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bear on understanding complex webs. The Poisson statistics of the graphical descrip-
tion of a web are like the normal statistics discussed earlier in that, although they do not
describe any empirical data from real complex webs, they do provide insight into the
inverse power law that replaces normal statistics. The random walk provides a frame-
work in which to examine the influence of long-time memory and spatial inhomogeneity
by modifying transition rates and studying how the resulting distributions are modified.
In this way Lévy flights are determined to be the solution to fractional differential equa-
tions in phase space and inverse power-law spectra are determined to be the solution to
fractional random walks. The fractional calculus is found to be a useful tool in modeling
the potential influence of what occurred long ago and far away on what might happen in
the here and now. The non-normal behavior of the world we experience is due to the fact
that simple differential equations, even stochastic ones, are not sufficient to determine
our future. Not all events from the past influence what we say and do in the same way.
Some occurred very long ago but still burn bright, whereas what occurred yesterday has
already disappeared into the forgettory. This is discussed more fully in a later chapter.
The linear dynamics from Chapter
3
were known to be inadequate for describing
complex webs even when they were first discussed. However, many of the concepts
introduced there bear fruit in the latter part of Chapter
4
through the notion of chaos. The
notions of randomness and chaos are closely linked, particularly how the chaos resulting
from the breakup of Hamiltonian orbits, nonlinear maps and strange attractors leads to
unpredictability in dynamical networks. The non-integrability of Hamiltonian systems
was used to define a booster, the source of statistical fluctuations in a web of interest.
The discussion of the relationship between chaos and random fluctuations in stochastic
equations of motion might seem like mathematical minutiae, but its importance in pro-
viding a mechanical basis for understanding randomness in physical networks cannot be
over-stated. If, on the other hand, you are content with the idea that we do not have com-
plete information about any complex web, so that in any dynamical model of the web we
should always include a little randomness, that acceptance will work as well. However,
one must exercise caution in introducing randomness into dynamical equations, because
the way uncertainty enters the description can modify the predicted outcome. Recall
that additive fluctuation led to the Gauss distribution whereas multiplicative fluctuations
gave rise to the inverse power law.
In Chapter
5
we describe the dynamics of complex webs in terms of non-
analytic functions. Functions of the kind that are continuous everywhere but nowhere
differentiable were invented by Weierstrass and were subsequently shown to have a
fractal dimension. We discussed non-analytic functions earlier in the context of fractal
processes, but in Chapter
5
we showed that the fractional calculus is required in order to
describe the underlying dynamics of the fractal processes described by these functions
[
10
]. The fractional calculus was briefly reviewed and used to describe determinis-
tic phenomena whose dynamics are non-local in time, that is, processes that depend
on memory. The solutions to dissipative fractional rate equations were shown to be
Mittag-Leffler functions, a Mittag-Leffler function being one that is a stretched expo-
nential at early times and an inverse power law asymptotically. This behavior was able
to generalize the Poisson distribution in which the exponential is replaced with the
