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.