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The experimental result shown in Figure
7.8
forces us make Pandora's choice for the
time-dependent rate:
r
0
g
(
t
)
=
r
1
t
,
(7.147)
1
+
which, in fact, as shown previously, yields the hyperbolic survival probability
T
t
μ
−
1
(
t
)
=
.
(7.148)
+
T
This is the proper time for us to use the stochastic rate defined in Section 3.5.2. Let
us adopt the more concise, and more precise, notation
r
(
t
)
=
g
(
t
−
t
i
),
(7.149)
where
t
i
is the random time of occurrence of the last event prior to time
t
. However, to
make the statistics more transparent, it is convenient to notice that we can write
exp
t
exp
t
t
),
−
r
(τ)
d
τ
=
−
g
(τ)
d
τ
=
(
t
−
(7.150)
t
t
when there are no events from time
t
to
t
and the last equality arises from the definition
of the survival probability. Thus, the survival probability
;
depends on the time
t
at
which observation begins:
t
t
e
−
t
r
(τ)
d
τ
=
e
−
t
g
(τ)
d
τ
t
)
=
dt
R
t
)
dt
R
t
)
(
t
,
(
(
0
0
t
dt
R
t
)(
t
).
=
(
t
−
(7.151)
0
In the case in which the laminar region between two consecutive events, occurring at
times
t
i
and
t
i
+
1
, is filled with values
ξ
s
drawn from a distribution with finite width,
it has been shown, see [
1
], that the autocorrelation function
C
t
)
(
t
,
reduces to the
t
)
age-dependent survival probability
(
t
,
, and hence the response function in (
7.144
)
becomes
d
dt
t
)
=
t
)
=
t
).
χ(
t
,
C
(
t
,
R
(
t
)(
t
−
(7.152)
Alternatively, the linear response function (
7.145
) becomes
t
d
dt
C
t
)
=−
t
)
=
dt
R
t
)ψ(
t
),
χ(
t
,
(
t
,
(
t
−
(7.153)
0
which shows explicitly how the choices of (
7.144
) and (
7.145
) depend on
R
.The
authors of [
35
,
43
,
60
,
61
] followed Sokolov [
59
], whose theory was proved [
4
,
7
]
to yield the choice of (
7.144
). The rationale for this choice is that the perturbation
does not influence the event-occurrence time, but only the drawing of the variable
(
t
)
ξ
s
to fill the time intervals between two consecutive events. We refer to this theory as
the
phenomenological
FDT. It has been shown [
4
,
7
] that the choice (
7.153
), called
the
dynamical
FDT, corresponds to the network response produced by the external
perturbation affecting the event-occurrence time according to the state of the web
S
.
