Note
Geoffrey West has his own opinions which are largely orthogonal to what the book is actually about. Here I collect them. When you read the book, feel free to subtract these opinions from the book. It would not affect your understanding of the central point of the book, which is the scaling theory.
- No scientific field is complete without some simple theories. All fields should be like physics. Big data, model-free theory, mere collection of data... those are not enough.
"All science is either physics or stamp collecting." -- Ernest Rutherford
- 21st century is the century of biology.
- Reductionism to the fundamental theory of everything is not enough. Because it is impractical to derive complex phenomena, like life, from the fundamental theories of particles, it is wrong to pay too much attention to it. A full theory of the universe needs several layers, all equally important.
- Most biologists are too interested in "stamp collecting". There are easy discoveries are missed because not enough biologists are thinking like physicists. The physicist way of science is to pick a few key ingredients, make a mathematical model out of those, that can successfully predict a lot. Once that's done, they add more ingredients ("perturbation theory", "second-order theory", etc), making the model more precise but less elegant.
Chap 1: The Big Picture
Remarkable facts
Justifying the title, West shows four scaling laws:- For organisms, the 3/4 metabolic rate law (Kleiber's law) and the 1 billion heartbeat per lifetime law (Rest Heart Rate and Life Expectancy, Levine, 1997).
- For cities, patent count scales with population (Invention in the City, Bettencourt et al, 2007).
- For companies, asset and net income scales with employee count at the same rate.
I remember hearing about the billion heartbeats law and was very amazed in around 2006. It is a beautiful fact, like a mystical communion with all animals.
The elements of scaling theory
Scaling theory explains these scaling laws using math and physics.
These complex systems are all approximately optimized for a balance between stability, metabolism, and self-repair, and since different organisms face similar constraints (the same physical laws), they develop complex strategies that nevertheless conform to simple scaling laws.
There are many ways for scaling laws to emerge, but the most important ways are:
- dimensional analysis: This is an old idea from classical physics, that even Galileo knew.
- fractals, scale-free, self-similarity: A cluster of very related newer ideas. The basic point is that a system looks the same at different scales, thus a small part looks the same as a big part. This is in fact already known to physicists in 19th century (see Rayleigh's principle of similitude), but the past physicists mostly dealt with smooth self-similar things. The serious study of fractals began with the recognition that many self-similar things are the opposite of smooth.
- networks: Also newer idea. Basically, it can be proven that optimal networks (optimal in some precise mathematical sense) are structured quite precisely, satisfy scaling laws. Also, even randomly grown networks can exhibit scaling laws (The Barabási–Albert model from 1999: Emergence of scaling in random networks).
More abstractly, scaling theory is a part of complex adaptive system theory, which studies how complex adaptive systems grow (like developmental biology) and what their adult-form looks like. The previous scaling laws were all about adult-form scaling between individuals. The following is about how there is a universal law of growth within each individual's lifetime:
West, Geoffrey B., James H. Brown, and Brian J. Enquist, ‘A General Model for Ontogenetic Growth’, Nature, 413.6856 (2001), 628–631 <https://doi.org/10/btq8xw>
Cities are different
Why do cities persist (even an atomic bomb did not kill Hiroshima) but organisms and companies die?
Because cities are different. Organisms and companies grow old, specialized, tame, and inflexible. Cities grow big, complex, and wild. Big companies grow bland and uniform in their product, filled with bureaucracy. Cities can grow and shrink, and very predictably, the bigger they grow, the more complex and creative they get. More jobs, different new jobs,
The bigger an animal (and company) gets, the less it metabolizes per unit mass. This is the 3/4 law. This is similar for cities: the bigger (size of city = population of city) it gets, the less power per capita it uses. This explains why supermarkets and gas stations are more crowded in big cities, and also why it's better for the climate to move into big cities.
However, for cities, it's not the 3/4 law, but more like the 0.85 law. This hints at something different.
And yes, something very different...
The bigger an animal (and company) gets, the slower it paces. Within any country, the bigger a city gets, the faster it paces in many aspects, all with exponent 1.15. So for example, double city population leads to $2^{1.15}=2.22$ times the amount of GDP, total wage, crime, patent... So in big cities, walking speed, crime rate, income per capita... are bigger than in small cities.
Froude was a famed engineer who in 1860s proposed Froude's number, the first dimensionless quantity, so important for properly scaled modeling. Isambard Kingdom Brunel's great failure SS Great Eastern could be avoided if he had used properly scaled models that has matched Froude's number. But sadly, he collaborated with John Russell, who dismissed Froude modelling methodology:
Because cities are different. Organisms and companies grow old, specialized, tame, and inflexible. Cities grow big, complex, and wild. Big companies grow bland and uniform in their product, filled with bureaucracy. Cities can grow and shrink, and very predictably, the bigger they grow, the more complex and creative they get. More jobs, different new jobs,
The bigger an animal (and company) gets, the less it metabolizes per unit mass. This is the 3/4 law. This is similar for cities: the bigger (size of city = population of city) it gets, the less power per capita it uses. This explains why supermarkets and gas stations are more crowded in big cities, and also why it's better for the climate to move into big cities.
However, for cities, it's not the 3/4 law, but more like the 0.85 law. This hints at something different.
And yes, something very different...
The bigger an animal (and company) gets, the slower it paces. Within any country, the bigger a city gets, the faster it paces in many aspects, all with exponent 1.15. So for example, double city population leads to $2^{1.15}=2.22$ times the amount of GDP, total wage, crime, patent... So in big cities, walking speed, crime rate, income per capita... are bigger than in small cities.
Bettencourt, Luis, and Geoffrey West, ‘A Unified Theory of Urban Living’, Nature, 467.7318 (2010), 912–13 <https://doi.org/10/fwng66>
Chap 2. The Measure of All Things
Scaling theory before 1900.
Galileo
Galileo in his Two New Sciences argued that scaling laws explain why giant animals, trees, etc would collapse under their own weight: weight scales like $volume = (length)^3$, but strength only scales as $(muscle cross section area) = (length)^2 = (weight)^{2/3}$, so $(relative strength) = (length)^{-1} = (weight)^{-1/3}$.
Drugs and body
Body mass index (BMI) is wrong, since mass doesn't scale like $(height)^2$, but $(height)^3$, but that's still wrong, since it does not control for age and sex. After controlling age and sex, finally, mass scales as $(height)^3$.
Dose-weight should have a nonlinear relation, the death of elephant Tusko from LSD overdose. Even to this day, some people still think that double the body weight = double the dose. This concerns Geoffrey West.
Ship-building
Back before 1900, ship-building, like any-building, was almost entirely based on tradition plus intuition. Nobody used properly scaled models, and they had no idea of wind tunnels.Froude was a famed engineer who in 1860s proposed Froude's number, the first dimensionless quantity, so important for properly scaled modeling. Isambard Kingdom Brunel's great failure SS Great Eastern could be avoided if he had used properly scaled models that has matched Froude's number. But sadly, he collaborated with John Russell, who dismissed Froude modelling methodology:
Therefore you will have on the small scale a series of beautiful, interesting little experiments, which I am sure will afford Mr. Froude infinite pleasure in the making of them... and will afford you infinite pleasure in the hearing of them; but which are quite remote from any practical results upon the large scale.
Lord Rayleigh
Rayleigh is a polymath who among other things proposed the principle of similitude, and used it to derive in a few lines the scaling law in Rayleigh scattering.Chap 3. The Simplicity, Unity, and Complexity of Life
Geoffrey West used to be a particle physicist, but became interested in biology in 1990s after the death of SSC project. He started also thinking about death, and was surprised that biology textbooks never give death a mention. There's developmental biology, but no decadent biology.
Then he found D'Arcy's On Growth and Form. D'Arcy's book promotes the viewpoint that there are physical laws which constrains the growth and form of organisms. West is inspired by this philosophy and set off to construct his scaling theory using math and physics.
k/4 laws, and (only) size matters
The 3/4 law isn't the only scaling law in biology. There are plenty, and they all have exponents close to k/4, where k is an integer. And they all scale with size (defined to be mature body mass).
almost all the physiological characteristics and life-history events of any organism are primarily determined simply by its size.
Not only organisms, but also super-organisms like ant colonies, scales by body mass, and with the same exponents!
Why is 4 so important?? Is it because we live in 3 dimensional space, and 4 = 3+1???
Yes.
Network theory of metabolism
Metabolism is defined as the flow and transformation of energy in a system. West explained metabolism scaling laws using network theory, which he developed in Santa Fe Institute, in 1990s, with James Brown, inventor of macroecology, and Brian Enquist, James Brown's student.
The network theory of metabolism is supposed to be a "zeroth-order theory", or "coarse-grained theory". I don't like the terminology. I'd rather call it "first-order theory", but whatever.
This is an inner portrait of the sad pony mathematician.
The theory goes like this:
- Metabolism scaling laws come from metabolic networks. All complex adaptive systems use networks to transport and transform its energy (doing metabolism), and thus explaining what networks are used in the systems would explain the scaling laws.
- The metabolic networks are divided into families, each family having members that are of the same type, but are different in shapes and sizes. For example
The families of metabolic networks must follow three axioms:
- All metabolic networks are space-filling.
- Each family of networks has some terminal units which are the same for all networks in the family.
- Each family of networks has an optimization criteria, and all networks in the family are (nearly) optimal among all networks with the same size.
Comment on axiom 1: The theory is, practically, only applied to 2-d and 3-d Euclidean spaces, but there's no reason it can't apply for other dimensions, or even other geometries... I believe that, in the future, after earth-based life colonize many planets, scaling theory that uses 2-d spherical spaces would become necessary to describe scaling laws across planet sizes, such as "planetary GDP scales with planet radius, with exponent 2/3, conditional on the same solar irradiance".
On axiom 2: Nothing real can be infinitely small, and moreover, there's often an optimal size for whatever kind of physical metabolism it does. For cells, it's a mitochondria. For lungs, it's an alveolus. For blood circulation system, it's a capillary tube. For electric grids, it's a plug. For human postal services, it's a mail-box.
On axiom 3: This is the result of "survival of the fittest". All observable networks are (nearly) the fittest for their purposes. But there are many ways to fitness, and that makes the optimality criteria different for different families of networks.
Also, since these networks are not mathematical networks, but physical networks, "size" is not defined by number of nodes, but by the volume/area/amount of space that it fills.
Applying network theory of metabolism to circulation systems in animals and in plants
It's a great fit! For details, check A General Model for the Origin of Allometric Scaling Laws in Biology, (1997).
For animals, each blood circulation system is a
- space-filling metabolic network: if a cell doesn't get blood, it dies.
- same size of terminal unit: blood vessels can't get smaller than one red blood cell, so that's a natural lower bound. The optimal size of a terminal blood vessel is determined by some kind of balance between fastest oxygen-diffusion, which favors small blood vessels that can get close to the cells, and minimizing friction, which favors large vessels.
I don't know if it's coincidental that it's almost exactly the size of one red blood cell...
Maybe... one red blood cell is optimized for the capillary! It's the biggest cell (because bigger is more efficient?? idk) that can still fit in the capillary. - optimal network: It optimizes efficiency, defined as the amount of energy rate the heart needs to do vs the blood flow volume rate.
Some
For plants, the physics is quite different, but it still comes out to follow some of the same scaling laws.
Fractals
Introduction to fractals. I didn't know about Richardson's pioneering research in fractals! Or his discovery of the scaling law of deadly quarrels! He's so amazing. A true inspiration. He was also locked out of academia due to his treasonous pacifism.
Benoit Mandelbrot also had a very unconventional academic journey. He got a tenure at Yale at 75 years old, the oldest professor in Yale's history to receive tenure.
West speculates that fringe scientists are more likely to discover radically new ways of thinking.
How long is the coastline of Britain? (mathematical comment)
My solution is simple: for a fractal curve with a well-defined fractal dimension $d$, its length is measurable by putting a measuring stick of length $x$ meters around it, and counting that $n(x)$ measuring sticks are used to get from end to end, and finding that $n(x)$ and $x$ has the following relation:
$$n(x) = C(1/x)^d$$
at the $x \to 0$ limit. More precisely, $\lim_{x \to 0} n(x)x^d = C$. In such a case, it is only reasonable to say that the curve has length $C$. For $n$ to be unitless (since it's just the number of sticks used, a naked number), $C$ must have units meters$^d$.
Similarly for other length units.
Wikipedia says:
If the coastline of Great Britain is measured using units 100 km long, then the length of the coastline is approximately 2,800 km. With 50 km units, the total length is approximately 3,400 km.
So solving for $n(100) = 28, n(50) = 68$ gives $n(x) = (1/x)^{1.28}$. So the coast of Britain has length 10166 km$^{1.28}$. Rejoice!
This is just an intuitive version of Hausdorff measure, which can be used to measure other wild things such as... volume of the Menger sponge, area of the Sierpinski gasket, etc.
One more fun fact: the coastline of South Africa is so flat it seems to have fractal dimension 1. It's probably the flattest coastline in the world.
Chap 4. The fourth dimension of life
The metabolic network theory makes very accurate predictions on many scaling law exponents. See Tables 1, 2.
Limits of mammal size
If a mammal is too small, it would have its aorta so narrow, that even in the aorta blood behaves more like molass than water (that is, it's dominated by viscosity rather than inertia). Then blood transport would be non-pulsate. Such a mammal would have no pulse, and be so inefficient as to go extinct.
This happens when the mammal has body weight ~1 g. This is about the same as that of a Etruscan shrew.
If a land mammal is too large, it would simply be crushed by its own weight. There's no such problem for a sea mammal, but there's still one problem: one of the derived scaling laws state that average distance between terminal units scales with body mass with exponent 1/12.
Universal growth curve
By a simple mathematical model, the universal growth curve can be derived, which matches the data:
Here it is again. It's such a beautiful curve.
The derivation is simple:
- Let adult body mass be $M$, body weight at $t$ be $m(t)$. So $M = \lim_{t\to\infty} m(t)$.
- By the 3/4 law, the total metabolic power at time $t$ is $P= A_0 m(t)^{3/4}$, where $A_0$ is some metabolic rate constant of the species.
- Metabolic power is used for only growth and maintenance.
- Throughout individual's life, each unit mass of body needs the same amount of metabolic power for maintenance. That is, everyday, one kilogram of any human baby's meat needs the same amount of energy to stay alive as one kilogram of any adult human's meat.
Let this maintenance power density be $B_0$. - Thus the maintenance power is
$$B_0 m$ - Let the amount of energy required to grow a unit mass of the body be $E_c$. It has units of joule/kilogram.
- Then at time $t$, the amount of metabolic energy devoted to gaining mass is...
...is...
$$E_c \frac{dm}{dt}$$
- Then, the metabolic rate balance equation is:
$$A_0 m^{3/4} = E_c \frac{dm}{dt} + B_0 m$$
which simplifies to
$$\frac{dm}{dt} = a m^{3/4} (1 - k m^{1/4})$$ - It's easy to see that as $m \to 1/k$, $dm/dt \to 0$, thus
$$\frac{dm}{dt} = a m^{3/4} \left(1 - \left(\frac{m}{M}\right)^{1/4}\right)$$ - Integrate to get
$$\left(\frac{m}{M}\right)^{1 / 4}=1-\left[1-\left(\frac{m_{0}}{M}\right)^{1 / 4}\right] e^{-a t / 4 M^{1 / 4}}$$
where $m_0 = m(0)$ is mass at birth.
This is the universal growth formula. To plot them nicely on the same curve, define the scaled mass ratio:
$$r = \left(\frac{m}{M}\right)^{1 / 4}$$
$$r = \left(\frac{m}{M}\right)^{1 / 4}$$
and the scaled-shifted time variable:
$$\tau = a t / 4 M^{1 / 4} - \ln{\left( 1 - \left( \frac{m_0}{M}\right)^{1/4}\right)}$$
to obtain
$$r = 1 - e^{-\tau}$$
WOW!
Given the growth curve of any species, you only need three numbers $M, m_0, a$, to fit it to the universal growth curve! $M$ and $m_0$ are easy to measure. The only hard part is $a$, which is defined as $A_0/E_c$, both of which are a bit more difficult to measure.
What is death?
Death is what happens when growth becomes pretty much zero. At that point, all of the metabolism in a body is devoted to maintenance, so that if there's just a bit of fluctuation, the body can't take it, and rapidly fails.
The human mortality age curve has been lifting rapidly, but the maximal age has never changed: it has always been around 115 - 125.
From Our World in Data
This suggests it to be the theoretical maximum of natural human age. To live longer than that, something very fundamental about a human's metabolism must change. Perhaps it can be done by increasing effective metabolic density (by, for example, injecting nanomachines that do maintanence job, powered by electromagnetic energy from outside the body). Perhaps it can be done by decreasing cost of maintenance (by, for example, scanning the body and selectively killing senescent cells).
Other ideas include eating less, cooling down, and in general, slowing down metabolism. Caloric restriction really works, so is keeping body temperature a bit low.
Chap 5. From the Anthropocene to the Urbanocene
Exponentials grow really fast. Exponential growth is what made human economy so remarkable.
Cities are the natural units of modern human economy, not nation states. Modernity is urbanity. People are rapidly moving into cities. By 2050 it’s projected that about 7 billion, more than 2/3 world population, will live in urban areas.
Modernity: suddenly cities!
Before the modern age, economic output was mostly based on amount of arable land. Modern cities started in the Industrial Age, around 1800s. The prototype is Manchester, from which the cultural meme of "cities are dirty and harsh and alienating" originated.
An error in the book
West thinks that the current economy is unsustainable because it's "closed", that is, humans dig out fossil fuel and use it, disregarding the sun. That's essentially right, but West made a mistake in concluding that, since closed systems are unsustainable by the second law of thermodynamics, nuclear power would not be sustainable, and only solar power is.
First, using solar power does not make us escape the spectre of closed system. You may have escaped earth, but just take all of solar system as the system, and suddenly it's closed again.
Second, fusion power is technically finite, just like the sun, but it could still last the world's energy needs for millions of years.
That is, assuming our world economy stays constant. It doesn't. It could keep growing exponentially, but again, assuming exponential growth, humans would soon enough utilize the entire output of the sun... So even the sun is not a sustainable source of energy.
In conclusion, calling nuclear fusion "unsustainable" is essentially the same as calling solar power "unsustainable", so West is wrong.
Chap 6. Prelude to a science of cities
Cities are the true drivers of modern and near-future economy. Nation states are not very functional or resilient compared to cities, as If Mayors Ruled the World (2013) argues.
Planned cities are bad because the planning theory used is wrong. A better planning theory is necessary because there is very rapid urbanization all over the world, and good theory or not, they must be planned because unplanned rapid growth leads to shanty towns. The best kind of urban growth is "organic", but the traditional way of growing an organic city is too slow.
Jane Jacobs is an economist who proposed that
macroeconomically, cities are the prime drivers of economic development, not the nation-state as is typically presumed by most classical economists.
Jacobs also proposed that the most productive cities are not shaped like the famous planned cities, such as the Garden City, Le Corbusier, or Brasília, because city productivity is based on human creativity, which in turn is based on extensive and self-motivated human-human interaction.
How to make a productive city
Shape matters: For a city to encourage human-human interaction, it should be shaped accordingly.
High modernism doesn't have the right shape. The right shape is not simple to describe, not Euclidean, and highly dependent on local geography and history. No straight lines or circular arcs. It should be less smooth, more fractal, and less symmetric.
Symmetry is bad: Symmetric space is great for fundamental physics, but for humans who navigate using landmarks on the ground, symmetric space makes them anxious and want to hide, like a mouse wanting to run towards the wall when placed in an open space.
Asymmetry is good: It allows people (and presumably, mammals in general?) to feel comfortable, navigate intuitively, and it allows specialization of different zones in a city. And specialization is efficient.
Personal rant
I really don't get what's wrong with planned cities, gated communities, and "typical high-rise apartment blocks". I grew up in Shenzhen, China, where gated communities are everywhere (they are called "小区", "little regions"), and most permanent residents live in those, rich or poor. That's not just in Shenzhen, but in most new Chinese cities. High-rise apartment blocks are also everywhere, and when Chinese people go to any non-New-York American city, they are often shocked to find the lack of high-rises, and say that America is a "giant farm" ("大农场").
Shenzhen -- my kind of city
South Korean, Taiwanese, and Singaporean people tend to have similar feelings. Perhaps the dislike of high-rises and gated communities is a Western cultural problem? Western culture simply doesn't make high-rises and gated communities "tick", but East Asian culture does.
The criticism of Canberra also struck me as annoying, since I found it a very nice place to live during my undergraduate years.
But maybe it's just my anti-humanism showing... I mean, my favorite kind of building is brutalist, like this:
Kunming brutalism
And I never got the human obsession with non-human nature or greenery... What can you do with them? Eat them? Relaxation? I'd much prefer relaxing in a library.
Anyway, I think this chapter is biased by Western culture. The cultural creativity of a city is not necessarily killed by high modernism -- it is killed by high modernism plus Western culture.
Even the author admitted that there is one very successful city that's heavily planned: Singapore, which is, of course, East Asian.
Besides, maybe geometric rational planning isn't functional for human living, but they are very good for some things, such as industrial production. I have in mind the giant automatic factories, such as Gigafactory.
Or maybe it's just me. Whereas normal people have overdeveloped human senses and underdeveloped math senses, I have the opposite. And math can be done with no nature or "buzz" of the city. All you need are books, a good brain, and a group of people to discuss with, all of which are available in a high-rise modern apartment, a hive for a mathematician-swarm.
Oh, and Brasília was chosen as a UNESCO World Heritage Site due to its modernist architecture and uniquely artistic urban planning. So there, UNESCO think it's a beautiful failure worth protecting, if it is a failure at all.
Chap 7. Toward a Science of Cities
Observation: two scaling laws of cities
Geoffrey West started searching for quantitative laws in cities in 2004, and found two laws:
- Infrastructure vs population scales with exponent 0.85.
- Productivity vs population scales with exponent 1.15.
That's it!
Note: Both laws are conditional on the cities being in the same country, and the infrastructure/productivity measurement being of the same type. So, to find the 0.85 exponent, you should only collect data on the total number of electric poles in each of many cities in China, not other countries, and not gasoline stations.
Explanation: applying metabolic network theory to cities
On pages 320-322, West claims that the 0.85 and the 1.15 exponents necessarily sum up to 2, because of some kind of coupling between social networks and infrastructural networks. I could not find any paper for this, although I'm sure it's somewhere.
The 0.85 exponent shows that infrastructural networks enjoy economies of scale. It is because these networks are "top heavy": like the circulatory system, there are aortas, dividing finer and finer down to terminal units. Flow and connectivity between terminal units is minimal, but the higher up the hierarchy you go, the more flow and connectivity.
The 1.15 exponent is because productivity is proportional to social ties, and social networks has the 1.15 scaling exponent. This is because social networks are "bottom heavy". Ties between people are very strong. Ties between families weaker. Ties between clans, tribes, companies... even weaker.
I didn't get the math of this theory of city scaling. Maybe later.
Another error in the book
West keeps saying the "15% rule", as "doubling the city population increases productivity per capita by 15%", but that's wrong. The correct rule is $2^{1.15}/2 - 1= 11%$, so it's in fact the "11% rule".
Similarly, the other "15% rule", as "doubling the city population decreases infrastructure per capita by 15%", is also wrong. The correct rule is $1- 2^{0.85}/2= 10%$, so it's the "10% rule".
Chap 8. Consequences and predictions
Marchetti's constant
Marchetti and Zahavi found that humans on average to spend one hour travelling every day. And since the extent of a city (at least a city with one center) is roughly defined as the area where people commute to its center for work every day, it means a city's radius is roughly 30 minutes * commuting speed. And it shows. As commuting tools gets faster, cities get larger:
The results confirmed the 1.15 scaling exponent: in each country, number of links among people in a city scales with the population of the city, with exponent 1.15 (not quite, but pretty close).
However, the average size of the clique of close friends for each person does not scale at all. Big city or small city, the clique of friendship stays the same size. This is reasonable, since human intimate social interaction ability is limited by their brains, and brains don't magically get better at keeping close friendships in big cities. All the technologies in the world so far cannot make people keep more true true friends (Facebook friends, though...).
From Citylab
Confirmation of the 1.15 exponent for social networks
In The scaling of human interactions with city size (2014), the authors studied the network of social interactions directly, using cell phone calls as proxy. Basically, every cell phone user is a node, and every pair of people who has called each other at least once during the study period are connected. This gives the social network of 1.6 million people in Portugal and UK across their cities.The results confirmed the 1.15 scaling exponent: in each country, number of links among people in a city scales with the population of the city, with exponent 1.15 (not quite, but pretty close).
However, the average size of the clique of close friends for each person does not scale at all. Big city or small city, the clique of friendship stays the same size. This is reasonable, since human intimate social interaction ability is limited by their brains, and brains don't magically get better at keeping close friendships in big cities. All the technologies in the world so far cannot make people keep more true true friends (Facebook friends, though...).
... YET!
Dunbar's number is ripe for technosocial disruption!
Scaling law for visitation rate (visiting velocity)
See The hidden universality of movement in cities (2020) for all the data used in this section.
West and others also found a strange scaling law: for each place-of-interest in each city, consider all its visitors. Each visitor $i$ lives a distance $d_i$ away from it, and visits it with frequency $f_i$. Multiply them together to get $v_i = d_i f_i$, the "visiting velocity" for visitor $i$. The amusing thing is that it's a velocity: it's got units of (distance/time). It can be interpreted as the average speed for the visitor, if the visitor does nothing except visiting that place-of-interest, going in a straight line between home and that place.
How to actually rank cities
Scaling theory states that size matters the most for cities. That's the first-order term. To get to second-order, one studies how cities deviate from what scaling theory predicts.
It is more interesting to ask if a small city is "punching above its weight", than to know that New York is a big city that does everything big. This idea, of factoring out the effect of size, gives the Scale-Adjusted Metropolitan Indicators (SAMIs), developed in Urban scaling and its deviations: Revealing the structure of wealth, innovation and crime across cities (2010).
Basically, if a city's productivity is $k$ times what scaling theory predicts, it gets a score of $k$ there. In this way,
The following diagrams are from the paper:
The blue line is scaling theory's prediction (first-order term). The red dots are real cities.
Subtract by the blue line to get the SAMI score (second-order term)!
The SAMI scores seem to have Laplace distribution... why the pointy tip?
New York is disappointingly lacking in patent production!
To explain the high-rankers:
- Corvallis is a small town (53,000), which has both Oregon State University, and HP Inc., which has a large printing research and development operation there.
- Burlington is also small (42000), with almost 1/4 employed by the University of Vermont.
- San Jose is big (1 million), but it has Stanford and Silicon Valley.
Diversity of business establishments in cities
In Scaling and universality in urban economic diversification (2015), a remarkable plot is shown
It's a plot of the number of "establishments" (a business unit, so for example, a restaurant, an outlet, a lawyer's office).
NAICS is is a system used by business and government to classify business establishments according to type of economic activity. Think of it as Dewey Decimal Classification for businesses. It gives each type of business a 6-digit hierarchical code, and each establishment gets one code.
In the paper, the authors took all NAICS codes of all establishments in many cities, and for each city, calculated the frequency-rank curve. The results, as shown above, exhibits a universality.
The shape of the snake-like curve is derived from assuming that
- the total number of establishments per person stays constant (Empirically, about 1 establishment per 22 people, no matter which city!)
- the growth of business types in a city follows a rich get richer process.
Another interesting fact is that, as cities grow, the rankings of establishments shift predictably. Some establishments are infrastructural (shops, gas stations, etc) and thus scale sublinearly, while others are social (bars, high-tech offices) and thus scale superlinearly. As a city grows, the social establishments climb in rank.
Chase the superexponential future! Bloody impaled on the singularity!
This part is based on Growth, Innovation, Scaling, and the Pace of Life in Cities (2007).
Since social network scales superlinearly, there is no economy of scale, but the opposite. The bigger the body gets, the slower its metabolism goes. The bigger the city gets, the faster its society runs. And since modernization is urbanization, the more modern the world gets, the faster the world runs.
It's not an illusion that the past was a chiller place. Ever since modernity, people started to really strain themselves to keep up, and it's only getting harder. Even I feel the strain and I'm only in college. Geoffrey West is 80 now. What a remarkable mind to make such breakthroughs at such an old age.
CHASE THE FUTURE!!
Now, since productivity in cities scale superlinearly, with exponent 1.15, it means that, assuming its maintenance cost scales linearly, cities would grow superexponentially.
It's simple: just like when we derived the growth curve, consider the metabolism budget:
$$\frac{dm}{dt} = a m^{1.15} - b m$$
which, if you solve it, gives finite-time blowup. It's simple: since productivity scales faster than maintenance, for a very big city, its maintenance shrinks to nothing, and so, for large $m$,
$$\frac{dm}{dt} \approx a m^{1.15}$$
which gives $m(t) = m_0(t_0 - t)^{-1/0.15}$, blowing up at $t = t_0$.
This can't be, because nothing infinite can happen in a finite time in this universe. If we tried, we'd fail, and die impaled on the spike of singularity. This will be elaborated in Chapter 10.
Chap 9. Toward a science of companies
The ageing of companies
Young companies rapidly grow in asset, but as they approach maturity, they grow "merely" exponentially. That sounds okay, until you adjust for total GDP, then you find that, relative to the whole economy, mature companies don't grow at all.
This explains why big companies fail: just like old organisms, mature companies spend all their resources just staying alive. It becomes very fragile, and fails upon market shocks.
Death is fair for companies: old and young, rich and poor
On average, conditional on total asset and age, all companies have the same death rate. That is, take all companies with the same total asset and age, and calculate their death rate. You get the same answer. The half-life of companies is 10 years. Capitalism kills.There are some long-lasting companies. Those tend to be Japanese, very small, extremely niche, and has a small but devoted customer base.
Chap 10. The Vision of a Grand Unified Theory of Sustainability
The singularity is near
A science of the cities, companies, and growth in general, is necessary to ensure sustainable human civilization. West believes that the human civilization growth is superexponential, and this would encounter an essential singularity soon (on the order of a century). This cannot happen, and could end in a superexponential collapse (Fig 76, 77). To avoid collapse, it's necessary to change to a more steady-state way of civilization.Notice that there's a change of subject. Whereas before, West was talking about cities, now he's talking about civilization as a whole, and that's a big difference. Still, the argument that civilization as a whole grows superexponentially seems credible, according to the data collected in The Singularity is Near (Kurzweil, 2005).
Anyway, suppose this superexponential growth is true, then as we found before, we have finite-time singularity! That can't be.
The following two graphs are from Growth, Innovation, Scaling, and the Pace of Life in Cities (2007).
(a) Growth driven by sublinear scaling eventually converges to the carrying capacity.
(b) Growth driven by linear scaling is exponential.
(c) Growth driven by superlinear scaling diverges within a finite time (dashed vertical line).
(d) Collapse characterizes superlinear dynamics when resources are scarce.
Our civilization can't keep doing this, of course. So it has to break out of an unsustainable superexponential curve, change course, slow down. If we run out of steampower, we start electricity, then nuclear, then solar... Every time, we break out of a paradigm just before its superexponential growth curve falters in the face of a finite reality, to start a new segment of superexponential growth. This is illustrated in graph a below:
Speedcore is the sound of modernity!
The scream as we crash, eyes wide open, into the singularity.
Possible error?
West claims that this pile-up of superexponentials would culminate in an "essential singularity", but I think that's wrong. For it to be essential, it has to blowup faster than any rational function. From what I can see on the graph, it seems that the blowup is something like $(t_0 - t)^{-\alpha}$, where $\alpha$ is some real number around $1 + 1/0.15$. That would not be an essential singularity.Back to the book
Geoffrey West is very disturbed by this singularity. He thinks that singularity won't happen, but only because the physical universe cannot support it. What would happen instead is that humanity would get close to singularity, exhaust itself, and crash.To avoid that, he wants to develop a grand unified theory of sustainability, and use that to change human history to a
Personal comments
I do not see how a change to steady-state is going to help, however. That seems like changing human civilization from superexponential growth, to being like organisms... which still die in finite time. Organism, company, or city, they all die in finite times, it's just that organisms and companies die by rotting, and superexponential growths die by supernova exploding.Indeed, that seems similar to how past empires usually die. They grow until they stopped growing, then rotted. I'd rather, just once, push human civilization right to the singularity and make history.
Sustainability be damned, because nothing can be sustained.
On another note, regarding civilizations as organisms has been done before, but without using math. See for example social cycle theories, especially The Decline of the West (Spengler, 1918).
Afterword
A brief history and idea of Santa Fe Institute, the place to do complexity and interdisciplinary science in.Big Data is no good if it doesn't have a good theory. Data cannot speak without bias, because there's just too much data, we have to be biased in data collection and selection. Thus, we better have a good bias, informed by a good theory.
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