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

by Geoffrey West

As social animals now living in cities we still need just a lightbulb equivalent of food to stay alive but, in addition, we now require homes, heating, lighting, automobiles, roads, airplanes, computers, and so on. Consequently, the amount of energy needed to support an average person living in the United States has risen to an astounding 11,000 watts. This social metabolic rate is equivalent to the entire needs of about a dozen elephants. Furthermore, in making this transition from the biological to the social our overall population has increased from just a few million to more than seven billion. No wonder there’s a looming energy and resource crisis.

Like death, taxes, and the Sword of Damocles, the Second Law of Thermodynamics hangs over all of us and everything around us. Dissipative forces, analogous to the production of disorganized heat by friction, are continually and inextricably at work leading to the degradation of all systems.

A typical complex system is composed of myriad individual constituents or agents that once aggregated take on collective characteristics that are usually not manifested in, nor could easily be predicted from, the properties of the individual components themselves.

To put it in simple terms, scaling implies that if a city is twice the size of another city in the same country (whether 40,000 vs. 20,000 or 4 million vs. 2 million), then its wages, wealth, number of patents, AIDS cases, violent crime, and educational institutions all increase by approximately the same degree (by about 15 percent above mere doubling), with similar savings in all of its infrastructure. The bigger the city, the more the average individual systematically owns, produces, and consumes, whether goods, resources, or ideas. The good, the bad, and the ugly are integrated in an approximately predictable package: a person may move to a bigger city drawn by more innovation, a greater sense of “action,” and higher wages, but she can also expect to confront an equivalent increase in the prevalence of crime and disease.

relative strength becomes progressively weaker as size increases. Or, as Galileo so graphically put it: “the smaller the body the greater its relative strength. Thus a small dog could probably carry on his back two or three dogs of his own size; but I believe that a horse could not carry even one of his own size.”

Drugs, like metabolites and oxygen, are typically transported across surface membranes, sometimes via diffusion and sometimes through network systems. As a result, the dose-determining factor is to a significant degree constrained by the scaling of surface areas rather than the total volume or weight of an organism, and these scale nonlinearly with weight. A simple calculation using the ⅔ scaling rule for areas as a function of weight shows that a more appropriate dose for elephants should be closer to a few milligrams of LSD rather than the several hundred that were actually administered.

Whales live in the ocean, elephants have trunks, giraffes have long necks, we walk on two legs, and dormice scurry around, yet despite these obvious differences, we are all, to a large degree, nonlinearly scaled versions of one another. If you tell me the size of a mammal, I can use the scaling laws to tell you almost everything about the average values of its measurable characteristics: how much food it needs to eat each day, what its heart rate is, how long it will take to mature, the length and radius of its aorta, its life span, how many offspring it will have, and so on. Given the extraordinary complexity and diversity of life, this is pretty amazing.

In trying to develop a theory, he hypothesized that the probability of war between neighboring states was proportional to the length of their common border. Driven by his passion to test his theory, he turned his attention to figuring out how the lengths of borders are measured . . . and in so doing inadvertently discovered fractals.

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. Equally surprising, Richardson discovered that this increase in length progressed in a systematic fashion. When he plotted the length of various borders and coastlines versus the resolution used to make the measurements on a logarithmic scale, it revealed a straight line indicative of the power law scaling we’ve seen in many other places

The take-home message is clear. In general, it is meaningless to quote the value of a measured length without stating the scale of the resolution used to make it. In principle, it is as meaningless as saying that a length is 543, 27, or 1.289176 without giving the units it’s measured in. Just as we need to know if it is in miles, centimeters, or angstroms, we also need to know the resolution that was used.

it turns out that the pattern of fluctuations in financial markets during an hour of trading is, on average, the same as that for a day, a month, a year, or a decade. They are simply nonlinearly scaled versions of one another. Thus if you are shown a typical plot of the Dow Jones average over some period of time, you can’t tell if it’s for the last hour or for the last five years—the distributions of dips, bumps, and spikes is pretty much the same, regardless of the time frame. In other words, the behavior of the stock market is a self-similar fractal pattern that repeats itself across all timescales following a power law that can be quantified by its exponent or, equivalently, its fractal dimension.

Healthy hearts have relatively high fractal dimensions, reflecting more spiky and ragged EKGs, whereas diseased hearts have low values with relatively smooth EKGs. In fact, those that are most seriously at risk have fractal dimensions close to one with an uncharacteristically smooth EKG. Thus the fractal dimension of the EKG provides a potentially powerful complementary diagnostic tool for quantifying heart disease and health.24 The reason that being healthy and robust equates with greater variance and larger fluctuations, and therefore a larger fractal dimension as in an EKG, is closely related to the resilience of such systems. Being overly rigid and constrained means that there isn’t sufficient flexibility for the necessary adjustments needed to withstand the inevitable small shocks and perturbations to which any system is subjected.

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, almost the distance from Los Angeles to Chicago, or London to Moscow.

the size of the smallest mammal of just a couple of grams, comparable to the mass of the Etruscan shrew, which is the smallest known mammal. It is only about 4 centimeters long, easily sitting on the palm of your hand. Its tiny heart beats at more than a thousand times a minute—about twenty times a second—as it pumps blood with the same pressure and speed as you do, and even more astounding, as does a blue whale. And all of this through its minuscule aorta, which is only a couple of millimeters long and an astonishingly couple of tenths of a millimeter wide, not much thicker than a hairbreadth. As I have remarked earlier, no wonder the poor creature doesn’t live very long.

a relatively small increase of only 10°C leads to a doubling of metabolic rate and therefore to a doubling of the rate of living. By the way, this is why you don’t see many insects in the morning when it’s cool—they have to wait till it warms up to increase their metabolism.

This is the curse of consciousness. We all know we are going to die. No other organism is burdened with the enormity of the conscious knowledge that it has a finite lifetime and that its individual existence is eventually and inevitably coming to an end. No creature, whether a bacterium, an ant, a rhododendron, or a salmon, “cares” or even “knows” about dying; they live and they die, participating in the continual struggle for existence by propagating their genes into future generations and playing the endless game of the survival of the fittest. So do we. But over the last few thousand years, we have emerged as the consciousness and conscience of the evolutionary process and begun the extraordinary adventure of contemplating its meaning by bringing ideas of morality, caring, rationality, the soul, the spirit, and the gods to the universe.

The fascination with death, coupled with the incessant questioning and search for any meaning to life, permeates human culture but has mostly been manifested and formalized in the multiplicity of religious institutions and experiences that humans have invented. Science has generally placed itself outside of such philosophical meanderings. However, many scientists have seen the quest for understanding and unraveling “the laws of nature,” the passion for wanting to know how things work and what they are made of, as an alternative journey in coming to terms with these big questions even if they themselves are neither “religious” nor particularly “philosophical.” Somewhere along the line, I realized that I was one of them, finding in science, or at least in physics and mathematics, some version of the spiritual sustenance that seems to be a universal need.

Unlike the predominantly positive images of many life-history events such as birth, growth, and maturity, most of us don’t want to confront aging and death. As Woody Allen pithily put it: “I’m not afraid of death, I just don’t want to be there when it happens.”

The city as the engine for social change and increasing well-being is one of the truly great triumphs of our amazing ability to form social groups and collectively take advantage of economies of scale.

Cells in larger animals are systematically processing energy at a slower rate than cells in smaller ones. So at the critical cellular level cells suffer systematically less damage at a slower rate the larger the animal, and this results in a correspondingly longer life span.

The growth of a system, whether an economy or a population, is often expressed in terms of a quantity called the doubling time, which is simply the time it takes for the size of the system to double. Exponential growth is characterized by having a constant doubling time, which also sounds fairly harmless until one realizes that it implies, for example, that it would take the same time for a population to double from ten thousand to twenty thousand, thereby adding just ten thousand people, as it would for it to double from 20 million to 40 million, thereby adding a humongous 20 million people. Amazingly, the doubling time of the global population has actually been getting systematically shorter and shorter as was indicated above: it took 300 years from 1500 to 1800 for the population to double from 500 million to a billion, but only 120 years to double to 2 billion, and only another 45 to double again to 4 billion. This is shown in Figure 31. So until relatively recently we have been increasing at an accelerating pace that is actually faster than a pure exponential! Although this acceleration has begun to slow down over the past 50 years, we are still effectively growing at an exponential rate.

I have met scant few economists who do not automatically dismiss traditional Malthusian-like ideas of eventual or imminent collapse as naive, simplistic, or just plain wrong. On the other hand, I have met scant few physicists or ecologists who think it’s nuts to believe otherwise. The late maverick economist Kenneth Boulding perhaps best summed it up when testifying before the U.S. Congress, declaring that “anyone who believes exponential growth can go on forever in a finite world is either a madman or an economist.”

To put it slightly differently, the rate at which we need to process energy to sustain our standard of living remained at just a few hundred watts for hundreds of thousands of years, until about ten thousand years ago when we began to form collective urban communities. This marked the beginning of the Anthropocene, in which our effective metabolic rate began its steady rise to its present level of more than 3,000 watts today. But this is just its average value taken across the entire planet. The rate at which energy is used in developed countries is far higher. In the United States it is almost a factor of four larger, at a whopping 11,000 watts, which is more than one hundred times larger than its “natural” biological value. This amount of power is not a lot smaller than the metabolic rate of a blue whale, which is more than one thousand times larger in mass than we are.

Nevertheless, his basic philosophy of a planned “town and country” community has persisted to this day and has left its mark not only on the many variants of garden cities that have since sprung up around the world but also in the design concept of almost every suburban development of every city. An interesting special case of this on a large scale is Singapore. Even as the city has grown to become a major global financial center with more than five million inhabitants and has continued to build the usual ostentatious steel and glass skyscrapers, its saving grace is that it has maintained the dream of being a garden city on a grand scale. This is primarily due to its visionary top-down leader the late Lee Kuan Yew, who had the foresight in 1967 to require that Singapore develop as a “city in a garden” with abundant vegetation, open green spaces, and a sense of tropical lushness despite its chronic shortage of land. Singapore may not be the world’s most exciting city, but its ambience of greenness is palpable.

He speculated that this apparent universality has its origins in the evolution of the cognitive structure of the brain: we simply do not have the computational capacity to manage social relationships effectively beyond this size. This suggests that increasing the group size beyond this number will result in significantly less social stability, coherence, and connectivity, ultimately leading to its disintegration. For situations where group identity and cohesiveness are perceived as central for the group to function successfully, recognizing this limitation and the broader implications of social network structure is clearly of great importance.

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. Putting all of this together we are led to the outrageous speculation that cities are effectively a scaled representation of the structure of the human brain. It’s a pretty wild conjecture, but it graphically incorporates the idea that there is a universal character to cities. In a nutshell: cities are a representation of how people interact with one another and this is encoded in our neural networks and therefore in the structure and organization of our brains.

This simple nonlinear quadratic relationship between the maximal number of links between people and the size of the group has all sorts of very interesting social consequences. For instance, my wife, Jacqueline, particularly enjoys dinner parties if a single conversation can be sustained by the entire group, so she is reluctant to participate in dinner parties larger than six. With six people there are 6 × 5 ÷ 2 = 15 possible pair-wise independent conversations that have to be “suppressed” for a single collective one to emerge and be maintained. This is just about possible, and it’s tempting to speculate that it’s because the number of other guests, five, corresponds to Dunbar’s number for the group size of an average individual’s inner circle. With ten at the table, there are a whopping forty-five such dyadic possibilities, which inevitably leads to a balkanization of the group as it disintegrates into two, three, or more separate conversations. Of course, many people prefer this modality but it is worth remembering that if you want a certain kind of group intimacy, having more than about six people is going to make it quite a challenge.

And in 1956, Sir Charles Darwin, grandson of the Charles Darwin, wrote an essay on the forthcoming Age of Leisure in the magazine New Scientist in which he argued: Take it that there are fifty hours a week of possible working time. The technologists, working for fifty hours a week, will be making inventions so the rest of the world need only work twenty-five hours a week. The more leisured members of the community will have to play games for the other twenty-five hours so they may be kept out of mischief. . . . Is the majority of mankind really able to face the choice of leisure enjoyments, or will it not be necessary to provide adults with something like the compulsory games of the schoolboy? They could not have been more wrong. The main challenge they foresaw was how to keep people occupied so that they wouldn’t become bored to death. Instead of giving us more time, “science and compound interest” driven by “technologists working for fifty hours a week” have, in fact, given us less time. The multiplicative compounding of socioeconomic interactivity engendered by urbanization has inevitably led to the contraction of time. Rather than being bored to death, our actual challenge is to avoid anxiety attacks, psychotic breakdowns, heart attacks, and strokes resulting from being accelerated to death.

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.”

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.

The time between the “Computer Age” and the “Information and Digital Age” was no more than about thirty years—to be compared with the thousands of years between the Stone, Bronze, and Iron ages. The clock by which we measure time on our watches and digital devices is very misleading; it is determined by the daily rotation of the Earth around its axis and its annual rotation around the sun. This astronomical time is linear and regular. But the actual clock by which we live our socioeconomic lives is an emergent phenomenon determined by the collective forces of social interaction: it is continually and systematically speeding up relative to objective astronomical time. We live our lives on the metaphorical accelerating socioeconomic treadmill. A major innovation that might have taken hundreds of years to evolve a thousand or more years ago may now take only thirty years. Soon it will have to take twenty-five, then twenty, then seventeen, and so on, and like Sisyphus we are destined to go on doing it, if we insist on continually growing and expanding. The resulting sequence of singularities, each of which threatens stagnation and collapse, will continue to pile up, leading to what mathematicians call an essential singularity—a sort of mother of all singularities.