Before concluding this unit,
I'd like to offer some thoughts

and reflections and opinions
on urban scaling.

Sort of take stock, kind of give you a sense 
of things, at least my opinion of them.

So first, studying properties of cities
is an important and really difficult issue.

They're definitely complex systems.

They're made up of a large number of
strongly interacting entities,

and those entities are often

heterogeneous. They're not all the same.

And entities, here, I'm thinking
not only of people, of course,

but also the networks and infrastructure,
social and physical,

that support those people.

So they're definitely complex systems,
and they're important because

I think the majority of us live in cities,

and all indications are that 
the world population

will become more and more urbanized
in the years ahead.

And cities give us good things,
like income, health, creativity, new ideas,

but they can also bring bad things:

disease and crime,
pollution and climate change,

and so understanding their properties
is important,

and just on a personal note,
maybe having grown up in a big city,

I find cities fascinating to think about.
I enjoy traversing through cities

(even though I live in a small town now)

and thinking about all the different 
things that makes up a city.

So, some thoughts on this work:

First, um, data in some level,
or reliable data,

is perhaps hard to come by.

It's very different than, say,
studying a physical system

where it's easier --- relatively easy ---

to measure properties,
and properties are static, and so on.

So, some of the things that I have in mind
when thinking about

economic outputs of cities,
I suppose that could be measured by taxes

and revenue and so on,

but an awful lot of economic activity
goes on in an informal...

it's in the informal sector

that never gets measured or captured
by any official...

any official metric.

So people might exchange services and 
goods without going through any sort of

official entity, and so those transactions
might not get recorded

and entered into this record,
but surely they're an important part

of the fabric of urban life.

Additionally, there are a lot of people
who, for one reason or another,

are undocumented, and may be uncounted
in these estimates of population.

Here in the U.S., we have a large population
that we refer to as 'undocumented'

or sometimes, unfortunately, as 'illegal',
and as I'm recording this,

the European... the migrant crisis in 
Europe is a huge issue.

Thousands and thousands and thousands
of displaced people,

and with continued civil war
and continued climate change,

we're witnessing more and more migration
of refugees across the world,

and those people aren't always counted.

So, um, sometimes an estimate of

a city population, it looks like 
a good hard number because

it's there in black and white on a piece of paper,
but it's hard to know how many people

actually live in cities.

So, anyway, these are some things I think about
as I look at this data.

Additionally, um, the data itself is messy,

which is not surprising because the world is messy
and cities are a little bit messy,

and by 'messy', I mean, if we plot the data
on our favorite way, a log-log plot,

we definitely see a trend 
(this thing here on the left)

There's definitely a clear trend,

but there's a lot of variation around that
trend as well. So we can

certainly fit a trend line and that's a

reasonable thing to do. But is this exactly

a power law? Well... surely not. Right? 
Because there's an awful lot of spread

around this. Is it even best described by

a power law? Maybe a log-normal is a
better... um... functional form to assume

for this. And I think there's some evidence
to suggest that actually

log-normals may be better. Or that it's 
difficult to state with confidence

that these really are power laws.

So there's some question about
data analysis as well.

On the other hand, one of the themes of
this course is that fractal is not

necessarily a yes/no quantity. Things are
more or less fractal-like

if they exhibit approximate scaling to a lesser
or greater degree.

So I think that it's certainly a reasonable

thing to do to fit a power law to this
and then to sort of draw

inferences from that. But there's some

questions about the data analysis for this

Is it really a power law, is that the best way

to think about this? I tend to think it's

an OK way to think about it... but it's an

area of some controversy, and there's certainly

not universal agreement. So, regardless,
there certainly is some trend

here and you can describe this trend with

various exponents. And that, at this

point is just an empirical relationship.

We notice that there is some pattern
between, say, population and

road length. Or population and economic

output, measured one of any of a number
of different ways.

So that observation, that 
empirical observation

in and of itself, I think is 
interesting intrinsically,

and is certainly useful as well. Because

it lets us make certain predictions, on 
average, about what might happen

as cities get larger. What sort of resources

will they use, what sort of benefits might

we see from those cities, what sort of

drawbacks might we see from those cities?

So, saying a relationship is only empirical

doesn't necessarily mean that it's not

a useful or interesting relationship. So

then the question is, though, can we

say more than, "Gee, this is an empirical

relationship"? Can we speak to the potential

mechanisms that would generate these

sorts of scaling behaviours, that are seen

more or less. And here I think that this is

very much work that is ongoing. Initial
efforts seem to me to be promising

but at least in my opinion, I don't think

it's sort of, the final word. There's not

a solidified theory the way there is for

metabolic scaling in the 
West-Brown-Lindquist theory.

It takes time to develop ideas, to sift

through what the important assumptions are

what the not important assumptions are, and

that process is ongoing. Additionally,

one thing that is maybe left out of this

analysis, and I think it's just worth

mentioning but not dwelling on,

is the idea of intentionality. That

cities, and particularly the people who

inhabit them, can exhibit choice,

intention, design, in how they build

cities. So there are clearly some constraints

almost metabolic constraints, in terms of

power lines and water mains and roads

and paths and so on. But there's also room for

for choices. I think that's worth 
keeping in mind

as well, that we're studying cities,

perhaps as if they were organisms, in

analogy with metabolic systems, and that

analogy is a powerful one, a very productive

one, but it's probably not the only way
to talk about cities, and it's useful

to remember - at least I hope - that we have
some choice in how cities develop

It's not necessarily inevitable. In any
event, I don't mean any of these

comments to be overly critical, or at
all negative towards urban scaling theory.

I think it's a really fascinating area. 
It's provocative, gets me to think

about cities in interesting ways, and I

think already has produced some

useful results, these empirical
regularities. Recognition, that is

that there are these empirical regularities

in how certain properties of cities scale

So, again, this is an ongoing area of

research. I'm excited to see how it

develops over the next few years, or
decades, and I hope that

this introduction has been useful for you

and that you can follow along and

maybe contribute to this effort as well.