Growth fluctuation in preferential attachment dynamics

TL;DR

This paper analyzes fluctuations in the Yule-Simon process, deriving the probability distribution of deviations from the mean-field growth, and confirms these with numerical experiments, enhancing understanding of discrete dynamics in preferential attachment models.

Contribution

It provides the first analytical derivation of the fluctuation distribution in the Yule-Simon process, accounting for discrete effects neglected in mean-field approximations.

Findings

Derived exact and approximate fluctuation distributions.

Confirmed theoretical results with numerical experiments.

Enhanced understanding of discrete dynamics in preferential attachment.

Abstract

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