The article extends the theoretical and applicative analysis of Zipf’s law. We are concerned with a set of properties of Zipf’s law that derive directly from the power law expression and from the discrete nature of the objects to which the law is applied, when the objects are words, lemmas, and the like. We also search for variations of Zipf’s law that can help explain the noisy results empirically reported in the literature and the departures of the empirically obtained nonlinear graph from the theoretical linear one, with the variants analyzed differing from Mandelbrot and lognormal distributions. A problem of interest that we deal with is that of mixtures of populations obeying Zipf’s law. The last problem has relevance in the analysis of texts with words with various etymologies. Computational aspects are also addressed.

This paper provides a detailed and long-period statistics of the use of synonyms in posts related to specific events on social networks (SNs), an extended analysis of the correlations of the flows of the synonyms in such posts, a study of the applicability of Zipf’s law to posts related to specific events on SNs, and an analysis of the dynamics of the fluxes of synonyms in the posts. The paper also introduces the study of the distances in the phase space for the characterization of the dynamics of the word fluxes on social networks. This article is a partial report on recent research performed for a deeper analysis of social networks and of processes developing on social networks, including used lexicon, dynamics of messages related to a specific type of topic, and relationships of the processes on SNs with external events.

There is an intimate theoretical relationship between Zipf’s law and the expected number of hapax legomena, dis-legomena, and n-legomena in general, as established in recent papers. The relationship was confirmed by the empirical analysis for very large texts. However, the known theoretical relationship was established under the assumption of very large vocabularies and very large texts, where the dimension of the vocabularies used does not limit the maximal rank value and thus the hapax legomena, or the dis-legomena etc. (n-legomena for small n values). In addition, the theoretical results were established under the hypothesis that the probabilities of the words are context-independent, which is not satisfied for small vocabularies and small texts, as typically used in fake news and in other classes of small texts over the Internet. We provide a theoretical analysis under rectified hypotheses, prove several results under relaxed hypotheses, and show examples of practical results.