The Hidden Structure of Scaling

Organizations, organisms, and communication networks all have similar growth dynamics

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Let’s envision a discussion among engineers. We won’t have to stretch our imaginations far: One engineer has proposed a new idea, and the others, typically, are finding faults with it. The engineer with the idea has rebuffed all the criticisms, until one of the critics plays her last card. “Yes—but does it scale?” she asks. There is an ensuing silence.

Systems are usually complex, nonlinear, and adaptive. And as they change in size, their behavior, as well as their economics, can change disproportionately. Geoffrey West, a theoretical physicist at the Santa Fe Institute, has recently published a book, titled Scale: The Universal Laws of Growth, Innovation, Sustainability, and the Pace of Life in Organisms, Cities, Economies, and Companies (Penguin Random House). This book engagingly describes West’s search for an underlying general mathematical description of why systems scale the way they do. He asks a number of fascinating questions along the way. For example, why do we keep eating, but stop growing? Why do almost all companies live for only a few years, while cities continue to grow and thrive? Why does the rate of innovation have to accelerate to sustain socioeconomic life?

Though these scaling questions would appear to have widely differing explanations, West proposes an underlying commonality that’s based on only a few factors and amenable to mathematical analysis. One of those factors is the necessity for the “space filling” required to feed all the end points of a system as it expands. In animals, it could be the blood distribution system of arteries, while in cities it could be the pipes, wires, and roads for power and water distribution, communications, and transportation.

A recurrent feature of these examples is the fractal nature of the systems—the self-similarity wherein the system branches out into ever-smaller arteries that resemble larger branches. There is a powerful scaling effect of this kind of topology. A fractal growth is able to fill an area, using the length of its branches. Normally, covering an area demands resources that scale in proportion to the square of the dimensions, while in the branching, fractal system, the resources scale linearly. Similarly, as in the case of blood distribution, a fractal arterial system can fill a volume using area. In such systems, there is an economy of scale wherein cost grows more slowly than size would otherwise dictate.

However, there are usually mitigating factors that diminish such scaling benefits. In the case of a communication network, for example, there is a need for increased maintenance as the network grows. And there are demands for continuous reconfiguration as the user base evolves.

But other effects can come into play. As a communication network snakes its way outward from the center of a city, end users radiate their own social networks of interconnection back toward the city and elsewhere. These social networks are also fractal, both in a geographic and logical sense, as diminishing links of importance branch from family to friends to acquaintances. Engineers may be familiar with Metcalfe’s Law—that the number of possible pairwise connections among n users scales as n2—or Reed’s Law, which holds that the number of possible groups scales as 2n. The reality is, however, that as the network grows, the fraction of these possible connections that are actually used becomes vanishingly small. Nonetheless, the accelerating interconnecting power of networks combined with the economies of infrastructure give cities the ability to scale with increasing vitality and productivity.

Unfortunately, this column hasn’t scaled very well. As I have compressed the issue, I’ve lost so much. I’m afraid you’ll have to read West’s book. It’s worth it.

This article appears in the September 2017 print issue as “The Structure of Scaling.”

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