Information Technology Reference
In-Depth Information
Upon implementing the uniform-back-injection assumption again we obtain the condi-
tional waiting-time distribution density
exp
t
+
t
t
)
=
γ(
t
)
t
)
dt
ψ(
t
|
t
+
−
γ(
,
(3.203)
t
from which it is clear that the memory in the waiting-time statistics is determined by the
time dependence of the perturbation. The function
t
)
ψ(
t
|
is the conditional probability
t
+
τ
that an event occurs at time
t
, given the fact that the sojourn in the interval
I
began at time
t
. On the other hand,
t
=
τ
1
+
τ
2
+···+
τ
n
. Thus, the event that
occurred at time
t
depends on an earlier event that occurred at time
=
τ
1
+
τ
2
+···+
τ
n
−
1
, and so on, so that by following this argument back the dependence on the initial
time
t
0 is established. This buried dependence on the initial state of the system is
the sense in which we mean that the perturbation induces memory into the stochastic
process. We discuss this more completely in subsequent sections.
The new waiting-time distribution function (
3.203
) is no longer exponential in time
and consequently the underlying process is no longer Poisson. The precise non-Poisson
form of the process depends on the nature of the time dependence of the perturbation.
The time dependence of the rate in the dynamical generator (
3.200
) has been intention-
ally left arbitrary and we explore certain choices for the perturbation later. There are
various ways in which perturbations of the rate can be realized: (1) by externally excit-
ing the web with a time-varying force; (2) by introducing nonlinear interactions into
the web dynamics; (3) by nonlinearly coupling the web to the environment; and (4) by
coupling the web to a second nonlinear dynamic web, to name a few.
Herein we use the non-Poisson nature of the statistical process as a working definition
of one type of complexity. We know full well that this definition is not acceptable to all
investigators, but we take solace in the fact that the properties of non-Poisson statistics
are sufficiently rich that such distributions can be used to explain the characteristics of
a large number of complex phenomena in the physical, social and life sciences.
=
3.4.3
Inverse power-law correlations
The covariance function for non-stationary time series is given by (
3.77
), which is inter-
esting as a mathematical expression but does not provide much guidance regarding how
to interpret the underlying data set. Suppose we have
N
discrete measurements given by
{
how might we go about determining whether there is a pattern
in the data? One measure is the correlation coefficient defined by
Q
j
}
,
j
=
1
,
2
,...,
N
;
N
j
=
1
(
−
n
[
1
/(
N
−
n
)
]
Q
j
−
Q
)(
Q
j
+
n
−
Q
)
Cov
n
Q
Va r
Q
=
r
n
=
,
(3.204)
Q
j
−
Q
2
N
j
=
1
−
n
(
1
/
N
)
the ratio of the covariance between elements in the discrete time series separated by
n
nearest neighbors and the variance of the time series. This is a standard definition
