Information Technology Reference
In-Depth Information
With increasing noise
D
,
g
0
increases and tends to its maximum value, which, in this
idealized representation, is
g
0
=
1. The logarithmic contribution to the signal-to-noise
ratio (
7.100
), which is negligible for
g
0
→
0, becomes active in the case of large values
of
g
0
, and makes the signal-to-noise ratio decrease to zero after a broad maximum, in
the region of
g
0
of the order of
ω
. This non-monotonic behavior of the signal-to-noise
ratio with increasing noise is why this process is called stochastic resonance. In this sub-
section we have followed the approach to stochastic resonance proposed by McNamara
and Wiesenfeld [
46
].
7.3
The Universality of LRT
In this section we discuss LRT from a perspective outside the stochastic differential
equations used in Chapter 3. The new approach extends the formulation and applica-
tion from physical processes driven by classical or quantum Liouville equations to the
dynamics of complex networks in a variety of venues. Such extensions are necessary
because the mechanisms that can be identified in physical webs might not be acting in
the webs of society or in those of life; the interactions between organizations or organs
may be less tangible than those in physical webs. This is an ambitious undertaking;
however, there are preliminary indications that the non-stationary LRT proposed in this
section may turn out to be a universal principle. This would make the more general
LRT one of the few principles applicable to the dynamics of complex webs that is not
mechanism-specific; another would be the generalized Onsager principle.
No general theory exists that successfully extends LRT from the condition of an equi-
librium web to that of a non-equilibrium web [
21
]. Although the question is interesting,
little progress has been made to explain some particular aspects of non-equilibrium
web response to perturbation; no theorem as general as the equilibrium fluctuation-
dissipation theorem (FDT) is yet available [
21
]. Furthermore, the recent theoretical
prediction on the lack of a steady response to external stimuli [
61
] may generate the
conviction that in the non-stationary case it is not possible to have any form of LRT.
The recent literature on non-Poisson renewal processes has increased the interest
regarding the action of non-ergodic renewal events [
19
,
45
,
54
], with a wide set of appli-
cations, ranging from quantum mechanics [
18
] to brain dynamics [
32
], thereby casting
significant doubt on the possibility of using conventional LRT to study the effects of
perturbation in these cases. This limitation is a consequence of two things: (i) it is very
difficult, if not impossible [
16
], to describe the time evolution of event-driven webs by
means of Hamiltonian operators (the classical or quantum Liouville equation); and (ii)
it is not yet well understood how to use the linear response structure of (
7.70
) when a
stationary autocorrelation function is not available, in spite of the fact that some pre-
scriptions for the non-stationary situation already exist [
4
,
7
]. These are probably the
reasons why Sokolov and Klafter [
60
,
61
] coined the term “death of linear response” to
denote the fading response of a complex web to a harmonic stimulus. This interesting
phenomenon piqued the interest of other researchers and also inspired a debate on the
best way to generate this response in surrogate sequences [
35
,
43
,
44
].