Information Technology Reference
In-Depth Information
We follow the CTRW prescription discussed in a previous chapter and make the
assumption that there exists a stochastic connection between the discrete time
n
and
the physical (continuous) time
t
. In the natural time scale, namely when we set the
condition that the elementary time step
1, the classical master equation has the
discrete form given by (
7.6
). However, now unlike the argument preceding (
7.6
), where
the statistics are determined by the toss of a coin, we want to incorporate the waiting-
time distribution into each occurrence of the
n
events. We assume that the time interval
τ(
t
=
n
)
=
t
(
n
+
1
)
−
t
(
n
)
is derived from a waiting-time distribution density (
3.239
). The
state
p
(
n
)
lasts from
t
(
n
)
to
t
(
n
+
1
)
. The event occurring at time
t
(
n
)
is an abrupt
change from the state
p
(
n
−
1
)
to the state
p
(
n
).
At a generic time
t
we can write the
probability as
t
∞
dt
ψ
n
(
t
)(
t
)(
n
p
p
(
t
)
=
t
−
I
+
K
)
(
0
).
(7.39)
0
n
=
0
Note that
dt
is the probability that
n
events have occurred, the last one occurring
at time
t
. The function
ψ
n
(
t
)
denotes the probability that no event occurs up to time
t
and is given by (
3.218
). The occurrence of an event corresponds to activating the
matrix
(
t
)
(
I
+
K
)
, so that activating
n
events transforms the initial condition
p
(
0
)
into
This form in (
7.39
)iskeptfrom
t
, at which time the last event occurs,
up to time
t
, the time interval from
t
to
t
being characterized by no event occurring. Of
course, the expression (
7.39
) takes into account that the number of possible events may
range from the no-event case to the case of infinitely many events. For this mathematical
idealization to be as realistic as possible, we have to assume that the waiting time
n
p
(
I
+
K
)
(
0
).
may
be arbitrarily small, as small as we wish. This is not quite realistic, insofar as there may
be a shortest truncation time. In real networks there exists also a largest truncation time.
We assume that the latter is so extended that it is possible to accommodate a very large
(virtually infinite) number of events.
Note that the renewal nature of the events driving the dynamics of the network of
interest
S
makes it possible to establish a hierarchy coupling the successive events:
τ
t
dt
ψ
n
−
1
(
t
)ψ
1
(
t
).
ψ
n
(
t
)
=
t
−
(7.40)
0
In fact, if at time
t
the
n
th event occurs, the
(
n
−
1
)
th event must have occurred at an
t
<
earlier time 0
<
t
. We make the assumption that at
t
=
0 an event certainly occurs,
namely
ψ
0
(
t
)
=
δ(
t
).
(7.41)
The waiting-time distribution density
ψ(
t
)
is identified with the probability of the first
event's occurrence at a time
t
>
0,
ψ
1
(
t
)
=
ψ(
t
),
(7.42)
