Simple Generative Mechanisms for Modeling Exponential, q-Exponential, and Power-Law Inter-Event Times of Words

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Abstract
The inter-event times of frequent words in real written texts have been shown to fall into three distributional classes: exponential, q-exponential, and power-law. Here, we define the inter-event times as the number of sentences between successive appearances of a given word. In a previous study, we proposed a pseudo-text generation algorithm in which each sentence is composed by one of two word-selection mechanisms, i.e., randomized selection and priority-based selection, and we further showed that the algorithm can reproduce synthetic texts in which words exhibiting the three distributional classes coexist as in real written texts. The present paper develops a revised and mathematically clearer formulation of that algorithm and introduces approximate stochastic models corresponding the algorithm to explain why the three classes emerge. For the randomized selection mechanism involved in the algorithm, we propose a corresponding stochastic model that approximately brings geometric inter-event times which can be regarded as a discrete exponential law. In the model, the exponential law can be derived when the local occurrence probability of a given word is supposed to vary slowly. For the priority-based selection mechanism, we introduce a coarse-grained local occurrence probability in a corresponding stochastic model and show that a mixture of geometric distributions proposed by the model naturally yields heavy-tailed behavior, including power-law asymptotics. The q-exponential regime is then interpreted as an intermediate regime produced by temporal variation in local occurrence probabilities and by the coexistence of the two selection mechanisms. These findings provide a logically smoother connection between the previous empirical study and the present theoretical development of the text generation algorithms and the corresponding stochastic models.

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