Information Technology Reference
In-Depth Information
The survival probability for a web with
N
=
10
3
nodes for various
K
values.
Figure 7.5.
Consequently, the probability density for the length of time the web remains in a given
global state is an inverse power law of the form
ψ(τ)
∝
τ
−
3
/
2
for an extended interval
of sojourn times.
The survival probability corresponding to the above argument is one having the
inverse power-law form
(τ)
∝
τ
−
1
/
2
for any fixed unperturbed value of the coupling
parameter
g
. In Figure
7.5
we see that as the number of nodes in the web increases
the time over which the survival probability has the predicted inverse power-law form
increases. However, in each case the distribution is eventually truncated by an expo-
nential and the dynamics are determined by solving the Langevin equation (
7.27
). This
behavior is predicted by Kramers' theory, the predicted shape being an inverse power
law, followed by a shoulder ending in an exponential.
7.1.4
The Yule process
In our earlier presentation of the BA model we pointed out that the preferential-
attachment mechanism central to their discussion had been identified previously under a
variety of names by a number of investigators, the first of whom was the mathematician
Yule. As a matter of fact the BA model is a limiting case of the more general argument
presented by Yule in 1925 [
71
] to explain the hyperbolic relation discovered by Willis
in his experimental investigations of speciation of flowering plants published with Yule
in 1922 [
70
]. The Yule process is reviewed by Newman [
49
] using a master equation, a
technique that had not been developed at the time of Yule's analysis. Although Yule and
subsequently Newman concluded the argument in terms of new species being added to
a genus or family, we follow Simon [
58
] and make the presentation in terms of words
in order to avoid confusion over biological jargon.
Simon presented the stochastic model in terms of word frequency. Consider the writ-
ing of a topic that has reached
n
words in length; the double-indexed quantity
p
k
,
n
is
the fraction of
different
words that have occurred exactly
k
times in these first
n
words.
The stochastic process was defined by the following two assumptions.
