Databases Reference
In-Depth Information
The Matthew Effect
Often, the existence of power laws with values of the exponent higher than
1 are explained by the Matthew effect. In the biblical Book of Matthew,
there is a verse about “the rich get richer.” Many phenomena exhibit this
behavior, where getting a high value of some property causes that very
property to increase. For example, if a Web page has many links in, then
people are more likely to find the page and may choose to link to it from
one of their pages as well. As another example, if a book is selling well
on Amazon, then it is likely to be advertised when customers go to the
Amazon site. Some of these people will choose to buy the topic as well,
thus increasing the sales of this topic.
rank 1000 the sales are a fraction of a book is too extreme, and we would in
fact expect the line to flatten out for ranks much higher than 1000.
2
The general form of a power law relating x and y is log y = b + a log x. If we
raise the base of the logarithm (which doesn't actually matter), say e, to the
values on both sides of this equation, we get y = e
b
e
a log x
= e
b
x
a
. Since e
b
is
just “some constant,” let us replace it by constant c. Thus, a power law can be
written as y = cx
a
for some constants a and c.
Example 1.8 : In Fig. 1.3 we see that when x = 1, y = 10
6
, and when x =
1000, y = 1. Making the first substitution, we see 10
6
= c. The second
substitution gives us 1 = c(1000)
a
. Since we now know c = 10
6
, the second
equation gives us 1 = 10
6
(1000)
a
, from which we see a =−2. That is, the law
expressed by Fig. 1.3 is y = 10
6
x
−2
, or y = 10
6
/x
2
.
2
We shall meet in this topic 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.
