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statistical properties of these subsets may be characterized by the distribution of fractal
dimensions
f
. In order to describe the scaling properties of multifractal signals it is
necessary to use many local
h
-exponents.
The incremental increases in web complexity have progressed under our eyes from
linear to nonlinear, then back to linear, but with fractional dynamics, and from ordinary
Langevin to generalized Langevin to fractional Langevin equations when the random
influence of the environment is taken into account. However, even these techniques
are not adequate for describing multifractal statistical processes. Multifractals have not
a single fractal dimension, but multiple fractal dimensions, which could, for exam-
ple, change with the development of the network in time. One strategy for modeling
a phenomenon whose fractal dimension changes over time, a multifractal process, is to
make the operator's fractional index a random variable. To make these ideas concrete,
consider the dissipation-free fractional Langevin equation
(
h
)
t
−
η
0
D
t
[
Q
(
t
)
]−
Q
(
0
)
=
ξ(
t
).
(5.79)
(
−
η)
1
Equation (
5.79
) has been shown to be derivable from the construction of a fractional
Langevin equation for a free particle coupled to a heat bath when the inertial term is
negligible [
13
]. The formal solution to this fractional Langevin equation is
t
Q
(
t
)
≡
Q
(
t
)
−
Q
(
0
)
=
K
α
(
t
−
τ)ξ(τ)
d
τ
;
α
=
1
−
η,
(5.80)
0
where the integral kernel is defined as
1
(α)
1
K
α
(
t
−
τ)
=
−
τ)
α
.
(5.81)
(
t
The form of (
5.80
) for multiplicative stochastic processes and its association with mul-
tifractals have been noted in the phenomenon of turbulent fluid flow through a space
rather than a time integration kernel [
1
].
The random-force term on the rhs of (
5.80
) is selected to be a zero-centered, Gaussian
random variable and therefore to scale as
H
ξ(λ
t
)
=
λ
ξ(
t
),
(5.82)
where the Hurst exponent is in the range 0
<
H
≤
1
.
In a similar way the kernel scales
as follows:
1
(α)
1
)
α
=
λ
−
α
K
K
α
(λ
t
)
=
α
(
t
),
(5.83)
(λ
t
so that the solution to the fractional multiplicative Langevin equation scales as
H
+
1
−
α
Q
(λ
t
)
=
λ
Q
(
t
).
(5.84)
In order for this solution to be multifractal we assume that the parameter
is a random
variable. To construct the traditional measures of multifractal stochastic processes we
calculate the
q
th moment of the solution by averaging over both the random force and
the random parameter to obtain
α
