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In the Yule-Simon process, selection of words follows the preferentialattachment mechanism, resulting in the power-law growth in the cumulativenumber of individual word occurrences. This is derived using mean-fieldapproximation, assuming a continuum limit of both the time and number of wordoccurrences. However, time and word occurrences are inherently discrete in theprocess, and it is natural to assume that the cumulative number of wordoccurrences has a certain fluctuation around the average behavior predicted bythe mean-field approximation. We derive the exact and approximate forms of theprobability distribution of such fluctuation analytically and confirm thatthose probability distributions are well supported by the numericalexperiments.

Growth fluctuation in preferential attachment dynamics