Information Technology Reference
In-Depth Information
.
dr
dW
ψ(
W
)
=
p
(
r
)
(2.65)
The wealth and rank variables are related by (
2.58
), which can be inverted to yield
K
W
1
/η
r
=
,
(2.66)
and thus (
2.65
) becomes
K
1
/η
η
1
W
1
/η
+
1
.
ψ(
W
)
=
p
(
r
)
(2.67)
On examining (
2.61
), this yields the power-law index
1
η
α
=
1
+
(2.68)
if
p
is independent of
r
and therefore independent of
W
. Note that earlier we adopted
the symbol
(
r
)
μ
for the index of the inverse power-law distribution density
ψ
.Wehave
already remarked that
μ
=
2 is an important value, because it corresponds to the bound-
ary between the region
μ
=
a
>
2, where the mean value of
W
can be defined, and the
region
μ
=
a
<
2, where this mean value diverges to infinity.
we must take into account that in a real language the number of possi-
ble words is finite, and we define that length with the symbol
L
. As a consequence the
maximum wealth
W
max
is given by
To define
p
(
r
)
K
L
η
.
W
max
=
(2.69)
We set the uniform distribution for randomly choosing a word to be
1
L
,
p
(
r
)
=
(2.70)
thereby yielding the normalization constant
K
1
/η
η
1
L
B
=
.
(2.71)
Now we can express Pareto's law (
2.59
) in terms of Zipf's law (
2.58
). In fact, using
(
2.63
) and (
2.68
), we have
1
η
,
k
=
μ
−
1
=
(2.72)
and, using (
2.62
), we have
K
1
/η
L
.
A
=
(2.73)
An example of the empirical relation between the probability and the probability
density is shown in Figure
2.18
, depicting the number of visitors to Internet sites through
the America On Line (AOL) server on a single day. The proportion of sites visited is
ψ(
W
)
,
W
is the number of visitors and the power-law index
μ
=
2
.
07 is given by the
best-fit slope to the data.
