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where we have also used the relation between the Laplace transform of the sur-
vival probability and the waiting-time distribution density. Taking the inverse Laplace
transform of (
7.54
) yields the remarkable result
(
t
)
)
=
(
t
).
(7.55)
(
0
A network that is initially prepared to be out of equilibrium decays back to the equi-
librium condition in a manner determined by the survival probability. But why is this
result of interest?
Let us go back to the equation for the elements of the transition matrix (
7.7
). Accord-
ing to this prescription the web reaches equilibrium in one step in the natural time
representation. Thus, it is reasonable that in the continuous-time representation the
regression to equilibrium of
(
)
must coincide with the probability that in this time
scale no event occurs up to time
t
. The waiting-time distribution
t
serves the purpose
of creating in the continuous time scale
t
a process subordinated to that occurring in the
natural time scale
n
. Therefore, we call the waiting-time distribution the
subordination
function
. When we adopt the waiting-time distribution density of the hyperbolic form
given by (
3.239
) as a subordination function we obtain the regression to equilibrium
to be
ψ(τ)
T
T
μ
−
1
(
t
)
)
=
(
t
)
=
,
(7.56)
(
0
+
t
which is much slower than the typical exponential regression that is assumed in much
of the dynamical modeling of physical networks.
Nearly a century ago Lars Onsager proved that physical systems that are out of
equilibrium relax back to their equilibrium state by means of a variety of transport
mechanisms. Reichl [
55
] explains that one assumption necessary to prove Onsager's
relations is that fluctuations about the equilibrium state in a physical network, on aver-
age, decay according to the same laws that govern the decay of macroscopic deviations
from equilibrium. Typically this relaxation is exponential with a macroscopic rate of
relaxation. Consequently, it is the non-exponential relaxation back to the equilibrium
state given by (
7.56
) that is so interesting.
What this form of relaxation means in the present context is that one ought not to
expect complex webs to relax exponentially in time from excited configurations back to
their equilibrium condition. The complexity of the network slows the process from an
exponential to an inverse power-law form when that process is determined by crucial
events.
7.2.3
Perturbing the GME
The time-convoluted form of the GME (
7.51
) is by now familiar. It should be borne in
mind that this is the form of the master equation that arises when the network's dynamics
are driven by events, including crucial events, of course, when the waiting-time distri-
bution
ψ(τ)
is not exponential. Recall that memory depends on the fact that the bath
