1.3. THINGS USEFUL TO KNOW
15
1
10
100 1000 10,000
1
10
100
1000
10,000
100,000
1,000,000
10,000,000
Figure 1.3: A power law with a slope of −2
Example 1.9 : In Fig. 1.3 we see that when x = 1, y = 106, and when x =
1000, y = 1.
Making the ﬁrst substitution, we see 106 = c.
The second
substitution gives us 1 = c(1000)a. Since we now know c = 106, the second
equation gives us 1 = 106(1000)a, from which we see a = −2. That is, the law
expressed by Fig. 1.3 is y = 106x−2, or y = 106/x2.
✷
We shall meet in this book many ways that power laws govern phenomena.
Here are some examples:
1. Node Degrees in the Web Graph: Order all pages by the number of in-
links to that page. Let x be the position of a page in this ordering, and
let y be the number of in-links to the xth page. Then y as a function of x
looks very much like Fig. 1.3. The exponent a is slightly larger than the
−2 shown there; it has been found closer to 2.1.
2. Sales of Products: Order products, say books at Amazon.com, by their
sales over the past year. Let y be the number of sales of the xth most pop-
ular book. Again, the function y(x) will look something like Fig. 1.3. we
shall discuss the consequences of this distribution of sales in Section 9.1.2,
where we take up the matter of the “long tail.”
3. Sizes of Web Sites: Count the number of pages at Web sites, and order
sites by the number of their pages. Let y be the number of pages at the
xth site. Again, the function y(x) follows a power law.
4. Zipf’s Law: This power law originally referred to the frequency of words
in a collection of documents. If you order words by frequency, and let y
