Astrophysics Corner, Part 5 – Asteroids, Stars, and Konrath’s Book Sales – The Power Law

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, best-selling 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 (0-5000)	1	77
5001-12000	2	37
12001-28800	3	23
28801-69120	4	13
69121-165888	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 self-published writers, we would undoubtedly get a result that conforms much more closely to a power law than Joe’s does.

 I computed the R-square 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 R-square = .99

·         Star visibility data R-square = .99

·         Konrath book sales R-square = .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 best-fit 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 human-related 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.