The deconstruction of a text: the permanence of the generalized Zipf law—the inter-textual relationship between entropy and effort amount

Abstract

Zipf’s law has intrigued people for a long time. This distribution models a certain type of statistical regularity observed in a text. George K. Zipf showed that, if a word is characterised by its frequency, then, rank and frequency are not independent and approximately verify the relationship:

$${\text{Rank }} \times {\text{ frequency}} \approx {\text{constant}}$$

Various explanations have been advanced to explain this law. In this article, we talk about the Mandelbrot process, which includes two very different approaches. In the first approach, Mandelbrot studies language generation as the transmission of a signal and bases it on information theory, using the entropy concept. In the second, geometric approach, he draws a parallel with the fractal theory, where each word of the text is a sequence of characters framed by two separators, meaning a simple geometric pattern. This leads us to hypothesise that, since the statistical regularities observed have several possible explanations, Zipf’s law carries other patterns. To verify this hypothesis, we chose a text, which we modified and degraded in several successive stages. We called T _i the text degraded at step i. We then segmented T _i into words. We found that rank and frequency were not independent and approximately verified the relationship:

$${\text{Rank}}\,\beta_{i} \, \times {\text{ frequency}} \approx {\text{constant}}\quad \beta_{i} \, > 1$$

The coefficient β _i increases with each step i. We call Eq. (1) the generalized Zipf law. We found statistical regularities in the deconstruction of the text. We notably observed a linear relationship between the entropy H _i and the amount of effort E _i of the various degraded texts T _i . To verify our assumptions, we degraded a text of approximately 200 pages. At each step, we calculated various parameters such as entropy, the amount of effort, and the coefficient. We observed an inter-textual relationship between entropy and the amount of effort. This paper therefore provides a proof of this relationship.

Appendices

Annex 1

Theorem 4.1

Let a pair $(S_{i} , M_{i} ) \in \left[ {1, \ldots ,\infty } \right[ \times {\mathbb{R}}^{ + }$, then there exists a unique real $x, x \ge 1$ such that:

$$\int_{1}^{S} {\frac{dr}{{r^{x} }}} = M$$

if and only if: $M \in \left[ {o, \ldots ,{\text{Ln}}\left( S \right)} \right]$.

Proof

Let the function

$$F\left( x \right) = \int_{1}^{x} {\frac{dr}{{r^{x} }}}$$

where x ∊ [1,…,∞[. This function is strictly decreasing and continuous. Then we have:

$$F\left( 1 \right) = {\text{Ln }}s.$$

For x > 1, we have:

$$F\left( x \right) = \frac{1}{x - 1} \cdot \left( {1 - \frac{1}{{S^{x - 1} }}} \right)$$

If x tends towards ∞ then F(x) tends towards 0. The theorem of intermediate values then allows us to say that there is a unique value x such that F(s) = M.

Proposition

We have the following results:

• If S _i a decreasing sequence and M _i  − S _i is a decreasing sequence, then β _i is an increasing sequence.

Proof

We show the identity:

$$Y = \int_{1}^{{S_{i + 1} }} {\left( {\frac{1}{{r^{{\beta_{i + 1} }} }} - \frac{1}{{r^{{\beta_{i} }} }}} \right)dr = M_{i + 1} - M_{i} + \int_{{S_{i + 1} }}^{{S_{i} }} {\frac{dr}{{r^{{\beta_{i} }} }}} }$$

we have also:

$$\int_{{S_{i + 1} }}^{{S_{i} }} {\frac{dr}{{r^{{\beta_{i} }} }} \le } \int_{{S_{i + 1} }}^{{S_{i} }} {dr} = S_{i} - S_{i + 1} < 0$$

S _i is decreasing and also M _i  − S _i .

Hence, we can write:

$$\begin{aligned} Y & \le M_{i + 1} - M_{i} + S_{i} - S_{i + 1} \\ Y & \le \left( {M_{i + 1} - S_{i + 1} } \right) - (M_{i} - S_{i} ) \le 0 \\ \end{aligned}$$

Then we have Y < 0 hence, we can write:

$$Y < 0\;{ \Rightarrow }\;\forall r \ge 1\frac{1}{{r^{{\beta_{i + 1} }} }} - \frac{1}{{r^{{\beta_{i} }} }} \le 0$$

Thus

$$\forall r \ge 1\;r^{{\beta_{i} }} \le r^{{\beta_{i + 1} }} \Leftrightarrow \beta_{i} \le \beta_{i + 1}$$

$$\beta_{i} \le \beta_{i + 1} .$$

Annex 2

Annex 3

Demonstration of the relationship between the amount of effort and entropy (9) in Sect. 4.2.3

If we calculate entropy $H_{i} = - \sum\nolimits_{r = 1}^{{S_{i} }} {\text{Ln}} (p_{i} (r)) \cdot p_{i} \left( r \right)$ assuming that the distributions are generalized Zipf law, using the equation

$$f_{i} \left( r \right) \approx \frac{{k_{i} }}{{r^{{\beta_{i} }} }}\quad k_{i} > 0 \quad \beta_{i} \ge 1\quad r = 1, \ldots ,S_{i}$$

with $h_{i} = \frac{{k_{i} }}{{M_{i} }}$ and $\int_{1}^{{S_{i} }} {\frac{dr}{{r^{{\beta_{i} }} }}} = M_{i} ;M_{i} = \frac{M}{{k_{i} }}$ we have:

$$H_{i} = - {\text{Ln}}\left( {h_{i} } \right) + \beta_{i} \mathop \sum \limits_{r = 1}^{{S_{i} }} \frac{{h_{i} }}{{r^{{\beta_{i} }} }}{\text{Ln}}\left( r \right)\quad \beta_{i} \ge 1, 0 < h_{i}$$

according to $C_{i} \left( r \right) = \frac{{{\text{Ln}}\left( r \right)}}{{{\text{Ln}}\left( {V_{i} } \right)}}$ we have:

$$H_{i} = - {\text{Ln}}\left( {h_{i} } \right) + \beta_{i} \cdot {\text{Ln}}\left( {V_{i} } \right) \cdot \mathop \sum \limits_{r = 1}^{{S_{i} }} \frac{{h_{i} }}{{r^{{\beta^{i} }} }} \cdot C_{i} \left( r \right)$$

according to $E_{i} = - \mathop \sum \limits_{r = 1}^{{S_{i} }} C_{i} \left( r \right) \cdot p_{i} \left( r \right)$ we have:

$$H_{i} = - {\text{Ln}}\left( {h_{i} } \right) + \beta_{i} \cdot {\text{Ln}}\left( {V_{i} } \right) \cdot E_{i}$$

we obtain a relationship between entropy and amount of effort:

$$H_{i} = \gamma_{i} + \delta_{i} \cdot E_{i} \quad \gamma_{i} > 0,\quad \delta_{i} = \beta_{i} \cdot {\text{Ln}}\left( {V_{i} } \right) > 0$$

(9)

Demonstration of the relationship (16) between the amount of effort and entropy in Sect. 6.3

$p_{i} \left( r \right) = B \cdot \exp \left( { - A \cdot F_{i} (r)} \right) \quad A > O \quad i = 1, \ldots ,I$ where F _i is a strictly increasing unbounded effort function we suppose:

$$\mathop \sum \limits_{i = 1}^{I} p_{i} \left( r \right) = 1$$

then $H_{i} = - {\text{Ln}}\left( B \right) + A \cdot E_{i}$.

Proof

$$\begin{aligned} H_{i} & = \mathop \sum \limits_{i = 1}^{I} {\text{Ln}}(p_{i} \left( r \right)) \cdot p_{i} \left( r \right) \\ H_{i} & = - \mathop \sum \limits_{i = 1}^{I} {\text{Ln}}(B \cdot \exp \left( { - A \cdot F_{i} \left( r \right)} \right) \cdot B \cdot \exp \left( { - A \cdot F_{i} \left( r \right)} \right) \\ H_{i} & = - \mathop \sum \limits_{i = 1}^{I} {\text{Ln}}\left( B \right) \cdot B \cdot \exp \left( { - A \cdot F_{i} \left( r \right)} \right) - \mathop \sum \limits_{i = 1}^{I} - A \cdot F_{i} \left( r \right) \cdot B \cdot { \exp }\left( { - A \cdot F_{i} \left( r \right)} \right) \\ H_{i} & = - \mathop \sum \limits_{i = 1}^{I} {\text{Ln}}\left( B \right) \cdot p_{i} \left( r \right) + \mathop \sum \limits_{i = 1}^{I} A \cdot F_{i} \left( r \right) \cdot p_{i} \left( r \right) \\ H_{i} & = - {\text{Ln}}\left( B \right)\mathop \sum \limits_{i = 1}^{I} \cdot p_{i} \left( r \right) + A \cdot \mathop \sum \limits_{i = 1}^{I} F_{i} \left( r \right) \cdot p_{i} \left( r \right) \\ E_{i} & = \mathop \sum \limits_{i = 1}^{I} F_{i} \left( r \right) \cdot p_{i} \left( r \right) \\ H_{i} & = - {\text{Ln}}\left( B \right) + A \cdot E_{i} \\ \end{aligned}$$

If $F_{i} \left( r \right) = \frac{{{\text{Ln}}\left( r \right)}}{{{\text{Ln}}\left( {V_{i} } \right)}}$ is the effort function defined by Mandelbrot:

$$p_{i} \left( r \right) = B \cdot \exp \left( { - A \cdot \frac{{{\text{Ln}}\left( r \right)}}{{V_{i} }}} \right) = B \cdot \frac{1}{{r^{{\beta_{i} }} }}\quad \beta_{i} = \frac{A}{{{\text{Ln}}\left( {V_{i} } \right)}}.$$

Annex 4

The text’s characteristic

• Total number of signs: 344,292

• Percentage of separators: 20.2 %

• Percentage of characters: 79.8 %

• Total number of segmented words: 61,086

• Number of identified sources: 5853

• Number of distinct characters: 75

• Adjustment coefficient β: 1.12

This text is available on line using the URL: http://fr.wikisource. org/wiki/La_Valeur_de_la_Science, website consulted in February 2015.