The Underlying Laws Binding Cities, Companies, and Living Systems

By James Case

Scale: The Universal Laws of Growth, Innovation, Sustainability, and the Pace of Life in Organisms, Cities, Economies, and Companies. By Geoffrey West. Courtesy of Penguin Press.

As a teenager in London, physicist and author Geoffrey West procured a temporary job in the research labs of International Computers Limited. It was a transformative experience, during which he decided to pursue a career in research. After obtaining an undergraduate degree in physics at the University of Cambridge and a Ph.D. from Stanford University, West eventually moved to Los Alamos National Laboratory and became the founder and group leader of the Elementary Particle Physics and Field Theory Group. He later joined the Santa Fe Institute (SFI), eventually serving a term as president. In 2006, he was listed among Time’s “100 Most Influential People in the World.”

Among the phenomena that initially piqued West’s interdisciplinary curiosity were the extraordinary number of documented power laws \(y=x^p\), which reduce to straight lines when both variables are measured on a logarithmic scale. He possesses a physicist’s appreciation of scaling arguments, and traces the development thereof from Galileo—who argued that the heights to which certain animals can grow is limited by the fact that areas increase as the square of height while volumes increase as the cube—to William Froude—who discovered the importance of the ratio¹ \(gV^2/L\) for ship design—to Lord Raleigh, who explored the advantages of expressing physical laws in terms of dimensionless variables.

Scale’s main story line—from which West digresses early and often—begins with four striking power law examples, illustrated by scatter plots of the following: the basal metabolic rates of animals against their body mass, the number of heartbeats in an animal’s lifetime (also against body mass), the number of patents held by residents of particular cities against their populations, and the income and/or assets of corporations against the number of their employees.

The basal metabolic rate of an organism is simply the rate at which it consumes energy while at rest (for such, \(p \approx 3/4\)). For heartbeats in a lifetime, \(p \approx 0\), indicating that the hearts of virtually every known species beat roughly one billion times between birth and death. Small animal hearts beat much faster than human hearts, while large animal hearts beat more slowly. Thus, dogs and cats live only a few short years, while whales and elephants survive significantly longer than humans. The fact that \(p \approx 3/4\) for metabolic rates implies an economy of scale whereby an animal weighing 100 times as much as another consumes only 32 times as much energy while both are at rest. For patents held by city residents, as for income and/or assets of corporations, \(p\) exceeds unity, meaning that a city or firm that is 100 times as large as another consumes more than 100 times as many resources. Therein, says West, lies a fundamental difference between biophysical and socioeconomic growth.

As intriguing as the ubiquity of power laws themselves is the tendency of the observed exponents \(p\) to cluster, at least in the biophysical realm, around multiples of \(1/4\). In collaboration with various SFI colleagues, West found an explanation for this propensity in the structure of so-called “vital networks,” the most obvious examples being the mammalian network of veins and arteries and the fiber bundle structure of plants and trees. Cities also rely on vital networks, such as transportation networks, natural gas and electrical networks, water mains, and sewer lines. Corporations have organization charts and communication networks that are often quite different. All such networks appear fractal in the sense that, by zooming in on successively smaller parts of the whole, one discovers a series of remarkably similar branching diagrams.

West and his coworkers soon concluded that vital networks share three essential characteristics: they are (i) space-filling, in that they reach every extremity of the host organism, (ii) equipped to serve their clients through standard interfaces—capillaries for vascular networks, wall plugs for electrical networks, faucets and toilets for water and sewer lines—that vary little from client to client, and (iii) evolved to perform their essential services with notable efficiency. It is in this efficiency that West finds the beginnings of an explanation for the quarter power laws that govern the vascular networks of plants and animals.

At a point where a single channel divides in two, a standard principle of fluid mechanical design decrees that the cross-sectional areas of the downstream branches should sum to that of the upstream branch, lest impeding waves be reflected back toward the source of the flow. This fact is an important part of the rather complex argument by which West and his colleagues are able to account for numerous properties of plant and animal life, including the ability of plants and trees to bend without breaking when stressed by wind. Due to the area-preserving property of vital networks, the cross-sectional area of a tree trunk must equal the sum of the cross-sectional areas of all the leaf stems (a fact already known to Leonardo da Vinci).

The insight that a \(3/4\) power law relates basal metabolic rates to body mass leads West to a remarkably simple explanation of the “growth curves” used in industry to predict, among other things, the food requirements of animals being raised for meat as they progress from birth to market weight. Growth is accomplished primarily through cell division, which proceeds at a rate roughly proportional to the number of cells in place, and hence to body mass \(m(t)\). West thus concludes that

\[\dot{m}=am^{p}-bm, \tag1 \]

where \(p=3/4\).

The first term on the right represents the metabolic rate, while the second represents cell division. One may deduce species-appropriate constants \(a\) and \(b\) from measurements taken in the field and/or laboratory. Solutions are generally S-shaped, rising first slowly, then steeply from a birth weight \(m_0\) near \(0\) before leveling off at the maximum sustainable level determined by setting \(\dot{m}(t)=0\). The substitution \(m=x^4\) reduces \((1)\) to a linear equivalent.

More remarkably, West has identified dimensionless variables \(\mu\) and \(\tau\), in terms of which many (possibly all) species share the common growth curve \(\mu=1-e^{-\tau}\). To justify such a universal claim, he assigns specific colors and shapes to different animal species to display their growth histories as shapes scattered along the proposed curve (see Figure 1).

Figure 1. Geoffrey West’s universal growth curve. Courtesy of Penguin Press.

West devotes the first part of Scale to biophysical growth, concluding with an attempt to parlay his understanding of the metabolic process into an estimate of the longest possible human life span. He points out that aging proceeds at a roughly linear rate beginning at or about the age of 20, and counters the oft-expressed hope that science can extend human life spans much beyond 125 years. While caloric restriction may increase the lifetimes of lab mice by as much as 50 percent, West does not believe that humans are likely to subject themselves to so Spartan a regimen.

In chapters 5-10, West approaches the study of growth in socioeconomic systems—including cities, companies, and economies—similarly to the study of biophysical systems. Since the dawn of the Industrial Revolution, the advancements of such entities has brought rising standards of living to many nations, and most of the population would like to see it continue.

As previously mentioned, the power laws found in the study of socioeconomic phenomena are prone to exhibit exponents \(p\) in excess of unity. Accordingly, the solutions of \((1)\) tend to escape to \(\infty\) as \(t\) approaches some finite \(t_0\). If \(p=8/7\), for instance, \((1)\) may be reduced to a simple quadrature via the substitution \(m=x^7\). This, says West, need not doom mankind to an imminent apocalypse, because if some new innovation should materialize before the witching hour \(t_0\), a new (slightly curvilinear) “hockey stick” can succeed the old one. And if that can occur once, it can happen many times, as suggested in Figure 2.

Figure 2. Successive innovations. Figure credit: James Case.

Unfortunately, these progress-preserving innovations must keep growing larger and more frequent, meaning that the gaps between the vertical asymptotes of Figure 2 will narrow with the passage of time. This lends credence to the fear expressed by John von Neumann shortly before his death in 1957: mankind seems to be approaching an “essential singularity” in time, beyond which it may become impossible to prolong the era of rapid progress initiated by the Industrial Revolution.

Geoffrey West’s Scale is a landmark volume full of interesting ideas and engaging digressions that endow the book with both entertainment and cognitive value.

¹ Where \(g\) is acceleration due to gravity, \(V\) is velocity, and \(L\) is the length of the vessel of interest.

James Case writes from Baltimore, Maryland.