I recently started reading some research from Geoffrey West, Luis
Bettencourt and others on scaling phenomena relating to averages (there are
always deviations). This is an incredibly
rich topic and one cannot possibly do justice to it in a single post, so I'll
merely highlight one or two findings from the research and provide links for
further reading.
A good place to start is this article on biological organisms in Physics Today from 2004, by Geoffrey
West and James Brown - "Life's
Universal Scaling Laws". A couple of key passages (all bold text corresponds
to my emphasis):
In marked contrast to the amazing diversity and complexity of living
organisms is the remarkable simplicity of the scaling behavior of key
biological processes over a broad spectrum of phenomena and an immense range
of energy and mass. Scaling as a manifestation of underlying dynamics
and geometry is familiar throughout physics. It has been instrumental in
helping scientists gain deeper insights into problems ranging across the
entire spectrum of science and technology, because scaling laws typically
reflect underlying generic features and physical principles that are
independent of detailed dynamics or specific characteristics of particular
models. Phase transitions, chaos, the unification of the fundamental forces
of nature, and the discovery of quarks are a few of the more significant
examples in which scaling has illuminated important universal principles or
structure.
In biology, the observed scaling is typically a simple power law: Y
= Y0Mb, where Y is some observable, Y0
a constant, and M the mass of the organism.1-3 Perhaps of
even greater significance, the exponent b almost invariably
approximates a simple multiple of 1/4. Among the many fundamental variables
that obey such scaling laws - termed "allometric" by Julian Huxley4
- are metabolic rate, life span, growth rate, heart rate, DNA nucleotide
substitution rate, lengths of aortas and genomes, tree height, mass of
cerebral grey matter, density of mitochondria, and concentration of RNA.
[...]
An intriguing consequence of these "quarter-power" scaling laws is the
emergence of invariant quantities,7 which physicists recognize as
usually reflecting fundamental underlying constraints. For example,
mammalian life span increases as approximately M1/4,
whereas heart rate decreases as M-1/4, so the number of
heartbeats per lifetime is approximately invariant (about 1.5 x 109),
independent of size...
Bettencourt and West discussed cities and their economics and laws of
scaling in their article in Scientific
American last year titled "Bigger
Cities Do More With Less". Here are some notable points:
By sifting through this flood of data, covering thousands of cities
around the world, we have unveiled several mathematical “laws” that explain
how concentrating people in one place affects economic activity, return on
infrastructure investment and social vitality. Despite the rich diversity of
metropolitan regions across the U.S., China, Brazil and other nations, we
found a remarkable universality in the way that socioeconomic
characteristics increase with a city’s population. For example, if
the population of a city is doubled, whether from 40,000 to 80,000 or from
four million to eight million, we systematically see an average increase of
around 15 percent in measures such as wages and patents produced per capita.
If eight million people all live in one city, their economic output will
typically be about 15 percent greater than if the same eight million people
lived in two cities of half the size. We call this effect “superlinear
scaling”: the socioeconomic properties of cities increase faster than a
direct (or linear) relation to their population would predict.
The data also reveal that cities’ use of resources follows a similar,
though inverted, law. When the size of a city doubles, its material
infrastructure—anything from the number of gas stations to the total length
of its pipes, roads or electrical wires—does not. Instead these quantities
rise more slowly than population size: a city of eight million typically
needs 15 percent less of the same infrastructure than do two cities of four
million each. This pattern is referred to as sublinear scaling. On average,
the bigger the city, the more efficient its use of infrastructure, leading
to important savings in materials, energy and emissions.
Our findings also show that these patterns of increased productivity and
decreased costs hold true across nations with very different levels of
development, technology and wealth. Although we have much more
information for cities in richer parts of the world, we are beginning to
obtain good data from rapidly developing countries as well, and they seem to
fit the same mold. The gross domestic product for cities in Brazil and
China, for instance, closely follows the same superlinear curve that western
European and North American cities exhibit, though starting from a lower
baseline. We believe that the pattern holds true because the same basic
social and economic processes are at work, whether in São Paulo’s favelas,
under Beijing’s smog-filled skies or along Copenhagen’s tidy streets.
Although urban superlinear scaling, which represents the average,
idealized behavior of a city of a given size, prevails around the globe,
actual cities deviate to varying degrees from the roughly 15 percent
enhancements that come with size. Detailed data covering 40 years show,
for example, that San Francisco and Boston are richer than their size would
indicate, whereas Phoenix or Riverside, Calif., are somewhat poorer.
Curiously, these deviations persist for decades: cities tend to stay
remarkably close to their overperforming or underperforming histories.
For example, cities that have attempted to improve their lot by creating
conditions for the “next Silicon Valley” have often had disappointing
results. Our research suggests that certain intangible qualities of
social dynamics—more than the development of material infrastructure—hold
the key to generating virtuous cycles of innovation and creation of wealth.
These processes, such as the development of a spirit of local
entrepreneurship, a reputation for cutting-edge novelty, and a culture of
excellence and competitiveness, are difficult to design through policy
because they rely on the dynamics of a city’s social fabric across many
dimensions. We expect the results of this exciting area of research will
lead to better “recipes” for sustainable socioeconomic development.
See the Physics Today article as well
this article for some discussion of West's work, including some of the
criticisms of the averages and how they miss various deviations from the
averages. A recent paper that critiques the Bettencourt/West research on cities
is this one by
Cosma Shalizi, who says: