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Figure 1: ¾ Metabolic Scaling over 27 Orders of Magnitude 27*

Allometric relationships between the size and some specific properties of a system have been observed in many physical and biological systems. In mammals, the following scaling relationship between their metabolic rate R (which is a measure of the rate of energy use in physical systems) and their mass M is observed: R ~ M3/4. This relationship is known as Kleiber’s law (1).

Even though controversy exists with respect to this equation and the value of the exponent, which varies according to the type of animal being consider (warm or cold-blooded organisms, etc.), the relationship is believed to hold over 27 orders of magnitude (see Fig. 1). In the last decades it has been suggested that a universal scaling law of the sort exists for urban systems (2,3):

A ~ Mβ

where A denotes an urban measure, and M the population size. The nature of the urban indicator will determine the regime in which the exponent β lies. For urban indicators leading to economies of scale, such as infrastructure, β<1. This brings an analogy with the efficiency observed for the metabolic rate for animals. The case β=1 corresponds to urban measures related to the basic needs of individuals, and hence these are proportional to the size of a city. The last regime corresponds to increasing returns: β>1. These are observed for indicators that are the outcome of social interactions, such as income, and crime.

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Figure 2: Heatmap for total number of patents for cities bigger than 1.5×105 people

Our research on UK cities suggests however that such scaling laws for urban indicators are not universal (4). Relationships between urban measures and city size are not as simple as originally thought. Most importantly, our work shows that in order to analyse agglomeration effects, it is crucial to have a consistent system of cities, since the exponent fluctuates greatly given different boundary definitions of cities (see Fig. 2). The work on constructing different definitions of cities is outlined in the section “Percolation and urban morphology”.


  1. M. Kleiber, Physiological Reviews 27, 511 (1947).
  2. L. M. A. Bettencourt et al, Proc. Natl. Acad. Sci. USA 104, 7301 (2007).
  3. L. M. A. Bettencourt, Science 340, 1438 (2013).
  4. E. Arcaute, E. Hatna, P. Ferguson, H. Youn, A. Johansson and M. Batty, arXiv:1301.1674 [physics.soc-ph] (2013).


* G. B.  West and J. H. Brown (2005), J. Exp. Biol. 208, 1575-1592.