Zipf's Law

Understanding neural scaling laws is increasingly important for theory and practice, yet first-principles explanations remain limited even in simple settings. In this work, we study capacity-limited scaling in a bottlenecked bigram model with power-law (Zipf) token statistics, viewed as an analytically tractable toy model of a large language model. Fixed random embeddings serve as a proxy for representational capacity and induce an explicit bottleneck of width, allowing us to derive scaling laws for prediction regret as a function of model width, vocabulary size, and the Zipf exponent. Our results complement recent theoretical accounts based on optimization dynamics by isolating scaling due purely to representational constraints. In the appropriate regime, the theory recovers the functional form of classical neural scaling laws and suggests that heavy-tailed language statistics may underlie both optimization- and capacity-driven scaling. Experiments natural language datasets (WikiText2, OpenWebText) closely match the predicted curves.

Probability estimation over large alphabets under log loss is a well-studied problem, with celebrated methods such as the Good-Turing estimator. We introduce and study a new Bayesian estimator with four notable properties. First, its construction is exceptionally simple: multiply independent uniform draws from the probability simplex coordinate-wise and renormalize. Depth is the only structural parameter, and averaging over depths eliminates the need to tune it. Second, the regret of the resulting mixture, the excess code length it pays relative to a code that knows the source, admits an explicit and efficiently computable expression. Third, despite its simplicity and lack of tuned constants, the estimator is competitive across a diverse set of synthetic and real-text benchmarks with substantially more specialized methods, including Good-Turing. Fourth, the tractability of its regret allows us to identify scaling laws in data, alphabet size, and depth. For Zipf targets with exponent above one, the regret has a simple reading as long as the sample reveals only a small fraction of the alphabet. It closely matches the description length of the set of discovered symbols, at one bit of code per bit of description, plus a further cost per symbol. The data exponent is therefore the rate at which new symbols are discovered.

We present a computational stylometric analysis of the Tipitaka across all three Pitakas in English translation, extending earlier work on the Sutta Pitaka alone. The corpus spans 134,831 segments from Bhikkhu Sujato's Sutta Pitaka (114,591 segments, CC0), Bhikkhu Brahmali's Vinaya Pitaka (7,923 segments, CC0 2026), I.B. Horner's 1938 Vinaya translation (2,826 segments), three English translations of the Abhidhammattha Sangaha compendium (2,077 segments), and cross-tradition Vinaya texts from the Dharmaguptaka and Mulasarvastivada schools. We compute Zipf rank-frequency distributions with OLS-fitted exponents, Moving Average TTR (MATTR-500), numeral-word density, and vocabulary overlap (Jaccard and Szymkiewicz-Simpson coefficients). Main findings: (1) all corpora show Zipf-consistent distributions (R2 > 0.989); the Vinaya is closest to ideal Zipf slope -1 and the Sangaha corpus deviates most, with 'consciousness' displacing grammatical particles at rank 8; (2) MATTR-500 shows the Sutta and Vinaya Theravada are nearly identical in lexical diversity (0.399 and 0.400), while the Sangaha corpus is genuinely more diverse (0.560), confirmed by size-controlled subsampling; (3) the Sangaha corpus has the highest numeral-word density (3.26%), consistent with its systematic enumeration of mental and material categories; (4) the Mulasarvastivada Vinaya shares 20.0% vocabulary (Jaccard) and 49.1% (overlap coefficient) with the Theravada Vinaya, reflecting shared legal heritage across two millennia; (5) two English translations of the same Vinaya source text share only 24.2% of their vocabulary across 88 years, with 'musing' versus 'absorption' for jhana and 'defeat' versus 'expulsion' for parajika as the most diagnostic shifts. All results are point estimates; no significance testing is conducted. Code and data are released as open-source extensions to the Darshana Graph corpus (arXiv:2606.18222).

Unsupervised term discovery involves segmenting unlabelled speech into word- or syllable-like units and clustering these into a lexicon of candidate types. True lexicons follow a Zipfian distribution, yet the dominant centre-based clustering approach -- K-means -- produces a more uniform distribution due to an inductive bias toward spherical clusters. In this paper we revisit graph-based clustering as a bottom-up alternative, where segment embeddings are connected by pairwise similarity and partitioned using the Leiden algorithm. We show that graph clustering substantially outperforms centre-based approaches (K-means, GMM, BIRCH) in both word- and syllable-level lexicon discovery across three languages, producing more Zipf-like distributions. Another bottom-up approach, agglomerative clustering with average linkage, also performs well, although it is computationally less efficient and allows for less control over the resulting distribution. Our work calls into question the dominance of centre-based clustering for term discovery, and promotes graph clustering as an attractive alternative.

Cooking is a cultural expression of human creativity that transcends geography and time through the orchestration of ingredients and techniques, much like languages do through words and syntax. Yet, beneath the apparent diversity of culinary traditions, whether recipes obey statistical laws comparable to those of other symbolic systems remains unknown. Here we analyze a large corpus of traditional recipes spanning global cuisines, annotated using a state-of-the-art named entity recognition algorithm into ingredients, cooking techniques, utensils, and other culinary attributes. We find that ingredient usage exhibits Zipf-like rank-frequency scaling, that culinary diversity grows sublinearly with corpus size in accordance with Heaps' law, and that recipe complexity follows Menzerath-Altmann-type relations between the number and average information of constituent units. Consistent with observations in packaged foods, macronutrient concentrations across recipes also display a log-normal signature. Minimal generative models based on preferential reuse, constrained sampling, and incremental modification recapitulate these regularities, suggesting generic processes that shape recipe architecture across cultures. Together, these findings establish recipes as a compositional symbolic system in which complex structure emerges from simple, constrained generative processes.

Stopwords and function words are relatively less informative for the content of a language and more often play a structural role in a sentence. Stopwords are ubiquitous words and may contain verbs, adjectives and adverbs. On the other hand, function words are strictly prepositions, conjunctions, pronouns, determiners, qualifiers, articles, interrogatives, and a limited number of auxiliary verbs. In contrast to the well known Zipf's law for rank-frequency plot for all words, the rank-frequency plots for stopwords or function words are best fitted by the Beta Rank Function (BRF). On the other hand, the rank-frequency plots of non-stopwords or non-function-words also deviate from the Zipf's law, but are better described by a quadratic function of log-token-count over log-rank than by BRF. Based on the observed rank of stopwords or function words in the full word list, we propose a stopword/function word/subset selection model that the probability for being selected, as a function of the word's rank $r$, is a decreasing Hill's function ($1/(1+(r/r_{mid})^γ)$); whereas the probability for not being selected is the standard Hill's function ($1/(1+(r_{mid}/r)^γ)$). We validate this selection probability model by a direct estimation from an independent collection of texts. We also show analytically that this model leads to a BRF rank-frequency distribution for stopwords or function words when the original full word list follows the Zipf's law, as well as explaining the quadratic fitting function for the non-stopwords or non-function-words. A corollary of these results is that Zipf's law is not expected to be true for telegraphic speech in early childhood language learners or in agrammatism patients.

We inspect the deductive connection between the neural scaling law and Zipf's law -- two statements discussed in machine learning and quantitative linguistics. The neural scaling law describes how the cross entropy rate of a foundation model -- such as a large language model -- changes with respect to the amount of training tokens, parameters, and compute. By contrast, Zipf's law posits that the distribution of tokens exhibits a power law tail. Whereas similar claims have been made in more specific settings, we show that the neural scaling law is a consequence of Zipf's law under certain broad assumptions that we reveal systematically. The derivation steps are as follows: We derive Heaps' law on the vocabulary growth from Zipf's law, Hilberg's hypothesis on the entropy scaling from Heaps' law, and the neural scaling from Hilberg's hypothesis. We illustrate these inference steps by a toy example of the Santa Fe process that satisfies all four statistical laws.

Several explicit stochastic processes are known to satisfy Hilberg's law, a power-law growth of block entropy conjectured for natural language and recently connected to the neural scaling law. Existing examples either possess a positive Shannon entropy rate, are non-ergodic, or require comparatively involved constructions. We introduce multiperiodic processes, a new class of stationary ergodic processes over the natural numbers generated by random shifts of deterministic multiperiodic sequences. Under mild conditions, multiperiodic processes have vanishing Shannon entropy rate and, under a suitable parameterization, they satisfy both Zipf's law for symbol frequencies and Hilberg's law for block entropy. Since multiperiodic processes are not mixing, we identify the open problem of constructing an elementary strongly mixing source with vanishing entropy rate and Hilberg's law.