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t
β/α
. We find this form of the solution
to be extremely useful in subsequent discussions. For example, West and Nonnen-
macher [
30
] have shown that a symmetric
E
-dimensional Lévy-stable distribution of
the form
in terms of the scaling or similarity variable
q
/
d
E
k
(
bt
β
|
|
α
]
P
L
(
q
,
t
)
=
E
exp
[
i
k
·
q
−
k
(5.169)
2
π)
satisfies the scaling relation (
5.168
). Consequently, the scaling solution (
5.168
)
embraces a broad range of scaling statistics, and we obtain the incidental result that
the Lévy statistical distribution
P
L
(
q
,
t
)
is a solution to the CTRW formalism.
5.5
Reiteration
This chapter encapsulates a number of approaches to describing the dynamics of com-
plex webs when the pieces of the network burst like bubbles or the present configuration
strongly depends on what happened in the distant past, or the measures of variability
overwhelm our ability to make predictions. Non-analytic dynamics examines ways of
studying the changes in web properties over time when the differential equations of
motion are no longer suitable.
The fractional calculus is one systematic method for determining how a fractal func-
tion changes in space and time. The ordinary derivative of a fractal function diverges
and consequently the process described by such a function cannot have ordinary dif-
ferential equations of motion. On the other hand, the fractional derivative of a fractal
function is another fractal function, suggesting that a fractal process described by such
a function may well have fractional equations of motion. In the deterministic case the
ordinary rate equation whose solution is an exponential is replaced by a fractional rate
equation whose solution is a Mittag-Leffler function (MLF). The memory incorporated
into the time dependence of the MLF depicts a process that has an asymptotic inverse
power-law relaxation. This is the slow response of taffy to chewing and the slow but
inexorable spreading of a fracture in a piece of material.
The fractional calculus opens the door to a wide variety of modeling strategies, from
the generalization of a simple Poisson process to the dynamics of the distribution of
cracks. Of course, there is also the extension of stochastic differential equations to
the fractional form, where the history-dependent dynamics determines the response to
present-day fluctuations. This means that the way a web responds to uncertainty today
is dependent on the dynamical history of the web. The longer the history the more mit-
igated is the present uncertainty; in particular, such dependence can explain anomalous
diffusion such as that observed in heart-rate variability.
The next step beyond fractal functions or processes is multifractals, that is, time series
in which the fractal dimension changes over time. The distribution of fractal dimensions
is necessary to capture the full dynamical complexity of some webs. One approach to
modeling multifractals is through the fractional calculus in which the fractional index is
itself a random quantity. The distribution of fractal dimensions can then be determined
through the scaling behavior of the solution to the filtering of the random force in the
fractional Langevin equation. This approach was useful in determining the statistical
