So let's summarise

what we've covered in unit 8.

Unit 8 was on on urban scaling.

And so the initial motivation

started by thinking about metabolic scaling

like we did in the last unit,

where we saw that the metabolic rate

of an animal

scales as its mass to the 3/4s power.

And we then saw that

there was a good theory suggesting,

quite convincingly I think,

that this scaling is a consequence

of the self-similarity

and optimality of the vascular network,

that provides nutrients and

removes wastes for the animal's body.

So then the idea is well, gee,

maybe we can think of cities

as in a sense like organisms,

that have some sort of metabolism,

that do things, and that these cities are

served by networks:

roads, electrical networks, water networks

and so on.

So, where does this point of view lead us?

Do we observe scaling?

So the answer to that question is, yes!

We do indeed observe scaling,

and here is one scaling plot among many.

And, so what emerges is that many

properties of cities are observed to

scale more or less; the data is fuzzy.

There's a lot of variation

in addition to this pattern or trend.

So on the left, this is a plot of

total road length in US cities

versus a population. Its a log-log plot,

there's definitely a trend here,

it's reasonably well-fit, but again,

there's a lot of variation, lots of points

are near the line,

and this exponent is about .85,

so, sub-linearly.

On the right, we have GDP, this is also of

US cities, how does that vary with

population? It's faster than linear.

On both of these plots, the dark line

is the linear line. So, here we see that

the curve is steeper than linear.

It's super-linear, the exponent turns out

to be around 1.13. So there's many many

empirical results like this for different

city properties, where one sees scaling to

more or less a good approximation,

but at the same time, some modest

variation about that trend line.

So if one looks at the scaling

behavior of different properties of cities,

the behaviors tend to fall into three

rather loose but nevertheless recognisable

catergories.

There's some behaviors that scale

super-linearly, so faster than linear with

an exponent greater than one.

Exponents tend to be around 1.15,

more or less,

and these are socio-economic outputs,

things like gross domestic product,

wages, total electrical consumption,

new patents, aids cases, other diseases,

crime, and so on.

Then there's some quantities that scale

linearly, with an exponent of 1.

So, that means that if you

double the city size,

you just double this quantity.

So, an example is total housing.

You double the number of people in the city

you're gonna double the number of houses

or apartments.

Similarly for total employment,

household water consumption,

household electrical consumption, and so on.

And then there's some quantities that scale

sub-linearly, less quickly than linear.

So, road lengths, electrical cables,

gas stations, and so on.

Where you'd double the population of a

city on average, the road length would

not double, it would less than double.

It goes 2^0.85, more or less.

So, then the question is, given these

observations, is there any theory or

possible mechanism of explanation for them?

And there's been some recent progress in this,

And here are some of the thinking behind this.

So first is, cities are served by

self-similar networks, just like

animals are served by self-similar networks.

But unlike the vascular networks for animals,

urban networks are not trees that

go, and branch, and go, and branch,

they're grids, they're self-similar,

but they're grids.

And these networks grow incrementally.

they can sort of grow from the inside out,

perhaps. They don't necessarily always

grow with the tips, they infill as well as

growing out. Additionally, the smallest

unit in urban systems, urban networks, is

the same, independent of the population

size. So in a big city and a small city,

the faucets are the same for water networks,

and the doors are the same for

transportation networks, and so on.

Additionally, cities are mixing,

cities exist to mix people up.

And so, put it by definition, a city is a

glomeration, a cluster of people,

all of which could, at least in principle,

interact with each other, that's what

makes it a city, and not two cities,

or a state or something.

So, it's possible, in principle at least,

for people to travel from

one side of the city to another.

And then, maybe the most fundamental

assumption, is that socio-economic outputs

are proportional to local interactions.

So, what drives GDP or wages or various

consumption measures, measures of creatvity,

and they tend to be interactions.

So, cities come together,

people come together in cities

because there's a benefit to this

interactions that one has,

those economical benefits,

social benefits, there also could be

drawbacks too, like disease and crime.

So, again, this is maybe the main

realization, or assertion, that

socio-economical outputs for cities are

proportional to local interactions.

So, where does this lead things,

taking stock?

Well, in my opinion, there's not yet a

fully-formed explanation for the observed

scaling patterns.

It's not as quite as tight a theory as

the West-Brown-Enquist theory for

metabolic scaling. It's entirely

understandable, this is very new results,

pretty new work, and I think some of the

mathematical details and assumptions

need to be teased out a little bit.

In addition, cities are much more

complicated a metabolic system,

and the data that we see, the patterns

that we see, are not as sharp.

I think there very clearly are power-law

trends, but they're only an approximation.

They're certainly not an exact result,

and there's a lot of variation in the data.

And that variation is interesting to study

as well.

So, nevertheless, it's interesting,

it's really intriguing to me, that there are

fairly robust scaling results for cities

across space and time.

So, cities in different continents seem to

have similar scaling results,

cities over time tend to

have similar scaling results, as well.

So that suggests to me even though

maybe there's reason to doubt any given

set of data, in its entirety,

these results suggests to me, and I think

to most pretty strongly, that there's

something interesting going on.

Yes, there's a lot of variation, but there

are some structural or network similarities

across all of these cities

that are worth paying attention to,

and that we can learn about.

So, where's is this all headed?

Well, we don't know.

It's an ongoing and active area of research.

And I think it's quite exciting.

To me, it's one of the most interesting

applications of these ideas of fractals

and scaling to a really difficult, but

important complex system.

So I'm quite excited to see where

this work goes, and how it develops

over the next several decades.

So, this brings us to the end of unit 8.

It's also the last full unit in the course.

In the next unit, I will briefly conclude

by reviewing some of the main results

and key themes that we've explored

over the last several months.

See you then.