  We investigate the possible origin of hierarchical structures in complex
systems describable in terms of a finite and small number of parameters which
control the behavioral pattern at each level of organization. We argue that the
limitation on the number of important parameters at each stage is a reflection
of the fact that Thom's classification of catastrophes, i.e., qualitative
changes, involve only a few parameters. In addition, we also point out that
even in systems with a large number of components, only a few may be of
statistically great significance, just as in Zipf's law the quantitative
measure of the important collections is inversely proportional to the rank. We
then consider the concept of relative degeneracies coming from change of
resolving power, at various scales, which too would vindicate the procedure of
coarse-graining in building up hierarchical organizations. We suggest that,
similar to the group-theoretical annihilation of dangling tensor indices due to
symmetry to minimize energy, even in more inexact contexts such as in biology
and the social sciences, similar attempts by the system to reduce frustration
may lead to cluster formation, which are semi-closed, and let leakage
interactions come into play at larger scales.

Adaptation and Self-Organizing Systems;Disordered Systems and Neural Networks;Physics and Society