Approaches to the classification of complex systems: Words, texts, and more

TL;DR

The chapter surveys physics-inspired methods to classify complex systems, focusing on texts and genomic sequences. It develops rank--frequency models (Zipf, Zipf–Mandelbrot) and introduces a temperature-like parameter with scaling laws to compare samples of varying sizes, extending these ideas to nucleotide strings in mtDNA and viral RNA. It further explores deformed-exponential fits (Tsallis and Kaniadakis) for frequency spectra in genomes and employs entropy alongside a suite of length- and frequency-based indicators to classify texts and viruses, including language evolution, text comprehensibility, and coronavirus genome analyses. The work demonstrates that such parameters can reveal meaningful, though sometimes fuzzy, structure across domains, advocating multi-parameter analyses to better discriminate complex systems.

Abstract

The Chapter starts with introductory information about quantitative linguistics notions, like rank--frequency dependence, Zipf's law, frequency spectra, etc. Similarities in distributions of words in texts with level occupation in quantum ensembles hint at a superficial analogy with statistical physics. This enables one to define various parameters for texts based on this physical analogy, including "temperature", "chemical potential", entropy, and some others. Such parameters provide a set of variables to classify texts serving as an example of complex systems. Moreover, texts are perhaps the easiest complex systems to collect and analyze. Similar approaches can be developed to study, for instance, genomes due to well-known linguistic analogies. We consider a couple of approaches to define nucleotide sequences in mitochondrial DNAs and viral RNAs and demonstrate their possible application as an auxiliary tool for comparative analysis of genomes. Finally, we discuss entropy as one of the parameters, which can be easily computed from rank--frequency dependences. Being a discriminating parameter in some problems of classification of complex systems, entropy can be given a proper interpretation only in a limited class of problems. Its overall role and significance remain an open issue so far.

Figures (15)

• Figure 1: Rank--frequency dependences for Alice's Adventures in Wonderland in English, Hawaian, and Ukrainian (own results). The relevant studies based in particular on this text were reported in Ref. Rovenchak:2015SLT,Rovenchak:2015CSCS
• Figure 2: Rank--frequency dependence for the Ukrainian translation of Alice's Adventures in Wonderland fitted using Zipf's law (\ref{['eq:Zipf']}) and the Zipf--Mandelbrot law (\ref{['eq:ZM']}). The fitting parameters for (\ref{['eq:Zipf']}) are $A = 1786\pm7$, $z = -0.952 \pm 0.001$, the fitting range is $r\ge10$. The fitting parameters for (\ref{['eq:ZM']}) are as follows: $C = 3524 \pm 50$, $M = 4.91 \pm 0.05$, $B = 1.083 \pm 0.004$ (own results)
• Figure 3: Frequency spectrum for the Swahili translation of Alice's Adventures in Wonderland fitted using (\ref{['eq:Nj-exp']}). The upper limit for fitting is $j_{\rm max}=14$. The values of the fitting parameters are as follows: $\alpha = 1.25 \pm 0.06$, $T = 446 \pm 20$ (own results)
• Figure 4: Translations of the Gospel of John in several languages (codes are given according to the ISO 639-3 standardISO:www, except for ISO 639-6 emen for the Early Modern English). Arrows demonstrate the evolution direction. Error bars show uncertainties in the determination of the fitting parameters. This figure contains partial data from previously published studiesRovenchak:2014Rovenchak:2019book
• Figure 5: Fitting results for the frequency spectrum of nucleotide sequences in human mtDNA. Curve 1 corresponds to the ordinary exponential ($T = 88.8 \pm 3.2$), curve 2 corresponds to the kappa-exponential ($T = 88.4 \pm 1.6$, $\varkappa = 4.3 \pm 1.1$), and curve 3 stands for the Tsallis $q$-exponential ($T = 86.4 \pm 1.8$, $q = -1.4 \pm 0.8$); all the data are own results