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which is still an inverse power law for the index in the range given above or equivalently
for the Hurst exponent in the range 0
1.
We cannot over-emphasize that the fractal dimension
D
and the Hurst exponent
H
can vary independently of one another as indicated above in the two asymptotic
regimes. From this example we see that the fractal dimension is a local property of time
series (
t
.
5
≤
H
≤
→
→∞
)
and, although not proven here, these are general properties of
D
and
H
[
21
]. It is in
this asymptotic regime that the fractal dimension and the Hurst index are related by
D
0), whereas the Hurst exponent is a global property of time series (
t
H
. Consequently, the scaling behavior of the second moment does not always
give the Hurst exponent as is claimed all too often in the literature and methods of
analysis have been developed to distinguish between the short-time and long-time
scaling properties of time series. We take this question up again at the appropriate
places in our discussion.
=
2
−
2.3
Measuring what we cannot know exactly
There is no general theory describing the properties of complex webs, or, more pre-
cisely, there is a large number of theories depending on whether the web is biological,
informational, social or physical. Part of the reason for the different theories has to
do with the assumptions about web properties which are systematically made so that
the discipline-specific problem being studied becomes mathematically tractable. At the
same time it is often argued that the observed properties of these webs should not
strongly depend on these assumptions, at least in some asymptotic limit. Two exam-
ples of where this asymptotic independence is almost universally believed to be true are
in the choices of initial and/or boundary conditions.
One similar assumption that appears to have only incidental significance, but which
will prove to be pivotal in our discussion, has to do with the preparation of a web and the
mathematical specification of that preparation. Consider a set of dynamical equations
describing some physical phenomenon, whose complete solution is given when one
specifies the initial state of the web. The initial state consists of the values for all the
dynamic variables in the web taken at the initial time
t
0
, a procedure that is completely
justified when solving deterministic equations. Note that this specification of the initial
state of the web together with the analytic solution to the equations of motion determines
the future evolution of the web. It is generally assumed that the initial state specifies
when the web is prepared as well as when the measurement of the web is initiated.
We find that neither of these assumptions is necessarily valid. While criticism of the
assumptions of classical determinism is widely known, and leads to the adoption of
stochastic approaches, the study of time lags between the preparation and observation
of physical quantities leads to the fairly new idea of
aging
.
In simple webs, such as physical networks described by Hamiltonians, it is well
known that the dynamical variables can be determined to any desired degree of accuracy,
including the initial state. A real web, on the other hand, is different. By real web we
mean one that does not necessarily satisfy the assumptions necessary for a Hamiltonian
