Information Technology Reference
In-Depth Information
we can immediately conclude that the probability
p
k
has an asymptotic power-law
distribution with exponent
μ
=
1
+
1
/θ,
(7.38)
thereby restricting the domain of the power law to
The empirical power-law
index obtained by Willis and Yule [
70
] for all flowering plants is
μ
≥
1
.
μ
=
2
.
5
±
0
.
1,
implying that the constant probability of a new species being generated is
.
It is worth pointing out that the inverse power-law index obtained here is not restricted
to the value
κ
=
1
/
3
3 as it was in the BAmodel. It would be a useful exercise for the student
to reinterpret the above argument in terms of the graph-theory parameters introduced in
our review of the BA model. Newman [
49
] also generalized the above argument to
networks composed of a collection of objects, such as genera, cities, papers published,
citations, web pages and so forth. He maintains that the Yule process may well explain
the reputed power laws of all these phenomena.
μ
=
7.2
General modeling strategy
The solution to the master equation has been written in general and the eigenvalue
spectrum determines the dynamics of the probability in the state space for the network
of interest. The question of how the web's dynamics responds to external perturbations
now arises. This is an issue of fundamental importance that has been widely studied
in the field of ordinary statistical physics, particularly in the non-Markov case of the
generalized master equation (GME). More specifically, the neural web of the human
brain is continually responding to external stimulation, as do the physiologic networks
of the human body to a changing environment and the economic markets to world news.
The theory of complex webs, concerning as it does non-ergodic networks, has forced
us to address the same issue from a more general perspective that should reduce to the
traditional prescriptions when we recover the ordinary equilibrium condition.
7.2.1
From crucial events to the GME
Let us now make the assumption that the events do not occur at regular times, but
instead the interval between two consecutive events is determined by the hyperbolic
distribution density
given by (
3.239
). It is important to notice that, in the case in
which the time fluctuations are characterized by a well-defined time scale, the departure
from the condition of event occurrence at regular times will not produce significant
effects, resulting in only a change of time scale. It is evident that interesting effects
emerge when the events are crucial, and especially when
ψ(
t
)
2. The reasons for this
will become clear subsequently. For the time being let us limit ourselves to noticing that
a set of interacting units located on the nodes of a complex network generates global
dynamics characterized by abrupt changes corresponding to those crucial events, which
are incompatible with ergodic theory, namely the crucial events occur with
μ<
μ<
2
.
