If you have read much astrophysics, you have probably come
across the term “power law”. Similarly
if you have followed blogs on books and publishing, you have probably come
across a phrase that goes something like “book sales follow a power law”. What exactly does this mean, and what is this
mysterious mathematical entity known as a power law?
For the purposes of this blog, I will try to avoid being
very theoretical and keep equations to a minimum. Most of the important understanding of this
subject can be explained with graphs and words and that’s usually what a
general audience prefers anyway. Astrophysical
(hi Scott!) and mathematical types are permitted a smile at the
oversimplification of some of the explanations below.
So, what is a power law?
Simply put, it is a mathematical function which is used to describe a
situation in which one variable varies as a power of another. A common example of a power law, that most of
us can recall from high school science, is the equation that describes the
force of gravity, where the gravitation force between two masses varies as the
inverse square of the distance between those masses. This is an example of a power law that is
underlain by a straightforward scientific theory, the theory of gravity.
Many phenomena in the physical world, especially the
astrophysical world, appear to be modelled well by power laws, though the
theoretical laws underpinning this are not always so obvious. One example is the size of asteroids in the
Asteroid Belt, as outlined below, showing the number of asteroids in each size
range (the data is from Wiki):
Description (Radius)

Size Categ

Num Aster

0  0.49 km

1

29,000,000

0.5  1.99 km

2

2,750,000

2  7.99 km

3

290,000

8  31.99 km

4

11,100

32  127.99 km

5

800

128 km  Up

6

39

The asteroid data has also been graphed below, to show the
typical shape of a power law, which will tend to be dominated by a few of the
categories, with most of the cases falling into these ranges. The second graph shows a nice feature of a
power law, which is that if we graph it using a logarithmic scale, we will get something
pretty close to a straight line. Note
that the vertical scale has been transformed into a logarithmic scale, with
equal intervals for each time the graph axis goes up by a factor of ten.
In fact, we can use the slope of this straight line to
estimate the actual value of the “power” in the power law. We won’t bother with that here, as it gets a
bit tricky and we just want to understand the main features of the power law at
a graphical and qualitative level.
Let’s look at another set of astrophysical data, because
it’s fun. Dodecahedron Books does mostly
science fiction, so we like stars and astrophysics. This table shows the number of stars visible
to the naked eye under good conditions (a dark sky, good seeing, and good
eyesight), by brightness category (stellar magnitude, lower for brighter stars,
and higher for dimmers stars, which happens to be the reverse of the asteroid
size scale we used earlier). This data
is from wiki, though I am sure similar data can be found in all kinds of astronomy textbooks.
Description

Stellar Magnitude

Num of Visible Stars

Dimmest

5

1,602

Fifth brightest

4

513

Fourth brightest

3

171

Third brightest

2

48

Second brightest

1

15

Brightest

0

5

Again, here are the two graphs: raw data and logarithmically
scaled. Note the nice straight line on
the bars of the latter graph, indicating a linear fit to the logarithmically
transformed data, which (after a bit of algebra transforming back to the
original data) indicates a power law (please note: blogger seems to be acting wonky, so I hope the graph shows up):
Power laws are frequently a good fit for phenomena in the
domain of human culture and economics.
We often want to answer questions like “what is the likelihood of a book
selling between 1000 and 5000 copies, given usual book sales statistics?” Unfortunately, it is not always easy to get
good data, but luckily for us, bestselling indie writer Joe Konrath recently shared his sale
statistics on his blog, so let’s see how that particular dataset conforms to a
power law. I binned his sales data into
six ranges, so that we could compare that data with our asteroid and star data
later. Each bin size was 2.4 times the
previous bin  that’s pretty nearly the same as how the magnitude scale for
stars works.
Description

Category

Unique Book Titles in Each Sales
Category

Fewest Sales (05000)

1

77

500112000

2

37

1200128800

3

23

2880169120

4

13

69121165888

5

6

Highest (165889 and up)

6

2

So, Konrath’s sales data conform pretty well to a power law,
though not quite as nicely as our asteroid and star brightness data. It looks like he writes more books than would
be expected at the higher sales categories and fewer books at the lower sales categories
 lucky him. That’s not a surprise,
given his reputation. If we had a set of
book sales from a random collection of writers, especially selfpublished
writers, we would undoubtedly get a result that conforms much more closely to a
power law than Joe’s does.
I computed the
Rsquare of the log transformed data for each dataset, to make a quantitative
comparison. The closer to 1 that the
value is, the better the data fits a power law function. You can also see this qualitatively if you
inspect the various graphs, noting how closely the lines sit on the tops of the
bars :
·
Asteroid data Rsquare = .99
·
Star visibility data Rsquare = .99
·
Konrath book sales Rsquare = .92
Alternatively, we can compare the three datasets
graphically, as below. This shows the
percentage of events falling into each of the six categories for the differing
datasets.
It is clear that the asteroid line is much more weighted to
the first category than the stars dataset or the Konrath book sales
dataset. Simply put, the smallest asteroids
make up a much larger proportion of all asteroids than the dimmest stars make
up a proportion of all visible stars, or that Konrath’s lowest selling titles
make up a proportion of his overall titles.
So, as far as his power function is concerned, Konrath is closer to the stars than he is to
the asteroids.
That’s also obvious
in the second graph below, which plots the logarithm of the respective
independent variables against their respective categories. The slope of the bestfit
straight line is lower for the book sales, indicating a flatter power law. You can also see that the book sales data
deviate more from a straight line than do the data for the stars and asteroids.
So, what’s behind these power law functions? That’s an active area of research in the
various disciplines.
The asteroid data are undoubtedly a result of the
complicated mechanisms that underlie the formation of the solar system from the
primordial nebula and its subsequent evolution.
This will obviously include the inverse square gravitational law, but
the mixing of particles, the resulting collisions and accretions, the
interaction with magnetic fields and other factors must also come into play. There are a lot of papers on the subject in
the scientific literature.
The stars data is probably a combination of how brightness
falls off as an inverse square law and the particular astrophysics and
astrophysical history of the sun’s part of the galaxy. The fact that the stellar magnitude scale is
related to the human eye’s logarithmic response to light also plays a role.
The book data (along with much other humanrelated power law
phenomena) is thought to be a manifestation of something called preferential
attachment theory. Basically, that’s a
variation of the principles known as “the rich get richer”, “nothing succeeds
like success” or “first mover advantage”.
But that’s another blog for another day.
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