Random texts do not exhibit the real Zipf's law-like rank distribution

Abstract

Background: Zipf's law states that the relationship between the frequency of a word in a text and its rank (the most frequent word has rank , the 2nd most frequent word has rank ,...) is approximately linear when plotted on a double logarithmic scale. It has been argued that the law is not a relevant or useful property of language because simple random texts - constructed by concatenating random characters including blanks behaving as word delimiters - exhibit a Zipf's law-like word rank distribution.

Methodology/principal findings: In this article, we examine the flaws of such putative good fits of random texts. We demonstrate - by means of three different statistical tests - that ranks derived from random texts and ranks derived from real texts are statistically inconsistent with the parameters employed to argue for such a good fit, even when the parameters are inferred from the target real text. Our findings are valid for both the simplest random texts composed of equally likely characters as well as more elaborate and realistic versions where character probabilities are borrowed from a real text.

Conclusions/significance: The good fit of random texts to real Zipf's law-like rank distributions has not yet been established. Therefore, we suggest that Zipf's law might in fact be a fundamental law in natural languages.

Figure 1. The rank histograms of English texts versus that of random texts ().

A comparison of the real rank histogram (thin black line) and two control curves with the upper and lower bounds of the expected histogram of a random text of the same length in words (dashed lines) involving four English texts. is the frequency of the word of rank . For the random text we use the model with alphabet size . The expected histogram of the random text is estimated averaging over the rank histograms of random texts. For ease of presentation, the expected histogram is cut off at expected frequencies below . AAW: Alice's adventures in wonderland. H: Hamlet. DC: David Crockett. OS: The origin of species.

Figure 2. The rank histograms of English texts versus that of random texts ().

The same as Fig. 1 for the model with alphabet size and probability of blank obtained from the real text.

Figure 3. The rank histograms of English texts versus that of random texts ().

The same as Fig. 1 for the model with alphabet size and character probabilities obtained from the real text.