Information Technology Reference
In-Depth Information
4.5
Précis
It is useful to observe that the central limit theorem and the normal distribution were the
backbone of social and medical theory for a couple of hundred years, from sociophysics
to homeostasis in the nineteenth century and well into the twentieth century. In the world
according to Gauss, phenomena are dominated by averages, as we observed earlier. It
is important to point out the fallacy of this view, but carping on this is not sufficient
for our purposes because the normal perspective is
insidious
. The acceptance of the
average as representative of a complex web is almost impossible to dislodge once it has
been adopted. The difficulty is due, in part, to the impossibility of remembering how
the phenomenon was understood before the average was taken as being representative.
The traditional random walk allows us to incrementally increase the complexity of
a web, while retaining the simplicity of Gaussian statistics. Incorporating spatial non-
locality and temporal anisotropy requires going beyond the historical random walks and
incorporating long-time memory and spatial heterogeneity into the stepping process,
which is done through the inclusion of fractal behavior in both space and time. The
fractal random walk is now a few decades old, but its utility in the interpretation of
complex webs is much closer to the present. The discrete form of the fractional calculus
has been shown to be a consistent formalism for taking these effects into account and
is one way to generate inverse power-law spectral densities for the web response to
a random environment. Of course, phenomenological random walks, even those that
are fractal, are not the only way for inverse power laws to enter into the dynamics of
complex webs. Chaos resulting from nonlinear interactions in the dynamic equations is
another.
One can find five functional categories to identify the role of randomness or chaos
in complex webs: (1) search, (2) defense, (3) maintenance, (4) cross-level effects and
(5) dissipation of disturbances, just as Conrad [
16
] suggested for chaos. The search
function might include such phenomena as mutation, recombination and related genetic
operations where randomness plays the role of generator of diversity. In addition to
diversity, search processes also contribute to behavior, say for non-conscious micro-
organisms. The function here is to enhance exploratory activity, whether dealing with
macromolecules, membrane excitability, the dynamics of the cytoskeleton or, on a con-
scious level, the human brain searching for creative solutions to difficult problems. The
random dynamics becomes part of a strategy for exploring the web's phase space.
The function of defense distinguishes the diversity of behavior used to avoid preda-
tors from that used to explore. An organism that moves about in an unpredictable way,
making unexpected changes in speed and direction, is certainly more difficult to ensnare
than one that moves in a straight line. This is apparent to any nature lover who has
watched on public television the graceful but erratic running of a gazelle to avoid being
takendownbyalion.
A third possible function for randomness is the prevention of entrainment in the peri-
odic activity of physiologic processes such as the beating of the heart. The “normal
sinus rhythm” taught in medical school is no more real than is normalcy; human heart
rates are not strictly periodic but contain a broad spectrum of time intervals between
