Zipf's law, power laws, and maximum entropy

TL;DR

This paper simplifies the explanation of power laws like Zipf's law by applying maximum entropy principles directly to Shannon entropy with a single constraint, challenging more complex existing models.

Contribution

It demonstrates that power laws can be derived more straightforwardly using maximum entropy with just the average log constraint, simplifying prior models.

Findings

Power laws can be derived from a single maximum entropy constraint.

The proposed method simplifies the explanation of Zipf's law.

It challenges the complexity of previous models like RGF.

Abstract

Zipf's law, and power laws in general, have attracted and continue to attract considerable attention in a wide variety of disciplines - from astronomy to demographics to software structure to economics to linguistics to zoology, and even warfare. A recent model of random group formation [RGF] attempts a general explanation of such phenomena based on Jaynes' notion of maximum entropy applied to a particular choice of cost function. In the present article I argue that the cost function used in the RGF model is in fact unnecessarily complicated, and that power laws can be obtained in a much simpler way by applying maximum entropy ideas directly to the Shannon entropy subject only to a single constraint: that the average of the logarithm of the observable quantity is specified.