I want to talk briefly about how to
visualize power law scaling on a plot.
That's done by noticing that there are
two equivalent mathematical expressions
that represent a power law. The first one
we have seen, y equals c, some constant,
times x raised to the power alpha.
You just saw that. If you take the
logarithm of both sides of this equation,
you get log of y equals alpha times
log of x plus log of c. Those are two
equivalent expressions. This one I
derived just by applying the rules of
logarithms we learned in an earlier unit.
If you don't see how I got this, just
believe me that that is true. That this
is equivalent, or go back and review those
rules or ask about it in the forum, if you
like. This is useful for the following
reason. If we plot y = c x ^ alpha, it
might look something like this,
for example, if alpha is negative, you
might notice if we look at this equation
with the logarithms, it's the equation of
a straight line, slope times x + some
constant. If we plot it on what's called
log-log plot, in which the y-axis is
log y and the x axis is log x. I am just
treating this as the y axis and this as
the x axis, and plotting this straight
line. The thing to remember is on a
regular plot, a power law looks something
like this. On a log-log plot, it looks
like a straight line where the slope is
the exponent alpha. Let's look at a
couple of examples, of famous power laws,
The Gutenberg-Richter law of earthquake
magnitudes, you've probably run into this.
You have heard of magnitude 1, 2, and 3
earthquakes. Well, magnitude is just the
logarithm of the energy of the earthquake.
What this means is that this is really a
log scale, where we are plotting the log
of the energy. Magnitude 2 earthquake
has 10 times more energy than a magnitude
one earthquake. Magnitude 3 earthquake
has 100 times more energy and so on.
This is a logarithm scale, and here on
this axis is the number of earthquakes per
year of a given magnitude. It is a
straight line. This is a logarithmic
scale also. This is a power law
because this is a straight line on a
log-log plot. These dots are the actual
data. This line is the best fit to the
data. It is a very good fit. It shows
that there are many, somewhere between
a hundred and thousand magnitude 1
earthquakes per year, but very few, say,
magnitude 5 or 6. These only happen
every few years. Earthquakes of even
higher magnitude are even rarer. This is
a very famous power law. One famous power
law that we're going to be talking about
in this unit is the law of metabolic
scaling in animals. If you plot the body
mass of an animal on this axis vs. the
metabolic rate, that is, the amount of
energy in the metabolism, measured in
Watts, of animals of different mass, you
can see these different animals, it falls
on a straight line. This has some very
surprising properties we'll be talking
about in more detail in a little while.
Finally, you'll hear very briefly about
some relatively new work about scaling in
cities. This plot is from a paper by
Bettencourt and West, that came out in
2010, a link to it is posted on our
course materials page, and it shows how
properties of cities such as crime, gross
domestic product, income and patents,
scale with the population of the city.
This is the city population divided by the
average of all city populations, and this
is the log of that, and the log of this,
and you can see that they all scale in a
very similar way, in a power law fashion.
That is very surprising and the
explanation for it isn't yet
well-understood. You'll hear a little
bit about that later on.