Information Technology Reference
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Number of users
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Figure 2.18.
The number of visitors to Internet sites through the AOL search engine on a December day in
1997. The plot corresponds to the density function ψ( W ) , with W being the number of visitors
in this case [ 1 ]. Adapted with permission.
2.3.2
Chaos, noise and probability
But from where does uncertainty emerge? A nonlinear web with only a few degrees
of freedom can have chaotic solutions and therefore may generate random patterns.
So we encounter the same restrictions on our ability to know and understand a web
when there are only a few dynamical elements as when there are a great many, but for
very different reasons. Let us refer to random processes, which are produced by the
unpredictable influence of the environment on the web of interest as noise. In this case
the environment is assumed to have an infinite number of elements, all of which we do
not know, but they are coupled to the web of interest and perturb it in a random, that is,
unknown, way [ 68 , 96 ]. By way of contrast, chaos is a consequence of the nonlinear,
deterministic interactions in isolated dynamical webs, resulting in the erratic behavior of
limited predictability. Chaos is an implicit property of complex dynamic webs, whereas
noise is a property of the environment in contact with such webs of interest. Chaos
can therefore be controlled and predicted over short time intervals, whereas noise can
neither be predicted nor be controlled except perhaps through the way the environment
is coupled to the web.
One of the formal mathematical constructs involving dynamical webs is ergodic the-
ory, a branch of mathematics that seeks to prove the equivalence between ensemble
averages and time averages. To determine a time average we imagine that the recurrence
time of a trajectory to its initial state is not exact, but has a certain error. The trajectory
is not really coming back to the same initial state, but is as close to that state as one may
want it to be. In this way, if the phase space is partitioned into boxes, every box may
contain an infinite number of recursions, and this property is independent of the box
size. Ergodic theory affirms that all the cells in the available phase space are “visited”
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