Zipf's Law

In 1932 George Zipf published an empirical rule describing a statistical regularity in the distribution of words used in any large text. Let us rank ($r$) words in a given text according to the frequency ($f$) of their occurence. So, the most frequently occuring word would be given the rank $r=1$, the next most frequently occuring word, $r=2$ and so on. Zipf's law states that

$$f(r) \sim r^{-\alpha}$$ (7.2)

with $\alpha \sim 1$. This power law has been tested over a large volume of literature and also different languages and is found to be accurate for words whose rank is not too low or too high! [9]. It has been argued [10] that for larger collection of texts, there are significant deviations from a single power law and that at large ranks there is a transition to a second power law regime. It is still unclear as to how one may explain the law within a model of human behaviour, or as to how one may use the generalised law to characterise and differentiate human writing from other froms of text. However it has ben argued [11] that the distribution of a word in text can be used to distinguish whether it is a noun or verb! Zipf also noted that a similar power law applied to the sizes of cities. That is, if the largest city in a country was assigned the rank $r=1$, the second largest $r=2$, then a power law as above is found. This has been found to be true for many countries but not for those with central planning. Thus one can hope that the explanation of the power-law of city size distribution can be found in the self-organising properties of a society. In reference [12] the authors consider a model of city formation and find that if the individuals in the model interact pairwise then Zipf's law emerges for larger cities. Other models have also been considered as discussed in that reference. Is Zip's power law an example of self-organised criticality ? The obvious tendency is to link any power law that occurs in a potentially self-organising system with the principle of self-organised criticality. However, persuasive models of self-organised criticality for Zipf's laws do not appear to have been constructed.