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Mittag-Leffler function. The solutions to fractional dissipative stochastic equations are
shown to scale, that is, to display anomalous diffusion, due to this dynamical mem-
ory. More importantly, the multifractal character of certain complex physiologic webs,
including migraines and cerebral blood flow, is successfully modeled with the derivative
index of such fractional equations being a random variable [ 11 ].
It is true that what occurs now and in the near future may be significantly influenced
by what happened in the distant past and fractional time derivatives are one way to
model that influence. However, when we look at such things as earthquakes, the natu-
ral habitat of ecological webs, or the terrain over which wars are fought, one is struck
by how the geography influences the propagation of information. Nothing moves in a
straight line for any substantial distance, but zigs and zags to accommodate the hills
and valleys. Random fields and the fractional propagation of influence on those fields
are also discussed in Chapter 5 , as is the first successful quantification of phenom-
ena with diverging central moments, through continuous-time random walks (CTRWs).
This theory has emerged as a systematic way to generalize the FPE to fractional form
and to account for the bursting behavior in time and the clustering in space observed in
many complex phenomena [ 10 ].
The promise of finding a new kind of science from the understanding of complex
webs has motivated and led a great deal of the research in complex webs over the last
decade. The literature on networks is much too vast for us to provide a review in one
brief chapter, so what we attempted to do in Chapter 6 was to show how some of the
major research themes fit into the flow of the general arguments associated with the
empirical hyperbolic distributions and their mathematical descriptions in the first five
chapters. The starting point is the easiest to understand since the paradigm was the
random network in which the statistics of the connectivity of the members of the web
are described by a Poisson distribution. The development of small-world theory and
the preferential attachment used to model the growth of networks was shown to be
closely related to certain historical ideas on how to understand the hyperbolic behavior
of complex webs. Various metrics developed in the context of networks were reviewed
and their values, such as the web size and clustering coefficient, for real webs were
compared with those from a random web. One of the important applications of the
scale-free modeling is the prediction of the distribution of failures both locally and
catastrophically, with the discovery that inverse power-law distributions are robust
against random attacks, but fragile with respect to attacks directed at hubs. The
assertion made at the turn of the century that this might be a property of the Internet
explains in part the subsequent explosion of interest in the science of networks. The
terrorist attack of 9/11 also provides a partial explanation, because of the need to
identify, disrupt and destroy terrorist webs.
Chapter 6 also introduced the idea of a deterministic web in which the complex
topology gave rise to the observed scaling behavior, with no statistics involved. This
geometric complexity relates back to the earlier discussion on fractals and non-analytic
functions.
In Chapter 7 we attempted to determine what is cause and what is effect in the many
distributions and models introduced in the topic. For example, the metrics reviewed in
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