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.