Oligo-parametric Hierarchical Structure of Complex Systems

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.