Hi Folks. In this unit we will look at

the topic of scaling. Scaling is the

study of how different aspects of the

system change as the size of the system

increases. It turns out that complex

systems often scale in ways that are

surprising and in many cases not

well-understood. Here we'll look at some

fascinating and somewhat controversial

results on scaling in biology and then

we'll take a brief look at some very new

work on scaling in cities. Let's start.

The field of scaling asks - How attributes

of a system change as the system's size

increases? For example, in biology we

might ask how attributes of organisms

change as their body mass increases.

Or in urban scaling, a very popular area

now days, how do attributes of cities

change as population increases?

The study of scaling has been very

important in the research community in

complex systems because many clues to how

complex systems work can be gained by

studying how these systems scale.

There is one simple mathematical concept

which is very important to understanding

scaling and that is the concept of

proportionality. Here is a simple

proportionality here. The quantity y is

said to be proportional to x,

if y is equal to some constant, c, times x.

For example, here is a proportionality

relationship. y is equal to 2 x.

Every time x is increased, y is increased

by double the amount. Here is another

example, y is equal to -1/3 x. That is,

say that x increases from 1 to 2, then

y will decrease from -1/3 to -2/3.

Proportionality is a linear relationship.

You can see that that is true because

formulas represent straight lines.

Let's look at the notion of scaling in a

very simple context, that of geometric

scaling. Suppose you want to move from

your current small bedroom to a larger

bedroom. Here is your current bedroom.

It has a certain length and the new

bedroom has twice that length. So, now,

you ask, how big a bed can I fit into my

new room? Well, the length of a bed you

can fit scales linearly with the length

of the room. That is, you have a

particular length of bed in your old room,

now you can fit twice that length into

your new room. So, we can see here that

bed length is proportional to room length.

You can plot it like this. As room length

gets bigger, size of the bed you can fit

scales in this linear way. Now, that's a

very simple example. Let's suppose you

have a rug in your original room. Now you

can get a bigger rug for your new room.

How much bigger? Well, the area of the

rug you can now fit in your new room

scales in a way that is quadratic with

the length of the room. Quadratic means

has the square of. Since area scales with

the square of the length, the area of the

rug you can fit scales with the room

length. If you double the size of the

room, that means you quadruple the square

feet or square meters of the rug that can

fill the room because 2^2 = 4.

Here's what the plot looks like, for

quadratic scaling. You can see that as

the room length is increased, the rug area

goes up, much faster than in the linear

scaling case. Finally, if the bedroom

belongs to a teenager, might be a teenager

who likes to pile laundry up to the

ceiling. I used to be that kind of

teenager myself. Now I am the mother of

one. As you get a bigger room, you can

pile more and more laundry up to the

ceiling. How does that scale with the

length of your room? The volume of

laundry you can pile up to the ceiling

scales cubically with the length of the

room. The volume of laundry is

proportional to the room length cubed.

That is because volume scales as length

cubed. You can see that by looking at

this plot. The volume of laundry goes up

very quickly as the room length increases.

If you double the size of your room, you

would be able to fit 8 times the amount of

laundry, because 8 is 2 cubed.

In general, many attributes have what we

call power-law scaling. We learned about

power-laws in the network section of this

course. Here the idea is that some

attribute, such as volume, or area, is

proportional to size, here we were using

room length, to the power of some constant

alpha. Notice the notation. This is the

proportionality symbol, and this is the

Greek letter alpha, the exponent, alpha is

a constant, so remember these are two

different symbols. This means that our

attribute, such as volume of laundry, is

equal to some constant, c, times size

raised to some power alpha, where both c

and alpha are constants. We can write

that in terms of y and x as y = c x ^alpha

This is what is called a power law.