Information Technology Reference
In-Depth Information
exp
4
uD
q
a
−
q
0
W
(
q
a
,
u
;
q
0
)
=
ν
−
ν
2
+
(6.43)
2
D
and here again the first-passage-time pdf is obtained by taking the inverse Laplace
transform. The form is given by
exp
2
D
(
LT
−
1
exp
t
√
−
(
q
a
−
q
0
)
W
(
q
a
,
t
;
q
0
)
=
q
a
−
q
0
)
θ
;
,
(6.44)
D
where the inverse Laplace transform is taken with respect to
θ
,
2
θ
=
ν
+
4
uD
.
(6.45)
Thus, using the inverse Laplace transform for exp
−
a
√
u
, we obtain
t
3
/
2
exp
2
q
a
−
q
0
1
−
(
q
a
−
q
0
−
ν
t
)
W
(
q
a
,
t
;
q
0
)
=
√
4
(6.46)
4
Dt
π
D
for a biased Gaussian distribution.
However, the task-priority model of Barabási is not universally accepted. Hong
et al
.[
21
] analyzed the time interval between two consecutive short messages (SMs)
via cell phone. They used eight volunteers (denoted by the letters A through H). These
volunteers consisted of one company manager (C) and seven university students (else).
The overall time spans of those phone records ranged from three to six months. As
shown in Figure
6.5
the inter-event distribution density
ψ(τ)
is well fit by the inverse
power-law function
1
τ
μ
,
ψ(τ)
∝
(6.47)
where the exponent
1. Each curve in Figure
6.5
has an
obvious peak appearing at approximately ten hours. This peak is related to the physi-
ologic need for humans to sleep. When it is time to sleep, individuals postpone their
response to phone messages for the duration of the sleep interval.
The time statistics of SM communications are similar to those observed in email [
8
]
and surface-mail [
27
] communications. However, the SM communication is directly
perceived through the senses. We cannot invoke the priority-task principle. We may
reply soon to urgent and important letters, and the ones that are not so important or
require a difficult reply may wait for a longer time before receiving a response. The same
ideas apply to emails. By contrast, we usually reply to a short message immediately. The
delay in replying here might be caused not by the fact that the received message is not
important, but by other motivating factors.
μ
is in the interval 1
.
5
<μ<
2
.
6.2
Graphic properties of webs
In this section we further compare random webs with real webs. We find that both
random and real webs are characterized by small length scales. The small-length-scale
