Zipf, Power-laws, and Pareto-a ranking tutorial

Abstract

Many man made and naturally occurring phenomena, including city sizes, incomes, word frequencies, and earthquake magnitudes, are distributed according to a power-law distribution. A power-law implies that small occurrences are extremely common, whereas large instances are extremely rare. This regularity or 'law ' is sometimes also referred to as Zipf and sometimes Pareto. To add to the confusion, the laws alternately refer to ranked and unranked distributions. Here we show that all three terms, Zipf, power-law, and Pareto, can refer to the same thing, and how to easily move from the ranked to the unranked distributions and relate their exponents. A line appears on a log-log plot. One hears shouts of "Zipf!","power-law!","Pareto"! Well, which one is it? The answer is that it's quite possibly all three. Let's try to disentangle some of the confusion surrounding these matters and then tie it all back neatly together. All three terms are used to describe phenomena where large events are rare, but small ones quite common. For example, there are few large earthquakes but many small ones. There are a few mega-cities, but many small towns. There are few words, such as 'and ' and 'the ' that occur very frequently, but many which occur rarely. Zipf's law usually refers to the 'size ' y of an occurrence of an event relative to it's rank r. George Kingsley Zipf, a Harvard linguistics professor, sought to determine the 'size ' of the 3rd or 8th or 100th most common word. Size here denotes the

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