ZIPFS LAW AND THE EFFECT OF RANKING ON PROBABILITY-DISTRIBUTIONS

Ranking procedures are widely used in the description of many differen t types of complex systems. Zipf's law is one of the most remarkable f requency-rank relationships and has been observed independently in phy sics, linguistics, biology, demography, etc. We show that ranking play s a crucial role in making it possible to detect empirical relationshi ps in systems that exist in one realization only, even when the statis tical ensemble to which the systems belong has a very broad probabilit y distribution. Analytical results and numerical simulations are prese nted which clarify the relations between the probability distributions and the behavior of expected values for unranked and ranked random va riables. This analysis is performed, in particular, for the evolutiona ry model presented in our previous papers which leads to Zipf's law an d reveals the underlying mechanism of this phenomenon in terms of a sy stem with interdependent and interacting components as opposed to the ''ideal gas'' models suggested by previous researchers. The ranking pr ocedure applied to this model leads to a new, unexpected phenomenon: a characteristic ''staircase'' behavior of the mean values of the ranke d variables (ranked occupation numbers). This result is due to the bro adness of the probability distributions for the occupation numbers and does not follow from the ''ideal gas'' model. Thus, it provides an op portunity by comparison with empirical data, to obtain evidence as to which model relates to reality.