Information Technology Reference
In-Depth Information
in science do not have a characteristic scale. We introduce that condition through a
function that scales, that is, the logarithm. The quantity to be varied is then
p
p
1
p
constant
I
=−
(
q
)
ln
p
(
q
)
dq
−
α
(
q
)
dq
−
−
β
(
q
)
ln
(
q
+
T
)
dq
−
,
(1.40)
where we have included the normalization and the average logarithm ln
(
q
+
T
)
as
constraints. The variation of
I
being set to zero yields the relation
ln
p
(
q
)
=
α
−
1
−
β
ln
(
q
+
T
),
which through the normalization condition yields the properly normalized distribution
on the domain
(
0
,
∞
)
:
T
β
−
1
)
=
(β
−
1
)
p
(
q
.
(1.41)
)
β
(
q
+
T
The entropy-maximization argument therefore determines that a distribution that has no
characteristic scale, is finite at the origin, and is maximally random with respect to the
rest of the universe is given by a hyperbolic distribution (
1.41
), which asymptotically
becomes an inverse power-law distribution
p
q
−
β
.
This hyperbolic distribution has the kind of heavy tail that is observed qualitatively
in the distribution of grades in the sciences. Consequently, it would be prudent to explore
the differences between the normal and hyperbolic distributions in order to understand
the peculiarities of the various phenomena they represent. But it is not just the grade
distribution that is described by a distribution with an inverse power-law tail but liter-
ally hundreds of complex webs in the physical, social and life sciences. This statistical
process is shown to be representative of many more phenomena than are described by
the distribution of Gauss. In fact, most of the situations where the normal distribution
has been applied historically are found to be the result of simplifying assumptions and
more often than not the use of the bell-shaped curve is not supported by data. These
are the phenomena we go after in this topic, to show that when it comes to describing
complex webs the normal distribution and nomalcy are myths.
(
q
)
∝
1.2
Empirical laws
The first investigator to recognize the existence of inverse power-law tails in complex
phenomena by looking at empirical data was the social economist Marquis Vilfredo
Frederico Damaso Pareto (1848-1923). Unlike most of his nineteenth-century contem-
poraries, he believed that the answers to questions about the nature of society could
be found in data and that social theory should be developed to explain information
extracted from such observations, rather than being founded on abstract moral prin-
ciples. As Montroll and Badger [
22
] explain, Pareto collected statistics on individual
income and wealth in many countries at various times in history and his analysis
convinced him of the following [
25
]:
