The starting point for urban scaling is similar

to the starting point for metabolic scaling.

In both we asked about how properties

of something depend on their size.

For urban scaling the size

it will be interested in is their population

and we’ll look at different properties.

wages, GDP, length of roads,

amount of electricity used and so on.

So in this video I want to just take an empirical look at this

what does the data suggest about this question.

So I’ll do so by showing you several plots

And as usual I’ll put the reference with the plots down here.

So here is the first one.

This is looking at population, down here.

And this is total wages and this is for US cities

And in this context the city is taken

to be a metropolitan statistical area,

So it might not be exactly the same as a formal city boundary,

often might includes suburbs.

If you have two cities are right next to each other

they will be considered the part of the same metropolitan area.

So any event, here is the data,

I think we got around 300 data points here

And this is on a log-log plot

And we can definitely see a linear trend.

We can calculate the slope,

which we know as the exponent in a power law

And that gives us a Beta at 1.12

So note here, this is larger than 1,

So what this means is

if we have a small city

and then we do compare to that city that twice as large

we might ask, well, how would the wages,

the total wages, total amount of money

made in both of those cities compared.

And you might think well,

the city that has twice as many people

should have wages that are twice as much.

What this says, it’s actually more than that,

it's faster than linear, super linear.

So that if you double the population

on average according to this trend,

you would more than double the total wages

It would go by to do though the 1.12 not to do the 1

Alright so, that’s sort of interesting I think

and others have thought.

Because we might expect that it would be linear

doubling population with double wages,

but that’s definitely now what we see

of course that can’t help but notice that

there is an awful a lot fuzz around this line.

so there’s a very clear trend

that’s pretty hard to deny

but it’s not an exact relationship like a physical law might be

there is even more scattered I think

than for most of the metabolic scaling plots.

So there’s a lot of variation among cities as well.

And there is a clear trend.

And as we talked about in metabolic scaling

the trend can be interesting

and the deviations from the trend can be interesting

and those two statements don’t need

to be in competition to each other.

Both can be interesting.

In this case I think both are interesting.

Ok, let’s look at a few other results.

And there are lots of lots of data sets like this

But I’ll show you a few more.

Alright, again we have population on the horizontal axis.

A log-log scale

This is log not a wages but it’s GDP, gross domestic product

And these are for Chinese cities

so this is measured in million Yuan.

And again we can see there is a very clear trend.

It’s certainly not a flat line. Beta is 1.12

But for this data set there is even more

variation about that trend.

But again there definitely is a trend line.

This plot here is for Germany, German cities

Again this is a log of GDP,

gross domestic product measured in Euros,

very clearly trend here. Beta in this case is 1.10

and some variations about the trend

but not as much as for China.

In both cases though this exponent is larger than 1

This is statistically significantly so indicating that

log of GDP or GDP grows faster than linearly with population.

So again in both these cases if you double population,

you more than double the GDP of the city

Alright, let’s look a one more this sort of plots

So here this is now the total road miles in the city

How many roads are there measured in miles

And again this is a log-log plot, population here

and in this case the exponent is 0.85

So that means the growth is slower than linear.

If you double the size of a population on average,

you don’t double the length the roads

It’s actually less than double it, through to the 0.85

So let me also explain what these lines are

This line here, this is the darkest line is a line with a slope of 1

And what this is showing is that

this data themselves are clearly

there is trend clearly less than 1

These two here, one of these lines is a fit line with that data.

The other is a line from the theory.

So it’s sort of theoretical fit

that I’ll explain in a subsequent video.

So note again here we see a quantity road miles

that’s not scaling linearly.

But in this case the exponent is less than 1

And here’s one more GDP plot

This is for US cities

again we’re seeing faster than linear growth.

This black line would indicate linear growth that’s a slope of 1

The measured data, the measured exponent is 1.13

That’s faster than linear.

And there are actually two lines here.

One is the measured exponent.

The other is that predicted by theory.

So there’s an urban scaling group at the Santa Fe Institute

lead by Luis Bettencourt, Geoffrey West and many others.

They produced a series of papers and are continuing to do so

with a lots of lots of plots like this.

So there are many, a lot more data we can look at

but for this video the main observation is that

there is evidence of scaling,

some sort of linear relationship on a log-log plot.

In some cases less than linear.

In some cases more than linear.

And there is a fair amount of fuzz around this,

It’s not an exact relationship, it’s a trend.

But there is still a fair amount of variation around this trend.