Testing Z ipf’s meaning-frequency law with wordnets as sense inventories (2019)

Abstract: According to George K. Zipf, more frequent words have more senses. We have tested this law using corpora and wordnets of English, Spanish, Portuguese, French, Polish, Japanese, Indonesian and Chinese. We have proved that the law works pretty well for all of these languages if we take - as Zipf did - mean values of meaning count and averaged ranks. On the other hand, the law disastrously fails in predicting the number of senses for a single lemma. We have also provided the evidence that slope coefficients of Zipfian log-log linear model may vary from language to language.

## introduction the dependency between meaning and frequency is undisputable. since zipf's discovery of the high correlation between mean sense count and mean rank @xcite , the law was confirmed by several research teams. among many zipfian laws, the modelling of the law of meaning-frequency dependency is probably the most fascinating one, because it directly concerns semantics. meaning strongly influences word frequency @xcite and it is clear that semantics precedes language form in text generation @xcite . 1 originally, zipf tested the law on thorndike's list of 20k most frequent words of standard english 2 and meanings taken from the thorndike-1 consider, for instance, a simple model of a random walk on an undirected graph of lexico-semantic relations. the stationary probabilities of lending in each vertex are proportional to the degree of the vertices (cf. @xcite , @xcite , that is to the number of sense relations, including the number of interconnected polysemous senses of the same lemma. as a result, one gains more polysemous words being chosen more frequently. ## related work despite the fact that most of zipf's laws, like the law of frequency-rank distribution or the law of abbreviation, were studied thoroughly, the meaningfrequency law itself gained relatively less attention @xcite . still, some attempts were made by several research teams. edmonds (2004) repeated zipf's experiment on the british national corpus with the use of princeton wordnet 2.0. he gathered lemmas in bins of 100 words each and estimated the γ coefficient at 0.40 (cf. ilgen and karaoglan (2007) tested the law on two turkish corpora (newspapers, novels, shortstories), one of which was manually tagged with word senses, while the second one was compared to an electronic turkish dictionary. the authors tested different frequency bin sizes ranging from 50 words up to 1000 words, showing gradual predictive power loss while moving from larger to smaller bins. they obtained the slope coefficient slightly lower than that of zipf's (0.42 and 0.39; in @xcite the above non-parametric approach was expanded to two other european languages: dutch and spanish; english is analysed again. the sources of frequency were the childes corpus and wikipidia, while the sources of sense inventories were wordnets: word-net, open dutch wordnet and the multilingual central repository for spanish. each wordnet is "a proxy for the number of meanings of a word" (ibidem). casas and colleagues @xcite also did some experiments with parametric modelling with wordnets as sense inventories and childes corpus as a source of frequencies. they calculated slope coefficients and r-squared values for bins of 100 and 500 words (cf. tab. 1). they also observed that smaller bins gave worse r-squared statistics. the criticism of the rank-frequency models was addressed by @xcite . piantadosi raised an important question of the explanatory validity of zipf's law derivation in various theoretical models. since there are dozens of different ways of deriving the zipf's equations (such as random-typing, stochastic models, semantic accounts, communicative accounts etc.), the derivation lacks its explanatory power and "[t]he key will be (...) to generate novel predictions and to test their underlying assumptions with more data than the law itself" (ibidem). @xcite argue that linguistic statistical models should be validated not only by measures of fit like r-squared determination coefficient, but also with additional measures of randomness of model residuals (they propose significance level set at 1%): "a low p-value is a strong indication that the null model is violated and may be used to refute the law (e.g., if p-@xmath0.01)." according to them, ordinary zipfian rank-frequency linear models unfortunately lack this randomness property (p-values ≪ 0.01). @xcite similarly points that rank-frequency models based on corpus data when analysed in a standard way (i.e., on the same sample), suffer from correlated errors, since the ranking is constructed out of the very same frequency distribution as frequency estimation itself. luckily, this argument cannot be applied to the same extent to meaning-frequency and meaning-rank distributions, since they are prepared with either frequency, or rank at once. especially, if the zipf's meaning-frequency law (or meaning-rank law) is modelled on the basis of different language resources (like a wordnet and a corpus) the problem vanishes. ## method we checked the validity of zipf's meaning-rank law by collating frequency counts and corresponding meaning counts. we did this by comparing general corpora, representing language in usage, and sense numbers taken from wordnets, which represent each language lexical system, cf. fellbaum (1998). another way to see these two sides of language reality is to compare frequencies and polysemy count in the very same text (in a widely sense tagged corpus). we did this on the richly annotated sherlock holmes subcorpus of nanyang technological university multilingual corpus, ntu-mc @xcite . ## conclusions we have presented novel, statistically valid, empirical evidence for the weak version of zipf's law of meaning distribution on eight languages from four distinct language families (indo-european, japonic, sino-tibetan and austronesian). zipf's law functions pretty well for mean values in terms of high determination coefficient rsquared, and non-zero slope coefficient γ (stable over the vast range of λ values, but changing while altering frequency lists). the law is, however, inefficient for individual lemmas, because of the lack of model residual normality, despite non-zero correlation coefficient r values. in the case of sherlock holmes stories, this zipfian catastrophe does not manifest itself only while shifting from bins to individual lemmas, but -surprisingly -also within each whole unabridged corpus containing hapax legomena, both for smaller and larger bins. slope coefficients that zipf tended to treat being close to -0.5, in fact, vary largely from language to language, and corpus to corpus, ranging from -0.5 to -0.1.