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:
...Re-analysis of the gross economic production and personal income for cities in the United States, however, shows that the data cannot distinguish between power laws and other functional forms, including logarithmic growth, and that size predicts relatively little of the variation between cities. The striking appearance of scaling in previous work is largely artifact of using extensive quantities (city-wide totals) rather than intensive ones (per-capita rates). The remaining dependence of productivity on city size is explained by concentration of specialist service industries, with high value-added per worker, in larger cities, in accordance with the long-standing economic notion of the “hierarchy of central places”.
Here are some papers that discuss the research on cities in more detail:
- Bettencourt et al., "Growth, innovation, scaling, and the pace of life in cities", Proc. Nat. Acad. Sci. 104 (2007), p. 7301
- Bettencourt et al., "Urban Scaling and Its Deviations: Revealing the Structure of Wealth, Innovation and Crime across Cities", PLoS One 5 (2010), p. e13541
Obviously a super-linear growth rate poses challenges. As Jenna Beck observed:
The superlinear growth rates also suggest that a city can grow indefinitely. When the researchers plugged the scaling exponents into an urban growth equation, they found that cities driven by economies of scale were destined to plateau, whereas those driven by innovation or wealth creation had the potential for unbounded growth. “Should a city have a finite size or should it grow forever? How should it grow? You would argue about it forever if you hadn’t measured,” Bettencourt says.
Sustaining that growth is the city’s chief challenge. To grow indefinitely, a city has to periodically reset its growth rate. Such “resetting” can come from innovations that revitalize the economy, or from outside factors, such as shifts in immigration. The pattern that an ever-growing city falls into is one of successive growth cycles—each one shorter than the last as the size of the city increases. “You’re on this treadmill and you’ve got to go on making these changes, these innovative changes, faster and faster because if you don’t you’ll stagnate and collapse,” West says.
How about corporations? West says corporations are closer to organisms in how they scale, rather than to cities. A couple of blog posts cover this topic. For example, Stewart Brand at The Long Now Foundation discusses a West lecture from July saying:
Are corporations more like animals or more like cities? They want to be like cities, with ever increasing productivity as they grow and potentially unbounded lifespans. Unfortunately, West et al.'s research on 22,000 companies shows that as they increase in size from 100 to 1,000,000 employees, their net income and assets (and 23 other metrics) per person increase only at a 4/5 ratio. Like animals and cities they do grow more efficient with size, but unlike cities, their innovation cannot keep pace as their systems gradually decay, requiring ever more costly repair until a fluctuation sinks them. Like animals, companies are sublinear and doomed to die.
The second major topic of the lecture was death, and why corporations die while cities appear immortal. In animals, death is the result of entropy. Metabolic processes create free radicals, which eventually overwhelm cellular repair mechanisms, and cause some vital organ to fail. Cities, compared to organisms, are incredibly resilient. They can be sacked, suffer industrial collapse, even get nuked, and still bounce back in a couple of decades. What Dr. West didn't elucidate were boundaries and conditions. Animals have well-defined boundaries between the organism and the world, and between alive and dead, while cities are far loser agglomerations. Is Roman Londinium the same city as British London? Cities have kept entropy at bay because the global population is continually increasing, and it's easier for a city to make new citizens than it is for an animal to make new cells.
Corporations are like cities, in that they are agglomerations of humans, but observationally, the data shows that like animals, corporations are sub-linear; larger corporations generate less income per employee. From this, Dr. West concludes that corporations are bound to die, which I think is an artifact of defining a corporation in terms of its legal charter, rather than the products it makes, the employees who work there, or equipment it uses. The names on the outside of a building are as relevant to the real business of business as the stripes on a leopard are to the business of a predatory cat. It's not the legal labels that matters, but the people, capital, and ideas. In that sense, the data is totally inadequate to explain corporate behavior.
In the context of corporations and what scaling laws they seem to follow on average and how that compares with cities and organisms, Geoffrey West's fascinating talk and charts at TED in July 2011 is a good one - also, read the accompanying interview with West here.
More readings on this topic:
1) Steven Strogatz in The Opinionator at NYT, "Math and the City" (May 2009)
2) Jim Giles in Scientific American, "Stubborn US cities rated in personality test" (Nov 2010)