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.