Information Technology Reference
In-Depth Information
It is convenient to rewrite (
7.151
) in the following form:
t
t
)
=
dt
R
t
)ψ(
t
),
ψ(
t
,
(
t
−
(7.154)
0
where we recall that
∞
R
(
t
)
=
0
ψ
n
(
t
).
(7.155)
n
=
The quantity
R
is the number of events per unit time produced by the ensemble of
complex networks prepared at time
t
(
t
)
=
0 and can easily be derived from
ψ(
t
)
for a
renewal process.
The double-indexed probability density
ψ
t
t
in (
7.154
) refers to a web that has
,
been prepared at time
t
0 when an event occurs, say one generated by an external
excitation. For a renewal process the probability density of an event occurring at time
t
given that the last event occurred at time
t
has the exact form [
68
]
=
t
0
ψ
n
(
∞
t
)
=
ψ(
t
)ψ(
t
)
dt
.
ψ(
t
,
t
)
+
t
−
(7.156)
n
=
1
In the physics literature
ψ(
t
)
dt
is the probability that an event occurs in the time inter-
. Since the observation process begins at time
t
>
,
+
val
0, the probability of
observing a new event after
t
depends on the last event occurring prior to
t
, at time
t
as expressed by the distribution density
(
t
t
dt
)
t
is
in general the last of a sequence of
n
events, occurring exactly at
t
, while the earlier
events can occur at any earlier time
t
t
)
. The event occurring at
t
<
ψ(
t
−
>
0. The probability for this last of
n
events is
t
)
ψ
n
(
1; this is the first term in the integrand of (
7.156
). The renewal nature
of the process is adopted to define the function
, with
n
≥
ψ
n
(
t
)
through the hierarchy
t
0
ψ
n
−
1
(
t
)ψ
1
(
t
)
dt
ψ
n
(
t
)
=
t
−
(7.157)
starting at the preparation
t
=
0, which is established by setting the condition
ψ
0
(
)
=
δ(
).
t
t
(7.158)
The hierarchy of (
7.157
) allows us to express the distribution densities
ψ
n
(
t
)
in terms
of
. The time-convolution structure of (
7.157
)isa
consequence of the renewal nature of the process and makes it convenient for us to use
the Laplace-transform method. It is straightforward to prove that (
7.157
)using(
7.158
)
yields the following relation among the Laplace-transformed functions:
ψ
1
(
t
)
, which is identified with
ψ(
t
)
ψ
n
(
)
=
( ψ(
n
u
u
))
.
(7.159)
From this algebraic form of the Laplace transform of the probability density for
n
events
it is evident that the time intervals between events in a renewal process are independent
of one another.
