencourt (p. 1438; see the cover; see the Perspective by Batty) developed a theory to explain the quantitative relationships observed between various aspects of cities and population size or land area.
Abstract
Despite the increasing importance of cities in human societies, our ability to understand them scientifically and manage them in practice has remained limited. The greatest difficulties to any scientific approach to cities have resulted from their many interdependent facets, as social, economic, infrastructural, and spatial complex systems that exist in similar but changing forms over a huge range of scales. Here, I show how all cities may evolve according to a small set of basic principles that operate locally. A theoretical framework was developed to predict the average social, spatial, and infrastructural properties of cities as a set of scaling relations that apply to all urban systems. Confirmation of these predictions was observed for thousands of cities worldwide, from many urban systems at different levels of development. Measures of urban efficiency, capturing the balance between socioeconomic outputs and infrastructural costs, were shown to be independent of city size and might be a useful means to evaluate urban planning strategies.
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Supplementary Material
Summary
Materials and Methods
Supplementary Text
Figs. S1 to S3
Tables S1 to S3
Resources
File (bettencourt.sm.pdf)
References and Notes
1
L. Mumford, The City in History: Its Origins, Its Transformations, and Its Prospects (Harcourt, Brace & World, Inc., New York, 1961).
2
I. Rodríguez-Iturbe, A. Rinaldo, Fractal River Basins: Chance and Self-Organization (Cambridge Univ. Press, New York, 1997).
3
E. N. Bacon, The Design of Cities (Penguin Group USA, New York, 1976).
4
West G. B., Brown J. H., Enquist B. J., A general model for the origin of allometric scaling laws in biology. Science 276, 122 (1997).
5
West G. B., Brown J. H., Enquist B. J., The fourth dimension of life: fractal geometry and allometric scaling of organisms. Science 284, 1677 (1999).
6
Samaniego H., Moses M. E., Cities as organisms: Allometric scaling of urban road networks. J. Transport. Land Use 1, 21 (summer 2008).
7
Waters J. S., Holbrook C. T., Fewell J. H., Harrison J. F., Allometric scaling of metabolism, growth, and activity in whole colonies of the seed-harvester ant Pogonomyrmex californicus. Am. Nat. 176, 501 (2010).
8
D. S. Dendrinos, H. Mullally, Urban Evolution: Studies in the Mathematical Ecology of Cities (Oxford Univ. Press, Oxford, 1985).
9
Sveikauskas L., The Productivity of Cities. Q. J. Econ. 89, 393 (1975).
10
Glaeser E. L., Sacerdote B., Why is there more crime in cities? J. Polit. Econ. 107, S225 (1999).
11
Bettencourt L. M. A., Lobo J., Strumsky D., Invention in the City: Increasing returns to patenting as a scaling function of metropolitan size. Res. Policy 36, 107 (2007).
12
Bettencourt L. M. A., Lobo J., Helbing D., Kühnert C., West G. B., Growth, innovation, scaling, and the pace of life in cities. Proc. Natl. Acad. Sci. U.S.A. 104, 7301 (2007).
13
Batty M., The size, scale, and shape of cities. Science 319, 769 (2008).
14
See supplementary materials for details.
15
R. Anderson, R. May, Infectious Diseases of Humans: Dynamics and Control (Oxford Univ. Press, Oxford, 1991).
16
Angel S., Parent J., Civco D. L., Blei A., Potere D., The dimensions of global urban expansion: Estimates and projections for all countries, 2000–2050. Prog. Plann. 75, 53 (2011).
17
D. Saunders, Arrival City (Pantheon Books, New York, 2010).
18
Nordbeck S., Urban allometric growth. Geogr. Ann. 53, 54 (1971).
19
Barthélemy M., Spatial Networks. Phys. Rep. 499, 1 (2011).
20
Milgram S., The experience of living in cities. Science 167, 1461 (1970).
21
M. Schläpfer et al., The Scaling of Human Interactions with City Size; http://arxiv.org/abs/1210.5215.
22
J. Jacobs, The Economy of Cities (Vintage Books, New York, 1969).
23
A. Marshall, Principles of Economics (MacMillan, London, 1920). See book IV, chapter X.
24
Glaeser E. L., Gottlieb J. D., The wealth of cities: Agglomeration economies and spatial equilibrium in the United States. J. Econ. Lit. 47, 983 (2009).
25
Gomez-Lievano A., Youn H., Bettencourt L. M. A., The statistics of urban scaling and their connection to Zipf’s law. PLoS ONE 7, e40393 (2012).
26
Alexander C., A City is Not a Tree. Archit. Forum 122, 58 (1965).
27
A. Downs, The law of peak-hour expressway congestion. Traffic Quarterly 16, 393 409 (1962).
28
Duranton G., Turner M. A., The fundamental law of road congestion: Evidence from US Cities. Am. Econ. Rev. 101, 2616 (2011).
29
S. Sassen, The Global City: New York, London, Tokyo (Princeton Univ. Press, Princeton NJ, 1991).
30
M. Fujita, P. Krugman, A. J. Venables, The Spatial Economy: Cities, Regions, and International Trade (MIT Press, Cambridge MA, 2001).
31
Lederbogen F., et al., City living and urban upbringing affect neural social stress processing in humans. Nature 474, 498 (2011).
32
J. Leskovec, J. Kleinberg, C. Faloutsos, “Graphs over time: Densification laws, shrinking diameters and possible explanations.” In KDD ’05 Proceedings of the 11th ACM SIGKDD International Conference on Knowledge Discovery in Data Mining, pp. 177–187 (2005); http://dl.acm.org/citation.cfm?id=1081893.
33
The smallest conceivable city has population size N = 2, as I assume that a city is intrinsically a social network and, as such, is predicated on the existence of social interactions.
34
L. M. A. Bettencourt, H. Samaniego, H. Youn, Professional diversity and the productivity of cities; http://arxiv.org/abs/1210.7335.
35
S. Angel, J. Parent, D. L. Civco, A. M. Blei, “The persistent decline in urban densities: Global and historical evidence of sprawl.” Lincoln Institute of Land Policy Working Paper; www.lincolninst.edu/pubs/1834 The-Persistent-Decline-in-Urban-Densities.
36
W. Alonso, Location and Land Use (Harvard Univ. Press, Cambridge MA, 1964).
37
M. Fujita, Urban Economic Theory (Cambridge Univ. Press, Cambridge, 1989).
38
Banavar J. R., Maritan A., Rinaldo A., Size and form in efficient transportation networks. Nature 399, 130 (1999).
39
Stewart J. Q., Warntz W., Physics of Population Distribution. J. Reg. Sci. 1, 99 (1958).
40
Tobler W. R., Satellite confirmation of settlement size coefficients. Area 1, 30 (1969).
41
Longley P. A., Batty M., Shepherd J., The size, shape and dimension of urban settlements. Trans. Inst. Br. Geogr. 16, 75 (1991).
42
Paulsen K., Yet even more evidence on the spatial size of cities: Urban spatial expansion in the US, 19802000. Reg. Sci. Urban Econ. 42, 561 (2012).
43
Bettencourt L. M. A., Lobo J., Strumsky D., West G. B., Urban scaling and its deviations: Revealing the structure of wealth, innovation and crime across cities. PLoS ONE 5, e13541 (2010).
44
Woldenberg M. J., An allometric analysis of urban land use in the United States. Ekistics 36, 282 (1973).
45
Dutton G. H., Criteria of growth in urban systems. Ekistics 215, 298 (1973).
46
Coffey W. J., Allometric growth in urban and regional social-economic systems. Can. J. Reg. Sci. 11, 49 (1979).
47
Sutton P., Roberts D., Elvidge C., Meij H., A comparison of nightime satellite imagery and population density in the United States. Photogramm. Eng. Remote Sensing 63, 1303 (1997).
48
Batty M., Ferguson P., Defining city size. Environment and Planning B 38, 753 (2011).
49
M. See, Batty, P. A. Longley, Fractal Cities: A Geometry of Form and Function (Academic Press, London UK, 1994).
50
M. Guerois, Les Formes des Villes Européenes vues du Ciel, thesis, Universite Paris I Pantheon-Sorbonne (2003).
51
See figures 5 and 6 and associated analysis in (63)
52
Aristotle, Politics, Book I, 1-2.
53
A. Pred, Urban Growth and City-Systems in the United States, 1940-1860 (Harvard Univ. Press, Cambridge MA, 1980).
54
A. Lösch, Die Räumliche Ordnung Der Wirtschaft (Gustav Fischer Verlag, Jena, Germany, 1940). Translated from German by W. H. Woglom, The Economics of Location (Yale Univ. Press, New Haven, CT, 1954).
55
W. Christaller, Die Zentralen Orte in S¨uddeutschland (Gustav Fischer Verlag, Jena, Gremany, 1933). Translated from German by C. W. Baskin, The Central Places of Southern Germany (Prentice-Hall, Englewood Cliffs, NJ, 1966).
56
P. Hall, Cities in Civilization (Pantheon Books, New York, 1998).
57
McCarthy J. D., Galle O. R., Zimmern W., Population density, social structure, and interpersonal violence: An intermetropolitan test of competing models. Am. Behav. Sci. 18, 771 (1975).
58
Szreter S., Mooney G., Urbanization, mortality and the standard of living debate: New estimates of the expectation of life at birth in nineteenth-century British cities. Econ. Hist. Rev. 51, 84 (1998).
59
R. Florida, The Rise of the Creative Class: And How Its Transforming Work, Leisure, Community and Everyday Life (Perseus Books, New York, 2002).
60
C. S. Fischer, To Dwell Among Friends: Personal Networks in Town and City (Univ. of Chicago Press, Chicago, 1982).
61
B. Wellman, Networks in the Global Village: Life in Contemporary Communities (Westview, Boulder CO, 1999).
62
L. M. A. Bettencourt, Research Notes. Available upon request from the author.
63
Chen Y., Characterizing growth and form of fractal cities with allometric scaling exponents. Discrete Dyn. Nat. Soc. 2010, 194715 (2010).
Information & Authors
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Published In
Science
Volume 340 | Issue 6139
21 June 2013
21 June 2013
Copyright
Copyright © 2013, American Association for the Advancement of Science.
Submission history
Received: 29 January 2013
Accepted: 29 April 2013
Published in print: 21 June 2013
Acknowledgments
I thank J. Lobo, G. West, and H. Youn for discussions. This research was supported by grants from the Rockefeller Foundation, James S. McDonnell Foundation (grant 220020195), NSF (grant 103522), Bill and Melinda Gates Foundation (grant OPP1076282), John Templeton Foundation (grant 15705), and Bryan J. and June B. Zwan Foundation.
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