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

by Geoffrey West

So who was Isambard Kingdom Brunel, and why is he famous? Many consider him the greatest engineer of the nineteenth century, a man whose vision and innovations, particularly concerning transport, helped make Britain the most powerful and richest nation in the world. He was a true engineering polymath who strongly resisted the trend toward specialization.

The scaling of metabolic rate has been known for more than eighty years. Although primitive versions of it were known before the end of the nineteenth century, its modern incarnation is credited to the distinguished physiologist Max Kleiber, who formalized it in a seminal paper published in an obscure Danish journal in 1932.5

In Figure 1 you can readily see that for every four orders of magnitude increase in mass (along the horizontal axis), metabolic rate increases by only three orders of magnitude (along the vertical axis), so the slope of the straight line is ¾, the famous exponent in Kleiber’s law.

More generally: if the mass is increased by any arbitrary factor at any scale (100, in the example), then the metabolic rate increases by the same factor (32, in the example) no matter what the value of the initial mass is, that is, whether it’s that of a mouse, cat, cow, or whale. This remarkably systematic repetitive behavior is called scale invariance or self-similarity and is a property inherent to power laws.

For example, the exponent for growth rates is very close to ¾, for lengths of aortas and genomes it’s ¼, for heights of trees ¼, for cross-sectional areas of both aortas and tree trunks ¾, for brain sizes ¾, for cerebral white and gray matter 5⁄4, for heart rates minus ¼, for mitochondrial densities in cells minus ¼, for rates of evolution minus ¼, for diffusion rates across membranes minus ¼, for life spans ¼ . . . and many, many more. The “minus” here simply indicates that the corresponding quantity decreases with size rather than increases, so, for instance, heart rates decrease with increasing body size following the ¼ power law, as shown in Figure 10.

Huxley’s term allometric was extended from its more restrictive geometric, morphological, and ontogenetic origins to describe the kinds of scaling laws that I discussed above, which include more dynamical phenomena such as how flows of energy and resources scale with body size, with metabolic rate being the prime example. All of these are now commonly referred to as allometric scaling laws.

The idea behind the concept of space filling is simple and intuitive. Roughly speaking, it means that the tentacles of the network have to extend everywhere throughout the system that it is serving, as is illustrated in the networks here.

II. The Invariance of Terminal Units

Similarly, there is relatively little variation in petioles, the last branch of a tree prior to the leaf, or even in the size of leaves themselves, during the growth of a tree from a tiny sapling to a mature tree that might be a hundred or more feet high. This is also true across species of trees: leaves do vary in size but by a relatively small factor, despite huge factors in the variation of their heights and masses.

In a similar fashion, the blood wave traveling along the aorta is partially reflected back when it meets the branch point, the remainder being transmitted down through the daughter arteries. These reflections have potentially very bad consequences because they mean that your heart is effectively pumping against itself. Furthermore, this effect gets hugely enhanced as blood flows down through the hierarchy of vessels because the same phenomenon occurs at each ensuing branch point in the network, resulting in a large amount of energy being expended by your heart just in overcoming these multiple reflections. This would be an extremely inefficient design, resulting in a huge burden on the heart and a huge waste of energy.

Because the cross-sectional area of any vessel is proportional to the square of its radius, another way of expressing this result is to say that the square of the radius of the parent tube has to be just twice the square of the radius of each of the daughters. So to ensure that there is no energy loss via reflections as one progresses down the network, the radii of successive vessels must scale in a regular self-similar fashion, decreasing by a constant factor of the square root of two (√2) with each successive branching.

It’s a lovely thought that the optimum design of our circulatory system obeys the same simple area-preserving branching rules that trees and plants do. It’s equally satisfying that the condition of nonreflectivity of waves at branch points in pulsatile networks is essentially identical to how national power grids are designed for the efficient transmission of electricity over long distances. This condition of nonreflectivity is called impedance matching. It has multiple applications not only in the working of your body but across a very broad spectrum of technologies that play an important part in your daily life. For example, telephone network systems use matched impedances to minimize echoes on long-distance lines;

The term impedance matching can be a very useful metaphor for connoting important aspects of social interactions. For example, the smooth and efficient functioning of social networks, whether in a society, a company, a group activity, and especially in relationships such as marriages and friendships, requires good communication in which information is faithfully transmitted between groups and individuals. When information is dissipated or “reflected,”

The invention of the AC induction motor and transformer in 1886 by the brilliant charismatic inventor and futurist Nikola Tesla marked a turning point and signaled the beginning of the “war of currents.” In the United States this turned into a battle royal between the Thomas Edison Company (later General Electric) and the George Westinghouse Company.

The effect of this energy loss is to progressively dampen the wave on its way down through the network hierarchy until it eventually loses its pulsatile character and turns into a steady flow. In other words, the nature of the flow makes a transition from being pulsatile in the larger vessels to being steady in the smaller ones. That’s why you feel a pulse only in your main arteries—there’s almost no vestige of it in your smaller vessels. In the language of electrical transmission, the nature of the blood flow changes from being AC to DC as it progresses down through the network.

You are very likely aware of this huge difference in speeds between capillaries and the aorta. If you prick your skin, blood oozes out very slowly from the capillaries with scant resulting damage, whereas if you cut a major artery such as your aorta, carotid, or femoral, blood gushes out and you can die in just a matter of minutes.

But what’s really surprising is that blood pressures are also predicted to be the same across all mammals, regardless of their size.

The resulting magic number four emerges as an effective extension of the usual three dimensions of the volume serviced by the network by an additional dimension resulting from the fractal nature of the network. I shall go into this in more detail in the following chapter, where I discuss the general concept of fractal dimension, but suffice it to say here that natural selection has taken advantage of the mathematical marvels of fractal networks to optimize their distribution of energy so that organisms operate as if they were in four dimensions, rather than the canonical three. In this sense the ubiquitous number four is actually 3 + 1.

Implicit in our measurement process, whatever it is, is the assumption that with increasing resolution the result converges to an increasingly accurate fixed number, which we call the length of the room, a presumably objective property of your living room.

However, to his great surprise, Richardson found that when he carried out this standard iterative procedure using calipers on detailed maps, this simply wasn’t the case. In fact, he discovered that the finer the resolution, and therefore the greater the expected accuracy, the longer the border got, rather than converging to some specific value! Unlike lengths of living rooms, the lengths of borders and coastlines continually get longer rather than converging to some fixed number, violating the basic laws of measurement that had implicitly been presumed for several thousand years.

To appreciate what these numbers mean in English, imagine increasing the resolution of the measurement by a factor of two; then, for instance, the measured length of the west coast of Britain would increase by about 25 percent and that of Norway by over 50 percent.

Natural selection has taken advantage of the fractal nature of space-filling networks to maximize the total effective surface area of these terminal units and thereby maximize metabolic output. Geometrically, the nested levels of continuous branching and crenulations inherent in fractal-like structures optimize the transport of information, energy, and resources by maximizing the surface areas across which these essential features of life flow. Because of their fractal nature, these effective surface areas are very much larger than their apparent physical size. Let me give you some remarkable examples from your own body to illustrate the point. Even though your lungs are only about the size of a football with a volume of about 5 to 6 liters (about one and a half gallons), the total surface area of the alveoli, which are the terminal units of the respiratory system where oxygen and carbon dioxide are exchanged with the blood, is almost the size of a tennis court and the total length of all the airways is about 2,500 kilometers,

This additional dimension, which arises from optimizing network performance, leads to organisms’ functioning as if they are operating in four dimensions. This is the geometric origin of the quarter power. Thus, instead of scaling with classic ⅓ exponents, as would be the case if they were smooth nonfractal Euclidean objects, they scale with ¼ exponents. Although living things occupy a three-dimensional space, their internal physiology and anatomy operate as if they were four-dimensional.

One of the surprising results that emerges is that the mortality rate for most organisms remains approximately unchanged with age. In other words, the relative number of individuals that die in any time period is the same at any age. So, for example, if 5 percent of the surviving population dies between ages five and six, then 5 percent of the surviving population will also die between ages forty-five and forty-six and between ninety-five and ninety-six. This sounds nonintuitive, but if we put it a different way it will make more sense. A constant mortality rate means that the number of individuals that die in some time period is directly proportional to how many have survived up until that time. If you go back to the earlier discussion on exponential behavior in chapter 3, you will discover that this is precisely the mathematical definition of the exponential function

I will not be discussing in any detail how the rates of physical and chemical processes depend on temperature, other than to reiterate

that they scale exponentially rather than as a power law. Consequently, the processes that govern the weather and the life history of plants and animals are exponentially sensitive to small changes in the temperature at which they operate. I remind you that a 2°C rise in average temperature leads to a whopping 20 percent increase in these rates.

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, on average, each gas station in a larger city serves more people and consequently sells more fuel per month than in a smaller one. To put it slightly differently, with each doubling of population size, a city needs only about 85 percent more gas stations—and not twice as many as might naively be expected—so there is a systematic savings of about 15 percent with each doubling.

So as far as their overall infrastructure is concerned, cities behave just like organisms—they scale sublinearly following simple power-law behavior, thereby exhibiting a systematic economy of scale, albeit to a lesser degree as represented by the different values of their exponents (0.75 for organisms vs. 0.85 for cities).

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 across the globe.

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.

There was, however, one major exception to this and that is a famous scaling law known as Zipf’s law for the ranking of cities in terms of their population size. This is shown graphically in Figure 39. It’s an intriguing observation: in its simplest form, it states that the rank order of a city is inversely proportional to its population size. Thus, the largest city in an urban system should be about twice the size of the second largest, three times the size of the third largest, four times the size of the fourth largest, and so on. So, for example, in the 2010 census, the biggest city in the United States was New York with a population of 8,491,079. According to Zipf’s law, the second largest city, Los Angeles, should have a population of approximately one half of this, namely 4,245,539, the third largest, Chicago, should have a population of about one third, or 2,830,359, and Houston, the fourth largest, should have a population of about one quarter, namely 2,122,769, and so on. The actual numbers were: Los Angeles, 3,928,864; Chicago, 2,722,389; and Houston, 2,239,558, all in reasonably good agreement with Zipf’s law to within less than 7 percent. Zipf’s law owes its name to the Harvard linguist George Kingsley Zipf, who popularized it through his fascinating book Human Behavior and the Principle of Least Effort, published in 1949.17 He first enunciated his law in 1935, not for cities but for the frequency of word use in languages.

In this spirit, individuals are considered to be the “invariant terminal units” of social networks, meaning that on average each person operates in roughly the same amount of social and physical space in a city. This is in keeping with the implications of a “universal” Dunbar number and the space-time limitations on mobile activity in cities that we just discussed. Recall that the physical space in which we operate is spanned by space-filling fractal networks, such as roads and utility lines that service infrastructural terminal units such as houses, stores, and office buildings where we reside, work, and interact, and between which we also have to move. 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. And as discussed earlier, it is this number that determines how socioeconomic activity scales with population size.

A crucial aspect of the scaling of companies is that many of their key metrics scale sublinearly like organisms rather than superlinearly like cities. This suggests that companies are more like organisms than cities and are dominated by a version of economies of scale rather than by increasing returns and innovation. This has profound implications for their life history and in particular for their growth and mortality. 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.

company can be thought of as its “metabolism” while expenses can be thought of as its “maintenance” costs. In biology, metabolic rate scales sublinearly with size, so as organisms increase in size the supply of energy cannot keep up with the maintenance demands of cells, leading to the eventual cessation of growth. On the other hand, the social metabolic rate in cities scales superlinearly, so as cities grow the creation of social capital increasingly outpaces the demands of maintenance, leading to faster and faster open-ended growth.

This is good news because, mathematically, linear scaling leads to exponential growth and this is what all companies strive for. Furthermore, this also shows why, on average, the economy continues to expand at an exponential rate because the overall performance of the market is effectively an average over the growth performances of all its individual participating companies. Although this may be good news for the overall economy, it sets a major challenge for each individual company because each one has to keep up with an exponentially expanding market. So even if a company is growing exponentially (the good news), this may not be sufficient for it to survive unless its expansion rate is at least that of the market (the bad news). This primitive version of the “survival of the fittest” for companies is the essence of the free market economy.

Furthermore, the upward trends of these older, slower-growing companies all follow an approximate straight line with similar shallow slopes. On this semilogarithmic plot, where the vertical axis (sales) is logarithmic but the horizontal one (time) is linear, a straight line means mathematically that sales are growing exponentially with time. Thus, on average, all surviving companies eventually settle down to a steady but slow exponential growth, as predicted.

After growing rapidly in their youth, almost all companies with sales over about $10 million end up floating on top of the ripples of the stock market. Of these, many operate with their metaphorical noses just above the surface. This is a precarious situation because if a big wave comes along they may well drown.

What is the special property of exponentials that they describe the decay of so many disparate systems? It’s simply that they arise whenever the death rate at any given time is directly proportional to the number that are still alive. This is equivalent to saying that the percentage of survivors that die within equal slices of time at any age remains the same. A simple example will make this clear: taking one year as the time slice, this says that the percentage of five-year-old companies that die before they reach six years old is the same as the percentage of fifty-year-old companies that die before they reach fifty-one. In other words: the risk of a company’s dying does not depend on its age or size.

Before addressing some of the consequences of this phenomenon, let me first elaborate on some of its salient features. Simple power laws and exponentials are continuously increasing functions that also eventually become infinitely large, but they take an infinite time to do so. Another way of saying this is that in these cases the “singularity” has been pushed off to an infinite time into the future, thereby rendering it “harmless” relative to the potential impact of a finite time singularity. In the case of growth driven by superlinear scaling, the approach to the finite time singularity, represented by the solid line in Figure 76, is faster than exponential. This is often referred to as superexponential, a term I’ve already used earlier when discussing the growth of cities.

I want to emphasize that this situation is qualitatively quite different from classic Malthusian dynamics, where this is no such singularity. The existence of a singularity signifies that there has to be a transition from one phase of the system to another having very different characteristics, analogous to the way the condensation of steam to water which subsequently freezes to ice epitomizes transitions between different phases of the same system, each having quite different physical properties. And indeed, underlying such familiar phase transitions are singularities in the thermodynamic variables characterizing the system (water) but in terms of temperature rather than time (0°C for freezing and 100°C for boiling). Unfortunately, for cities and socioeconomic systems the phase transition stimulated by the finite time singularity is from superexponential growth to stagnation and collapse, and this could lead to potentially devastating consequences.

A major innovation effectively resets the clock by changing the conditions under which the system has been operating and growth occurring. Thus, to avoid collapse a new innovation must be initiated that resets the clock, allowing growth to continue and the impending singularity to be avoided

There’s yet another major catch, and it’s a big one. The theory dictates that to sustain continuous growth the time between successive innovations has to get shorter and shorter. Thus paradigm-shifting discoveries, adaptations, and innovations must occur at an increasingly accelerated pace. Not only does the general pace of life inevitably quicken, but we must innovate at a faster and faster rate!

The theory explains and predicts this inverse relationship between the successive shortening of the time between innovations and how long ago they happened and is quantitatively consistent with the lines drawn on both graphs.

Until it was popularized by Kurzweil in his 2005 book The Singularity Is Near: When Humans Transcend Biology, “singularity” was a term that was hardly used colloquially.

Building on an earlier idea of a “technological singularity” introduced in 1993 by the science-fiction writer and computer scientist Vernor Vinge, Kurzweil proposed that we are approaching a singularity in which our bodies and brains will be augmented by genetic alterations, nanotechnology, and artificial intelligence to become hybrid cyborgs no longer bound by the constraints of biology. It was suggested that this would result in a collective intelligence that will be enormously more powerful than all present human intelligence combined.