Multifractal hopscotch in "Hopscotch" by Julio Cortazar

TL;DR

This study investigates the sentence-length variability (SLV) in Julio Cortázar's Hopscotch across Spanish, English, and Polish, analyzing multiple chapter-orderings to uncover long-range correlations and multifractal structure. Using SLV time series, the authors compare histograms to a discrete Weibull model and apply multifractal detrended fluctuation analysis (MFDFA) to extract generalized Hurst exponents $h(q)$ and the singularity spectrum $f(\alpha)$, supported by surrogate tests to validate nonlinear memory. They find persistent memory in SLV, exponential-like tails in SLV distributions (with PMDV better fit by $\beta>1$), and robust multifractality with a left-skewed spectrum $f(\alpha)$ across all languages and orders; non-printed and random chapter orders generally broaden the spectrum and enhance asymmetry, while translations largely preserve these properties. The work highlights the potential for applying multifractal analysis to literary texts and suggests implications for linguistic theory and the design of large language models, where exploiting multiscale structure could improve efficiency and reduce parameter requirements.

Abstract

Punctuation is the main factor introducing correlations in natural language written texts and it crucially impacts their overall effectiveness, expressiveness, and readability. Punctuation marks at the end of sentences are of particular importance as their distribution can determine various complexity features of written natural language. Here, the sentence length variability (SLV) time series representing "Hopscotch" by Julio Cortazar are subjected to quantitative analysis with an attempt to identify their distribution type, long-memory effects, and potential multiscale patterns. The analyzed novel is an important and innovative piece of literature whose essential property is freedom of movement between its building blocks given to a reader by the author. The statistical consequences of this freedom are closely investigated in both the original, Spanish version of the novel, and its translations into English and Polish. Clear evidence of rich multifractality in the SLV dynamics, with a left-sided asymmetry, however, is observed in all three language versions as well as in the versions with differently ordered chapters.

Figures (7)

• Figure S1: Time series of sentence length variability (SLV) measured in words for the original Spanish text of Hopscotch (top) and its translations into English (middle) and Polish (bottom). In order to ensure sufficient readability and preserve the vertical scale in all panels, the bars representing a few sentences with excessive length in each text have been rescaled by a factor of 1/3 (black arrows). Time series of SLV are presented in alternative orders of chapters: the printed order (left) and the order recommended by the author (right).
• Figure S2: Autocorrelation function for SLV time series for two chapter orders: the printed order ("-p", left) and the recommended order ("-r", right), and for three languages: Spanish (ES, top), English (EN, middle), and Polish (PL, bottom). Noise level is indicated by a red dashed line in each panel. Note the double logarithmic scale of the plots.
• Figure S3: Histograms of the SLV time series (dark colour) for three different language versions of Hopscotch: Spanish (top left), English (top right), and Polish (bottom) together with the histograms of the distances (in words) between any two consecutive punctuation marks (both the sentence-ending ones and the intra-sentence ones, light colour). Discrete Weibull distributions that have been least-square fitted to the histograms of both types are also shown in each panel (dashed and solid lines, respectively). In each case, values of the $p$ and $\beta$ parameters of the fits are explicitly given in legend boxes. (Insets) Histogram of the SLV time series with the fitted discrete Weibull distribution presented on the half-logarithmic scale.
• Figure S4: The generalised Hurst exponent $h(q)$ for the SLV time series representing Hopscotch in different languages: Spanish (top), English (middle), and Polish (bottom). The printed order (left) and the recommended order (right) of chapters are shown separately. The difference between extreme values of $h(q)$ for $-7 \le q \le 7$ is given by $\Delta h$ in each case.
• Figure S5: (Main) $q$-dependent fluctuation functions $F_q(s)$ calculated for time series of SLV in the Spanish original text of Hopscotch (top) and its two translations into English (middle) and Polish (bottom). The printed order of chapters is considered. The plots of $F_q(s)$ for particular values of $q$ are indicated by arrows. (Side) Singularity spectra $f(\alpha)$ associated with the exponents $h(q)$ calculated from the scaling regions of the respective functions $F_q(s)$ for three intervals: $q \in \mathopen{}\mathclose{\left[ -2, 2\right]$ (top right), $q \in \mathopen{}\mathclose{\left[ -4, 4\right]$ (middle right), and $q \in \mathopen{}\mathclose{\left[ -7, 7\right]$ (bottom right). In each case the asymmetry coefficient $A_{\alpha}$ and the singularity spectrum width $\Delta \alpha$ are also given.