Database Reference
In-Depth Information
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
.
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
□
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
a
is slightly larger than the −2 shown there; it has been found closer to 2.1.
(2)
Sales of Products
: Order products, say topics at
Amazon.com
, by their sales over the
past year. Let
y
be the number of sales of the
x
th most popular topic. Again, the func-
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
x
th 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 collec-
tion of documents. If you order words by frequency, and let
y
be the number of times
the
x
th word in the order appears, then you get a power law, although with a much
ingly, a number of other kinds of data follow this particular power law. For example,
if we order states in the US by population and let
y
be the population of the
x
th most
populous state, then
x
and
y
obey Zipf's law approximately.
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
Topic 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