RE: The Power of Power Laws

Some views on power law (you can find more detailed description of these
views in the paper presented at this e-conference with corresponding
references)

It seems that via power law we can distinguish an internal reflective
process from the external non-generic phenomenon. There should be something
in the generated structure, which really is a limit of iteration that
exhibits an internal process. Any internal (subjective) choice exhibits a
structure of the semantic paradox, arising to Epimenides. The paradox
results from mixing the notion of indicating an element with the act of
indicating a set consisting of elements. We will formalize this approach via
introducing the operation of determination of A and A-, denoted by F which
can be expressed as an infinite recursion, x = F(F(F( �F(x)� ))), by mapping
x = F(x) onto x = F(x). It can be considered as a point in a two dimensional
space. The operation of F is the contraction in a two-dimensional domain,
indicating either A or A-.

Following the approach of Y.-P. Gunji (see his papers in BioSystems and
other journals) we come to the following: because F(x) * x implies a
self-similar set, x is expressed as a particular infinitely iterated s-tuple
branching tree. In this x, the length of the kth branch, which is denoted by
x, is n/sk, where n is the length of the first branch, and the number or the
frequency of branch x, which is denoted by f(x), is sk. Therefore we get
x*f(x) = (n/sk)*sk = n (constant). Roughly speaking, this idea can be
generalized to get one of solutions for F(x). If faithfulness of A is
denoted by m, the invariance of faithfulness with respect to contraction is
expressed as f(m)*m = constant, where m is the value of faithfulness and
f(m) is the probability of m. If distribution of f(m) does not have an
off-set peak, m directly means the rank. Then f(m)*m=c represents what is
called the Zipf�s law, i.e. log (f(m)) = -m+c.

The similar formula was introduced by Mandelbrot (1982) for the fractal
structure, actually fractal is an iteration arising from the set of complex
numbers by squaring them, i.e. by reflecting them to the two-dimensional
space. An observer cannot detect the Zipf�s law until some tool appears,
which is a hyperlink between the other objects. It allows realization of the
combinatorial game between the objects connected by the hyperlink. The third
dimension is a reflection over this 2D domain. It appears if we estimate the
actualization domain (brane) for the error-correction. This is possible only
through the introduction of the internal time of observer. As a result, the
3D+T structure appears.

According to the Zipf's law, the probability of occurrence of words or other
items starts high and tapers off exponentially. Thus a few occur very often
while many others occur rarely. The distribution of words is often an
inverse exponential like e-an. The power laws can be indicative of the
self-organized criticality. The linear iteration with the power law leads to
the golden ratio limit. The golden ratio appears as a threshold for
establishing a connection between local and global periods of the word. The
local period at any position in the word is defined as the shortest
repetition (a square) �centered� in that position. The shortest repetition
from that position is described by the golden ratio.

The power law and the fractal structure appear in the systems exhibiting
reflective control.

Best regards

Andrei Igamberdiev

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Received on Mon Nov 4 02:54:30 2002