Published in IEEE Spectrum Magazine, Sept 2017

There is a discussion among engineers.  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 his last card.  “Yes, but does it scale?” he asks.  There is an ensuing silence.

Systems are usually complex, nonlinear, and adaptive, and as they change in size, their behavior, as well as economics, can change disproportionately.  Jeffrey West, a theoretical physicist at the Santa Fe Institute, has published a new book, simply entitled Scale, that engagingly describes his search for an underlying general mathematical description of why systems scale the way they do.  His analysis is applied to biological systems as well as technological ones.  He asks a number of fascinating questions related to scaling.  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, amenable to mathematics, based on only a few factors.  One of those factors is the necessity for “space filling” to feed all the end points as a system is expanded.  In biology it could be the blood distribution system of arteries, while in technology it could be the pipes, wires, and roads for power and water distribution, communications, and transportation. A recurrent commonality in these examples is the fractal nature of the growing systems – the self similarity wherein the system branches out into ever-smaller arteries that, except for size, topologically resemble earlier branching.  There is a powerful scaling effect of this topology.  A fractal growth is able on a plane to fill an area, using length.  The former scales as the square, while the latter scales 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.

However, there are usually mitigating factors that diminish this theoretical gain.  In the case of a communication network the need for increased maintenance as the network grows, and perhaps more importantly, the demand for continuous reconfiguration as the user base evolves, bring the scaling exponent to about 0.85.  That is, when cost is plotted against size on a log-log chart, it is a straight line with slope .85. As the network snakes it way outwards from the center of a city, the end users in turn radiate their own social networks of interconnection back towards 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 that the number of possible groups scales as 2N.  However, as the network scales, the fraction of these possible connections that are used becomes vanishingly small.  Nonetheless, the accelerating interconnecting power of networks together with the economy of infrastructure gives 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.