Scale: The Universal Laws of Growth, Innovation, Sustainability, and the Pace of Life, in Organisms, Cities, Economies, and Companies

by Geoffrey West

A survivorship curve simply represents the probability that an individual will live to a given age and is determined by plotting the percentage of survivors in a given population as a function of their age. Its converse is called a mortality curve and is the percentage of people who have died at a given age, representing the probability that an individual will die at that age. Biologists, actuaries, and gerontologists have coined the term mortality or death rate to denote the number of deaths in a population that occur in some given period of time (a month, say) relative to the number that are still alive.

Thus, shrews have heart rates of roughly 1,500 beats a minute and live for about two years, whereas heart rates of elephants are only about 30 beats a minute but they live for about seventy-five years. Despite their vast difference in size, both of their hearts beat approximately one and a half billion times during an average lifetime. This invariance is approximately true for all mammals, even though there are large fluctuations for the reasons I outlined above. The

The existence of approximate invariant quantities as well as of scaling laws in the complex process of aging and death provides important hints that these processes are not arbitrary and suggests that there might be coarse-grained laws and principles at play. Even more tantalizing is that the scaling laws of longevity have the same quarter-power structure as all other physiological and life-history events.

As we begin to lose the multiple localized battles against entropy we age, ultimately losing the war and succumbing to death. Entropy kills.

The tension between form and function, between town and country, between organic evolutionary development and the parsimony of unadorned steel and concrete, and between the complexity of fractal-like curves and surfaces and the simplicity of Euclidean geometry, remains an ongoing debate with no simple resolution or any easy answer.

Furthermore, the slope of the straight line, which is the exponent of the power law, is about 0.85, a little bit higher than the 0.75 (the famous ¾) we saw for the metabolic rate of organisms (Figure 1). Equally intriguing is that this exponent takes on approximately the same value for how gasoline stations scale across all of the countries shown in the figure. This value of around 0.85 is smaller than 1, so in the language developed earlier, the scaling is sublinear, indicating a systematic economy of scale, meaning that the bigger the city the fewer the number of gas stations needed on a per capita basis.

Thus these metrics not only scale in an extremely simple fashion following classic power law behavior, but they all do it in approximately the same way with a similar exponent of approximately 1.15 regardless of the urban system. So in marked contrast to infrastructure, which scales sublinearly

with population size, socioeconomic quantities—the very essence of a city—scale superlinearly, thereby manifesting systematic increasing returns to scale. The larger the city, the higher the wages, the greater the GDP, the more crime, the more cases of AIDS and flu, the more restaurants, the more patents produced, and so on, all following the

“15

percent rule” on a per capita basis in urban systems ...

Thus the larger the city the more innovative “social capital” is created, and consequently, the more the average citizen owns, produces, and consumes, whether it’s goods, resources, or ideas.

Typically, the fractal dimension of a healthy robust city steadily increases as it grows and develops, reflecting a greater complexity as more and more infrastructure is built to accommodate an expanding population engaging in more and more diverse and intricate activities. But conversely, its fractal dimension decreases when it goes through difficult economic times or when it temporarily contracts.

He and his collaborators found that at the lowest level of the hierarchy the number of people with whom the average individual has his or her strongest relationships is, at any one time, only about five.

The next level up contains those you usually refer to as close friends with whom you enjoy spending meaningful time and might still turn to in time of need even if they are not on as intimate terms with you as your inner circle. This typically comprises around fifteen people. In the level above this are people you might still call friends though you would only rarely invite them to dinner but would likely invite them to a party or gathering.

these successive levels of the group hierarchy—5, 15, 50, 150—are sequentially related to each other by an approximately constant scaling factor of about three.

The important point for our purposes is that viewed through a coarse-grained lens, social networks exhibit an approximate fractal-like pattern and that this seems to hold true across a broad spectrum of different social organizations.

Furthermore, because the geometry of white and gray matter in our brains, which forms the neural circuitry responsible for all of our cognitive functions, is itself a fractal-like hierarchical network, this suggests that the hidden fractal nature of social networks is actually a representation of the physical structure of our brains. This speculation can be taken one step further by invoking the idea that the structure and organization of cities are determined by the structure and dynamics of social networks, in which case the universal fractality of cities can be viewed as a projection of the universal fractality of social networks.

In economics, Zipf’s law actually predates Zipf. Much earlier it had been discovered by the influential Italian economist Vilfredo Pareto, who expressed it as a frequency distribution of incomes in a population rather than in terms of their ranking. This distribution, which is valid for many other economic metrics like income, wealth, and the size of companies, follows a simple power law with an exponent of approximately -2.

Many of the most interesting phenomena that we have touched upon fall into this category, including the occurrence of disasters such as earthquakes, financial market crashes, and forest fires. All of these have fat-tail distributions with many more rare events, such as enormous earthquakes, large market crashes, and raging forest fires, than would have been predicted by assuming that they were random events following a classic Gaussian distribution.

The two dominant components that constitute a city, its physical infrastructure and its socioeconomic activity, can both be conceptualized as approximately self-similar fractal-like network structures.

This exercise shows that there is a natural explanation for why social connectivity and therefore socioeconomic quantities scale superlinearly with population size. Socioeconomic quantities are the sum of the interactions or links between people and therefore depend on how correlated they are.

The integration of these two kinds of networks, namely, the requirement that socioeconomic interaction represented by space-filling fractal-like social networks must be anchored to the physicality of a city as represented by space-filling fractal-like infrastructural networks, determines the number of interactions an average urban dweller can sustain in a city.

Consequently, the exponent controlling social interactions, and therefore all socioeconomic metrics—the universal 15 percent rule for how the good, the bad, and the ugly scale with city size—is bigger than 1 (1.15) to the same degree that the exponent controlling infrastructure and flows of energy and resources is less than 1 (0.85), as observed in the data. Pictorially, the degree to which all of the slopes in Figures 34–38 exceed 1 is the same as the degree to which they are less than 1 in Figure 33.

The systematic increase in social interaction is the essential driver of socioeconomic activity in cities: wealth creation, innovation, violent crime, and a greater sense of buzz and opportunity are all propagated and enhanced through social networks and greater interpersonal interaction.

the sublinearity of infrastructure and energy use is the exact inverse of the superlinearity of socioeconomic activity.

This effective speeding up of time is an emergent phenomenon generated by the continuous positive feedback mechanisms inherent in social networks in which social interactions beget ever more interactions, ideas stimulate yet more ideas, and wealth creates more wealth as size increases.

Amusingly, data confirm that walking speeds do indeed increase with city size following an approximate power law, though its exponent is somewhat less than the canonical 0.15, being closer to 0.10 (see Figure 42).

the total number of contacts between people in a city over an extended period of time is plotted logarithmically versus the population size of the city. As you can see, a classic straight line is revealed for both sets of data, indicating power law scaling with the exponent in both cases having the same value very close to the predicted 1.15, in spectacular agreement with the hypothesis.

The scaling of four disparate urban metrics—income, GDP, crime, and patents—rescaled from Figures 34–38 to show how they all scale with a similar exponent of about 1.15. (45) The scaling of the connectivity between people as measured by the number of reciprocal phone calls between individuals across cities in Portugal and the United Kingdom showing a similar exponent confirming the prediction of the theory. (46) The size of modular groups of friends of individuals is approximately the same regardless of the size of the city.

It is an example of an amazing general symmetry of mobility: if the distance traveled multiplied by the frequency of visits to any specific location is kept the same, then the

number of people visiting also remains the same.

The proportionality constant is 21.6, meaning that there is approximately one establishment for about every 22 people in a city, regardless of the city size.

The universality is driven by the constraint that the sum total of all the different businesses in a city scales linearly with population size, regardless of the detailed composition of business types or of the city.

It is often conveniently forgotten that Jesus was actually referring to knowledge of the mysteries of the kingdom of heaven and not to material wealth. He was expressing a spiritual version of the very essence of diligent study, knowledge accumulation, and research and education as expressed by the ancient rabbis: He who does not increase his knowledge decreases it.

The general rule is that business types whose abundances scale superlinearly with population size systematically rise in their rankings, whereas those that scale sublinearly systematically decrease.

traditional sectors such as agriculture, mining, and utilities scale sublinearly; the theory predicts that the rankings and relative abundances of these industries decrease as cities get larger. On the other hand, informational and service businesses such as professional, scientific, and technical services, and management of companies and enterprises, scale superlinearly and are consequently predicted to increase disproportionally with city size, as observed.

Even though the conceptual and mathematical structure of the growth equation is the same for organisms, social insect communities, and cities, the consequences are quite different: sublinear scaling and economies of scale that dominate biology lead to stable bounded growth and the slowing down of the pace of life, whereas superlinear scaling and increasing returns to scale that dominate socioeconomic activity lead to unbounded growth and to an accelerating pace of life.

It recognizes up front that the economy is typically not in equilibrium but is an evolving system with emergent properties that result from the underlying interactions between its multiple constituent parts.

scaling laws are a consequence of the optimization of the network structures that sustain these various systems resulting from the continuous feedback mechanisms inherent in natural selection and the “survival of the fittest.”

In the case of cities, we would therefore expect the emergent scaling laws to exhibit much greater variance around idealized power laws than organisms do, because the time over which evolutionary forces have acted is so much shorter. Comparing fits to scaling in the two cases, such as Figure 1 for animal metabolic rates versus Figure 3 for the patent production in cities, confirms this prediction: there is a consistently larger spread around the fits for cities than for organisms. Extrapolating this to companies where “evolutionary” timescales are even shorter suggests that if they do indeed scale, there should be an even greater spread in the data around idealized scaling curves than for cities and organisms.

As we saw in chapter 4, sublinear scaling in biology leads to bounded growth and a finite life span, whereas in chapter 8 we saw that the superlinear scaling of cities (and of economies) leads to open-ended growth.

Intriguingly, companies manifest yet another variation on this general theme by following a path that sits at the cusp between organisms and cities. Their effective metabolic rate is neither sub- nor superlinear but falls right in the middle by being linear. This is illustrated in Figures 63 and 64, where sales are plotted

all large mature companies have stopped growing. Their growth curves when corrected for both inflation and the expansion of the market now look just like typical sigmoidal growth curves of organisms in which growth ceases at maturity, as illustrated in Figures 15–18 of chapter 4.

Of the 28,853 companies that have traded on U.S. markets since 1950, 22,469 (78 percent) had died by 2009. Of these 45 percent were acquired by or merged with other companies, while only about 9 percent went bankrupt or were liquidated; 3 percent privatized, 0.5 percent underwent leveraged buyouts, 0.5 percent went through reverse acquisitions, and the remainder disappeared for “other reasons.”

The half-life of U.S. publicly traded companies was found to be close to 10.5 years, meaning that half of all companies that began trading in any given year have disappeared in 10.5 years.

According to the Bank of Korea, of the 5,586 companies that were more than two hundred years old in 2008, over half (3,146 to be precise) were Japanese, 837 German, 222 Dutch, and 196 French. Furthermore, 90 percent of those that were more than one hundred years old had fewer than three hundred employees.

In biology, the network principles underlying economies of scale and sublinear scaling have two profound consequences. They constrain the pace of life—big animals live longer, evolve more slowly, and have slower heart rates, all to the same degree—and limit growth.

In contrast, cities and economies are driven by social interactions whose feedback mechanisms lead to the opposite behavior. The pace of life systematically increases with population size: diseases spread faster, businesses are born and die more often, and people even walk faster in larger cities, all by approximately the same 15 percent rule.

This can be restated as a sort of “theorem”: to sustain open-ended growth in light of resource limitation requires continuous cycles of paradigm-shifting innovations, as illustrated in Figure 78.

Qualitatively, this extreme version of reductionism may indeed have some partial validity, though I’m not sure to what extent anyone actually believes it—but, in any case, something is missing. The “something” includes many of the concepts and ideas implicit in a lot of the problems and questions considered in this book: concepts like information, emergence, accidents, historical contingency, adaptation, and selection, all characteristics of complex adaptive systems whether organisms, societies, ecosystems, or economies. These are composed of myriad individual constituents or agents that take on collective characteristics that are generally unpredictable in detail from their underlying components even if the dynamics of their interactions are known.