So here’s where things stand.
There is some fairly robust statistical evidence that
socio-economic outputs of cities scale super linearly
with an exponent greater than 1
So we’ve seen income, GDP, number of new patents
quantities like these tend to scale with an exponent greater than 1
And many of these exponent values are clustered around 1.15
Many infrastructural properties of cities, total length of roads
length of water pipes, number of gas stations
tends to scale sub linearly, less quickly than linearly
If you double the size of the city, you don’t double length of the roads.
It’s less than double. So the exponent here less than 1
And these exponent values tend to clustered around 0.85
So there is some empirical regularities that one observes
The question then is can we explain these empirical regularities
Why do we see this pattern?
And I am gonna discuss some thought along these lines in this video.
So the situation was similar for Kleiber’s law.
It was well established that the metabolic rate depended on
scaled with a creature’s mass to the three quarters power.
And than that beg an explanation and the West Brown Enquist
scaling theory provides a pretty convincing explanation
for why that particular pattern, that behavior is observed
for the metabolic rate of animals.
is there a similar thing for cities, in my opinion the answer is not yet
but I think we are getting close.
So, relatively recent paper by Luis Bettencourt in 2013
puts forth possible explanation for observed scaling in cities
And this is still ongoing work, I think it’s not settled,
it’s not in my opinion fully developed
But it’s promising and it’s certainly in my opinion interesting to think about
So in this video, I wanna describe some of the elements
towards this theory some ways you might think about cities
that could explain some of the scaling laws that we see.
So our starting point again is going to think about networks
So, for the vascular system, we said that the animals have to be served
by a space-filling vascular network,
So this is a obviously very crude drawing of this
It started with a heart and then it branches
in the three in the three in the three
And this has to be space or volume filling
because every part of you needs to be served
by the farthest out unit of this.
There needs to be capillaries everywhere in your body
in order to deliver oxygen, nutrients and to take wastes away.
So an initial thought is to think about a city
as also being served by a network.
And I think that makes sense, we need side walks and then roads
we need water and electricity and all of those delivered to us by networks.
However the networks for urban infrastructure may be self-similar fractal
the way the vascular system is but it’s not a tree.
So this is tree because it goes and branches, goes and branches
It doesn’t have any link sort of cross like this.
This is a very rough artist’s rendition or not very artistic rendition
of what urban infrastructure might look like.
So the idea here is this could really big road may be a high way
and then there are medium size roads, so these things here in blue
and then there are smaller roads here in red
or whatever color it is and so on and so on.
So we would see some sort of fractal or self-similar structure
grids laid on top of grids laid on top of grids.
But it doesn’t have this tree sort of behavior to it.
So we would describe it a little bit differently, just mathematics of it
Some other assumptions that go into this theory
So for the metabolic systems we assume that capillaries are the same size
and all creatures and then I think it’s a pretty well accepted assumption
based on the basic sort of biochemical properties that capillaries
have to perform and we can make a similar assumption here
It’s probably not as on as solid footing
but it seems fairly reasonable that say for a water network
distribution network of water pipes the smallest network
the smallest unit there, isn’t a capillary but it might be a faucet.
and it seems reasonable to say that the size of your faucet,
size of faucets and houses independent of the size of the city
So in New York City or Mexico City some the biggest cities around
faucets are probably about this big.
I live in a really small town, my faucet is about this big.
So faucets are about the same size everywhere.
There might be a fundamental unit for transportation.
If we’re talking about sidewalks, it might be
the width of a single person, right,
there is no, not in small towns you don’t have smaller sidewalk,
smaller than width of a person.
Everybody doorway might be about the same.
So anyway, so that’s another assumption that might go
into a theory of urban infrastructure
and then another network property I wanna think about
is that if you imagine things flowing across here
again we are gonna have some conservation of flow
flow conservation and optimization thereof went into
thinking about this and here also flow would have to be conserved.
if these are cars, the road network, the amount of cars that flow in
has to equal the amount of cars that flow out.
Their speed might be different things,
might be traffic, they will be pack together
But flow in is gonna have to equal flow out.
So these all give us, these fairly reasonable assumptions give us
some mathematical tools that we can use to begin to think about
how and why we might see scaling in these urban systems.
There are some other assumptions or ways of thinking about cities
that enter into the possible theory or explanation for scaling and cities
that Luis put forth in this 2013 paper.
The first is the idea that cities exist to mix people up.
That is city is mixing that if you’re in the same city, we’re in the same city
we can travel to see each other, the infrastructural network exist to do that
May be we live in the different side of the city. we don’t see each other everyday
But we could travel to see each other
And that extends almost from the definition of the city
A city is an agglomeration of people, a bunch of people living together
That can at least, that can all kinda travel to from again
from one side to other could at least in principal interact spatially.
Again the entire city doesn’t mix itself up everyday,
but it’s a mass of people that are somehow connected
connected by some sort of a transportation network.
So another assumption is that this transportation network,
well, there is some cost associated with it as well
cost might be related to socio-economic outputs
and that would scale not with the area of the city
but would transversal length of the city
how far it is to get from the one side to another.
So that sets up some scaling relationships there.
Additionally another consideration when thinking about cities
that helps lead to some scaling results
is that these infrastructure networks
that I described previously grow incrementally
so you might have a network like this
and then a new house comes and it’s in between two other houses
and then it’s gonna need water and electricity
and maybe a path or a road to it.
So networks rather than sort of growing like trees,
they grow incrementally and often sort of get in,
get filled in from the middle.
So that leads to networks that get more dense over time.
That’s definitely a difference from the tree like networks
that we have for vascular systems.
So that’s another consideration to think about in terms of
how these networks to play off of each other.
And then the last assumption or key idea
this is in some ways the most fundamental is that socio-economic
outputs are related to interactions.
So that what makes a city a productive place economically culturally
is not just a mere effect that there a whole bunch of people living close together
but those people interact, they share ideas, cultures, language, food,
all sort of stuff and that in turn leads to certain types of productivity
new ideas in terms of new patents, additional economic activity and so on.
So one of the base assumptions is that socio-economic outputs
are proportional to the amount of local interactions people would have.
So we take those ideas and work with a mathematically
It’s possible to come up with arguments for scaling relationships for cities.
And in particular there is an argument that suggests
that socio-economic outputs patents, GDP and so on
should scale as the population size to the seven sixth the power
So that’s a number that’s a little bit larger than 1,
1.15, 1.16 that kinda tends to be around many of these exponents cluster.
Similarly infrastructure network properties length of roads, areas of roads
things like that will scale as population size to the five sixth,
that’s the number less than 1
About around .85 which is sort of around what we see.
So as I’ve said this theory is still young.
I think it’s sort of the first steps towards a more solid theory
And I’m really interested to see how the theory progresses,
as people think more about the mathematics behind it
and assumptions behind it
where it leads over the next several years.
To me it’s interesting to think about I think it holds great promise
It’s not as solid as West Brown Enquist theory may not be able to be
because cities are different than metabolic systems.
But it’s an interesting sort of powerful provocative ideas and shows
I think some of what happens some of the good things that can happen
interesting thought provoking things that can happen when some of these ideas
of scaling and complex systems are applied
to these complicated social systems like cities.