Information Technology Reference
In-Depth Information
By inserting the distribution
(
g
)
of (
3.251
)into(
3.250
), replacing the integration
variable
g
with
x
=
g
(
t
+
T
), using the definition of the gamma function and the well-
known property
it is possible to prove that the distribution density
resulting from the integration in (
3.250
) becomes identical to the hyperbolic distribution
density (
3.239
).
The interpretation of this result is that the emergence of a hyperbolic law can be
caused by the superposition of infinitely many exponential relaxation functions. How-
ever, this approach to complexity has caused some confusion in the literature. In fact,
as pointed out by Allegrini
et al
.[
3
], for this approach to make physical sense we have
to imagine that the web remains in a state in the neighborhood of a given rate
g
for an
extended period of time. For example, it is possible that the distribution of rates yields
(
z
+
1
)
=
z
(
z
)
1
dg
(
g
)
∝
dt
,
(3.252)
/
in which case the renewal condition for the statistics is lost. This is especially evident
in the case where
g
is a monotonically decreasing function of time. The time inter-
val between two consecutive events tends to become longer and longer, whereas in a
renewal process a very long time interval between two consecutive events may appear
on any time scale.
The Weibull distribution function for
(
t
)
0 corresponds to a singularity that is
responsible for the emergence of the hyperbolic behavior so ubiquitous in the field of
complexity. This is reminiscent of the myth of Pandora's box. After Prometheus' theft
of the secret of fire, Zeus ordered Hephaestus to create the woman Pandora as part of
the punishment for mankind. Pandora was given many seductive gifts from Aphrodite,
Hermes, Charites and Horae (according to
Works and Days
[
25
]). For fear of additional
reprisals, Prometheus warned his brother Epimetheus not to accept any gifts from Zeus,
but Epimetheus did not listen and married Pandora. Pandora had been given a large
jar and instructions by Zeus to keep it closed, but she had also been given the gift of
curiosity, and ultimately opened it. When she opened it, all the evils, ills, diseases and
burdensome labor that mankind had not known previously escaped from the jar, but the
one thing that was good that came out of the jar was hope. Our box is given by (
3.236
).
In this topic, under the influence of curiosity we explore the singularity in (
3.236
). If we
move from this representation to the conventional Weibull picture, the condition
β
=
β
1
e
−
α
t
.Thisis
equivalent to not opening Pandora's box. If we do open it, we see the emergence of the
complex world of the inverse power law. Of course, we are giving an optimistic interpre-
tation of the myth of Pandora, as the discoverer of complexity, namely of the phenomena
of organized webs, including diseases and the spreading of epidemics. Understanding
their origin may contribute to improving the conditions of the world.
What is the origin of this inverse power-law behavior? We have seen two distinct
approaches to this condition. The superstatistics approach [
6
], in which an ensemble of
exponentials with different rates is assumed to govern the process, implies a deviation
from the renewal condition. In our discussion we focus more on the renewal approach,
apparently generates an extremely slow exponential relaxation
(
t
)
=
