Fractals and Scaling: Reflections on urban scaling


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.




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