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

by Geoffrey West

Metropolitan Oklahoma City, with a population of about 1.2 million people, has a GDP of about $60 billion, so its per capita GDP ($60 billion divided by 1.2 million) is indeed close to the average for the United States, namely $50,000. Extrapolating this to a city with a population ten times larger, having 12 million people, would predict its GDP to be $600 billion (obtained by multiplying the $50,000 per capita by the 12 million people), ten times larger than Oklahoma City. However, metropolitan Los Angeles, which is indeed ten times larger than Oklahoma City with 12 million inhabitants, has a GDP that is actually more than $700 billion, which is more than 15 percent larger than the “predicted” value obtained by the linear extrapolation implicit in using a per capita measure.

GDP, like almost any other quantifiable characteristic of a city, or indeed of almost any complex system, typically scales nonlinearly.

Surprisingly, an animal that is twice the size of another, and therefore composed of about twice as many cells, requires only about 75 percent more food and energy each day, rather than 100 percent more, as might naively have been expected from a linear extrapolation.

To which should have been added the critical phrase “unless something changes.” Change and, by implication, innovation, must occur in order to continue growing and avoid collapse.

This way of thinking about innovation, which relates it to the drive or need to grow bigger, to expand horizons and compete in ever-larger markets with its inevitable confrontation with potential limitations imposed by physical constraints, will form the paradigm later in the book for addressing similar kinds of innovation in the larger context of biological and socioeconomic adaptive systems.

This scale invariance expresses something absolute about the quantities they represent in that the dependence on the arbitrariness of the human choice of units and measurement has been removed.

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.

They show that almost all the physiological characteristics and life-history events of any organism are primarily determined simply by its size.

because the structures of biological networks are so varied and stand in marked contrast to the uniformity of the scaling laws, their generic properties must be independent of their specific evolved design. In other words, there must be a common set of network properties that transcends whether they are constructed of tubes as in mammalian circulatory systems, fibers as in plants and trees, or diffusive pathways as in cells.

Optimization principles lie at the very heart of all of the fundamental laws of nature, whether Newton’s laws, Maxwell’s electromagnetic theory, quantum mechanics, Einstein’s theory of relativity, or the grand unified theories of the elementary particles.

area-preserving branching

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.

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.

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.

In general, it is meaningless to quote the value of a measured length without stating the scale of the resolution used to make it.

qualification principle. what about dimensionless meaurements?

This curious gain of an effective additional dimension is a general feature of space-filling curves, to which I will return in the next chapter.

crenulated,

Consequently, most physical objects have no absolute objective length, and it is crucial to quote the resolution when stating the measurement.

This has stimulated the development of a new transdisciplinary subfield of finance called econophysics and motivated investment companies to hire physicists, mathematicians, and computer scientists to use these sorts of ideas to develop novel investment strategies.

Its fractality effectively endows it with an additional dimension.

However, driven by the forces of natural selection to maximize exchange surfaces, biological networks do achieve maximal space filling and consequently scale like three-dimensional volumes rather than two-dimensional Euclidean surfaces. This additional dimension, which arises from optimizing network performance, leads to organisms’ functioning as if they are operating in four dimensions.

Although living things occupy a three-dimensional space, their internal physiology and anatomy operate as if they were four-dimensional.

Consequently, no evolutionary advantage would be conferred by increasing size.

Anton Chekhov poignantly remarked, “Only entropy comes easy.”

A population that is growing exponentially is defined mathematically as one in which the rate of increase in its size (per minute, per day, or per year, for instance) is directly proportional to the size of the population that’s already there. Thus the growth rate itself grows even faster the bigger the population.

Under a constant percentage growth rate everybody is still getting richer and more prosperous, so no wonder we are hooked on the open-ended steroidlike exponential growth drug. It is truly a real high and is an explicit manifestation of the huge success of our economic dynamic.

This is illustrated in one of the great epic poems in world literature, the Shahnameh, written about one thousand years ago by the revered Persian poet Ferdowsi.

collieries

“every generation has perceived the limits to growth that finite resources and undesirable side effects would pose if no new recipes or ideas were discovered. And every generation has underestimated the potential for finding new recipes and ideas. We consistently fail to grasp how many ideas remain to be discovered. Possibilities do not add up. They multiply.”

Commenting on the U.S. budget, the leader of Senate Republicans from 1959 to 1969, Senator Everett Dirksen, was reputed to have said, “A billion here, a billion there, and pretty soon you’re talking about real money.”

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.

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.

We typically think of each city, and especially the one we live in, as being unique, with its own history, geography, and culture, having its own special individuality and character that we feel we recognize.

within their own urban systems they are approximately scaled versions of one another, at least as far as almost anything that you can measure about them is concerned?

various metrics, such as wages, crime, patent production, and total road lengths, depends on the overall economy, culture, and individuality of each national urban system.

The deviations from these lines are a measure of the residual footprint of the unique history, geography, and culture of each individual city

the group size of social primates scales with the neocortex volume of their brains as a classic power law.

This presumed connection between brain size and the ability to form social groups is called the social brain hypothesis.

Furthermore, because the geometry of white and gray matter in our brains, which forms the neural circuitry responsible for all of our cognitive functions, is itself a fractal-like hierarchical network, this suggests that the hidden fractal nature of social networks is actually a representation of the physical structure of our brains.

Nevertheless, the fact that these Zipf-like distributions are found across such a diverse set of phenomena suggests that they express some general systemic property that is independent of the character and detailed dynamics of the specific entities under consideration.

Furthermore, because these are approximately self-similar processes, the same dynamics occur at all scales. Thus the same generic mechanism that leads to a small adjustment in a financial market is at play when that market suffers a major crash.

A common metric used to address risk, whether in financial markets, industrial project failures, legal liabilities, credit loans, accidents, earthquakes, fire, terrorism, and so on, is the composite risk index, which is defined as the impact of the risk event multiplied by the probability of its occurrence.

enhanced socioeconomic activity and infrastructural economies of scale,

Just as biological time systematically and predictably expands as size increases following quarter-power scaling laws, so socioeconomic time contracts following the 15 percent scaling laws, both following mathematical rules determined by underlying network geometry and dynamics.

In this sense cities are time accelerator machines.

The contraction of socioeconomic time is one of the most remarkable and far-reaching features of modern existence.

This is just one graphic example of how the time required for travel has dramatically shrunk over the last couple of hundred years. It is often expressed with the platitude that the world has shrunk. Obviously it hasn’t shrunk—the distance between London and Los Angeles is still 5,470 miles—what has shrunk is time, and this has had profound consequences for every aspect of life from the personal to the geopolitical.

So the increase in transportation speed resulting from the marvelous innovations of the past couple of hundred years has not been used to reduce commuting time but instead has been used to increase commuting distances.

Marchetti’s constant,

This unforeseen consequence of mobile phones and other IT devices