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 paper I argue that the specific 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.
Highlights
Zipf’s law [1, 2, 3], and power laws in general [4, 5, 6], have and continue to attract considerable attention in a wide variety of disciplines — from astronomy to demographics to software structure to economics to zoology, and even to warfare [7]
In the present article I shall argue that the specific cost function used in the RGF model is unnecessarily complicated, ( RGF most typically leads to a hybrid geometric/power law, not a pure power law), and that power laws can be obtained in a much simpler way by applying maximum entropy ideas directly to the Shannon entropy itself [20, 21] subject only to a single constraint: that the average of the logarithm of the observable quantity is specified
Apart from the issues raised above, one could in addition explicitly restrict the state space to be finite, adding yet another free parameter, (M in the language of reference [8], N in the language of this note), but there is little purpose in doing so — the key insight is this: Once the data are assigned to ordinal boxes, hybrid geometric/power laws drop out automatically and straightforwardly by maximizing the Shannon entropy subject to the two very simple constraints ln n = χ and n = μ
Summary
Zipf’s law [1, 2, 3], and power laws in general [4, 5, 6], have and continue to attract considerable attention in a wide variety of disciplines — from astronomy to demographics to software structure to economics to zoology, and even to warfare [7]. I would argue that (at least as long as the main issue one is interested in is “merely” the minimum requirements for obtaining a power law) the appeal to a fractal framework and the iterative model adopted by [19] is unnecessarily complicated. To place this observation in perspective, I will explore several variations on this theme, modifying both the relevant state space and the number of constraints, and will briefly discuss the relevant special functions of mathematical physics that one encounters (zeta functions, harmonic series, poly-logarithms). I shall discuss an extremely general Gibbs-like model, and the use of non-Shannon entropies (the Renyi [22] and Tsallis [23] entropies and their generalizations.) There is a very definite trade-off between simplicity and generality, and I shall very much focus on keeping the discussion as technically simple as possible, and on identifying the simplest model with minimalist assumptions
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