Information Technology Reference
In-Depth Information
in the rank of the quantity, say city size or word usage; and those that follow the law of
Pareto, where the relative frequency is an inverse power law in the amount of the quan-
tity, say income level or the grade obtained in a science class. Moreover, the two kinds
of distribution are formally related and describe the scaling properties of a wide variety
of physiologic networks, resulting in fractal heartbeats, fractal breaths and fractal steps.
In Chapter
2
we list nearly fifty complex webs having hyperbolic distributions,
residing in disciplines from anthropology to sociology and including information
theory. This list is intended to demonstrate how creative scientists have been in rep-
resenting the complexity of webs in a variety of different ways that capture the intrinsic
variability of the underlying dynamics. We mentioned the relative frequency in terms of
rank or magnitude of a random variable, but there are also distributions in the time inter-
val between events such as earthquakes and solar flares, the power spectrum in terms
of frequency such as in music and DNA sequences, and even psychophysical distribu-
tions in terms of the stimulus intensity or the number of trials. Each distribution is a
testament to the fact that the complexity of the web does not lend itself to interpretation
using normal statistics. For example, there is no one number, such as the average reac-
tion time, that can be used to characterize learning using the method of trial and error.
The entire inverse power-law distribution is necessary to capture this psychophysical
phenomenon; a similar effect was also observed in the cardiovascular, respiratory and
motor-control webs.
In this chapter we also discussed the difference between the observed variation of the
data and that which results in normal statistics. An ever-expanding set of data is accumu-
lated with holes between regions of activity and whose overall size is determined by the
inverse power-law index. The values of the web variable out in the tail of the distribution
dominate the behavior of the process. In the spatial domain the inverse power law is
manifest through data points that resemble clumps of sheep grazing in a meadow as
opposed to the regular spacing of trees in an orchard. In the time domain the heavy-tailed
phenomena give rise to a pattern of intermittency. However, even in regions of activity,
magnification reveals irregularly spaced gaps between times of activity having the same
intermittent behavior down to smaller and smaller time scales.
The purpose of scientific theory is to understand natural phenomena through
prediction and testing, not to just describe what is observed. This is part of what made
the normal distribution so attractive; even though the future could not be predicted with
absolute certainty, a most probable future in terms of the average could be forecast. The
quality of the forecast could then be estimated using the standard deviation or width of
the distribution. As long as two successive predictions were not too far apart this seemed
like a good strategy. However, when patients with heart rates in the normal range have
heart attacks and die, one begins to wonder [
9
]. In the first two chapters the properties
of data are discussed, as are methods for data analysis and description. Chapter
3
begins
the description of the dynamics of webs that could possibly explain the properties of
the data presented earlier and, since we should walk before we run, the beginning for-
malism of the dynamics is primarily linear. We started with simple linear dynamical
webs and systematically increased the complexity of the web to capture various of the
complicated properties observed.
