Geoffrey West: The surprising math of cities and corporations

City infrastructure scales sublinearly with population: for example, the number of petrol stations per capita decreases as a city grows, and this economy of scale holds to the same degree everywhere in the world (Europe, Japan, China, Colombia), and applies to all infrastructure such as length of roads and electrical lines.

“The slope of that is less than linear. There is an economy of scale. Less petrol stations per capita the bigger you are... It scales in the same way everywhere... any infrastructure you look at -- whether it's the length of roads, length of electrical lines -- anything you look at has the same economy of scale scaling in the same way.”

Life is extraordinarily scalable — the same principles, dynamics, and organization operate across all mammals and can scale over a range of 100 million in size — and this scalability is one of the main reasons life is so resilient and robust.

“The same principles, the same dynamics, the same organization is at work in all of these, including us, and it can scale over a range of 100 million in size. And that is one of the main reasons life is so resilient and robust -- scalability.”

Biological metabolic rate scales sublinearly with mass at a slope of roughly three-quarters (less than one), which means doubling an organism's size requires only about 75 percent more energy rather than double, expressing an extraordinary economy of scale: the bigger you are, the less energy per capita you need.

“It's three-quarters, roughly, which is less than one -- and we call that sublinear... if you double the size of the organism, you actually only need 75 percent more energy. So a wonderful thing about all of biology is that it expresses an extraordinary economy of scale.”

When metabolic rate (energy needed per day to stay alive) is plotted against body mass across the full diversity of organisms, all species lie on the same straight line, revealing extraordinary underlying simplicity despite biology being the most complex and diverse system in the universe.

“what you see if you plot it in this slightly curious way is that everybody lies on the same line. Despite the fact that this is the most complex and diverse system in the universe, there's an extraordinary simplicity being expressed by this.”

All of life is controlled by networks — from the intracellular through the multicellular to the ecosystem level — and applying universal, mathematizable principles to these networks derives all the observed scalings and constraints, including the circulatory system, forests, and within cells.

“All of life is controlled by networks... If you take those networks... and you apply universal principles, mathematizable, universal principles, all of these scalings and all of these constraints follow”

To avoid the collapse inherent in superlinear growth, a major innovation must take place each time the system approaches a limit, resetting growth; sustaining growth therefore requires a continuous cycle of innovations that must occur faster and faster, forcing society onto an ever-accelerating treadmill that raises the question of whether socio-economic beings can avoid a systemic 'heart attack'.

“as we grow and we approach the collapse, a major innovation takes place and we start over again... there's this continuous cycle of innovation that is necessary in order to sustain growth and avoid collapse. The catch, however... is that you have to innovate faster and faster and faster... Can we, as socio-economic beings, avoid a heart attack?”

Given the size of a mammal, one can predict at the 90 percent level everything about its physiology and life history, because any physiological variable or life-history event plotted against size shows extraordinary regularity.

“you tell me the size of a mammal, I can tell you at the 90 percent level everything about it in terms of its physiology, life history, etc.”

As organisms get bigger, the pace of life systematically decreases: heart rates are slower, lifespans are longer, and diffusion of oxygen and resources across membranes is slower.

“systematically, the pace of life decreases as you get bigger. Heart rates are slower; you live longer; diffusion of oxygen and resources across membranes is slower, etc.”

Because cities are governed by superlinear social networks (more per capita as they grow), the theory predicts that the pace of life increases with city size — the bigger the city, the faster life gets — as shown by walking speed increasing with population across European cities, mirroring the reverse pattern of heart rate slowing in biology.

“If it's social networks with super-linear scaling -- more per capita -- then the theory says that you increase the pace of life. The bigger you are, life gets faster... On the right is the speed of walking in a bunch of European cities, showing that increase.”

Because companies scale sublinearly, the theory predicts sigmoidal growth for them; Walmart's growth from 1994 to 2008 matched the theory's predicted red line, and this pattern repeats across all 23,000 companies — they all start as hockey sticks, then bend over and die like organisms.

“if I'd have done this in 1994, I could have predicted what Walmart would be now. And then this is repeated across the entire spectrum of companies... They all start looking like hockey sticks, they all bend over, and they all die like you and me.”

Doubling the size of a city — from 100,000 to 200,000 or from 10 to 20 million — systematically produces a 15 percent per-capita increase in wages, wealth, patents, crime, police, and AIDS cases, together with a 15 percent per-capita savings on infrastructure; this 15 percent rule holds universally regardless of location.

“If you double the size of a city... then systematically you get a 15 percent increase in wages, wealth, number of AIDS cases, number of police, anything you can think of. It goes up by 15 percent, and you have a 15 percent savings on the infrastructure.”

Across 23,000 US companies, income and assets scale sublinearly with size (number of employees) like biology, indicating companies become dominated not by superlinear innovation and ideas but by economies of scale — namely bureaucracy and administration.

“we've looked at 23,000 companies in the United States... What is astonishing about companies is that they scale sublinearly like biology, indicating that they're dominated, not by super-linear innovation and ideas; they become dominated by economies of scale... by bureaucracy and administration”

City socio-economic quantities that have no analog in biology — wages, number of super-creative people, patents, crime, and disease cases such as AIDS and flu — scale superlinearly with population, with an exponent of about 1.15 to 1.2, meaning the bigger the city, the more of each you get per capita.

“the exponent, the analog to that three-quarters for the metabolic rate, is bigger than one -- it's about 1.15 to 1.2... the bigger you are the more you have per capita, unlike biology -- higher wages, more super-creative people per capita as you get bigger, more patents per capita, more crime per capita.”

Superlinear scaling from wealth creation and innovation produces, from the same theory, a rising exponential growth curve for cities and economies that fits the data well — but this system is destined to collapse for Malthusian reasons because it runs out of resources.

“if you have super-linear scaling from wealth creation and innovation, then indeed you get... a beautiful rising exponential curve... But it has a terrible catch, and the catch is that this system is destined to collapse... kind of Malthusian reasons -- that you run out of resources.”

Even though each organism, cell type, and gene evolved in its own unique niche through Darwinian evolution and natural selection, they have all been constrained to lie on the same scaling line, indicating that something beyond natural selection is at work.

“despite all of that Darwinian evolution and natural selection, they've been constrained to lie on a line. Something else is going on.”

Cities have a dual nature: while they are the source of major problems, they are also the solution because they act as vacuum cleaners and magnets that attract creative people and generate ideas, innovation, and wealth.

“cities are the vacuum cleaners and the magnets that have sucked up creative people, creating ideas, innovation, wealth and so on. So we have this kind of dual nature.”

There is an urgent need for a serious scientific theory of cities, meaning a quantifiable framework relying on underlying generic principles that can be made predictive, analogous to how biology can be modeled with universal laws.

“my provocative statement is that we desperately need a serious scientific theory of cities. And scientific theory means quantifiable -- relying on underlying generic principles that can be made into a predictive framework.”

It is very hard to kill a city — you could drop an atom bomb on a city and 30 years later it survives, and very few cities fail — whereas all companies eventually die, suggesting cities and companies obey fundamentally different dynamics.

“how come that it's very hard to kill a city? You could drop an atom bomb on a city, and 30 years later it's surviving. Very few cities fail. All companies die, all companies.”

A serious scientific theory of companies should be able to predict when a company such as Google will go bust.

“if you have a serious theory, you should be able to predict when Google is going to go bust.”

Organisms grow rapidly and then stop, following a sigmoidal curve; this bounded growth is very good for biology (a reason for its resilience) but very bad for economies, companies, and cities under the current growth paradigm, which instead demands hockey-stick exponential growth.

“You grow very quickly and then you stop... Very, very good for biology -- also one of the reasons for its great resilience. Very, very bad for economies and companies and cities in our present paradigm.”

The 15 percent scaling rule is the reason a million people a week gather in cities, because they are attracted by the wonderful things — creative people, wealth, income — while forgetting about the accompanying ugly and bad outcomes that scale up equally.

“This, no doubt, is the reason why a million people a week are gathering in cities. Because they think that all those wonderful things... is what attracts them, forgetting about the ugly and the bad.”

Cities are networks, and the most important network of a city is the population itself — cities are simply the physical manifestation of human interactions and the clustering and grouping of individuals.

“the most important network of cities is you. Cities are just a physical manifestation of your interactions, our interactions, and the clustering and grouping of individuals.”

The universal 15 percent scaling behavior arises from social networks — the interactions and clustering of people — which is why it holds identically across cities that evolved completely independently on different continents.

“underlying this is the social networks, because this is a universal phenomenon. This 15 percent rule is true no matter where you are on the planet... despite the fact that these cities have evolved independently. Something universal is going on. The universality, to repeat, is us”

Biology has a mathematical framework based on generic universal principles that can answer quantitative questions about a tree — how many branches or leaves of a given size, the energy flowing through each branch, canopy size, growth, and mortality.

“We have a mathematical framework based on generic universal principles that can answer those questions. And the idea is can we do the same for this?”

The tsunami of sustainability problems humanity faces is actually a reflection of the exponential increase in urbanization, because cities are the origins of global warming, pollution, disease, and the concentration of finance, economies, and energy demands.

“the tsunami of problems that we feel we're facing in terms of sustainability questions are actually a reflection of the exponential increase in urbanization across the planet.”

The United States went from less than a few percent urbanized 200 years ago to more than 82 percent today, the planet crossed the 50 percent urbanization mark a few years ago, and China is building 300 new cities over the next 20 years.

“Two hundred years ago, the United States was less than a few percent urbanized. It's now more than 82 percent. The planet has crossed the halfway mark a few years ago. China's building 300 new cities in the next 20 years.”

Every week until 2050, more than a million people are being added to the world's cities, an ongoing phenomenon that will affect everything.

“Every week for the foreseeable future, until 2050, every week more than a million people are being added to our cities.”