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Summary
This summary is machine-generated.

The Yule-Simon process, modeling word growth, shows fluctuations around average behavior. This study analytically derives and numerically confirms the probability distributions of these word occurrence fluctuations.

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Area of Science:

  • Computational Linguistics
  • Statistical Physics
  • Network Science

Background:

  • The Yule-Simon process models vocabulary growth using preferential attachment.
  • Previous models assumed continuous time and word counts, using mean-field approximation.
  • This overlooks inherent discreteness in time and word occurrences.

Purpose of the Study:

  • To analytically derive the probability distribution of fluctuations in word occurrences within the Yule-Simon process.
  • To investigate deviations from the mean-field approximation due to discrete variables.
  • To validate analytical findings with numerical simulations.

Main Methods:

  • Analytical derivation of exact and approximate probability distributions for fluctuations.
  • Utilizing mathematical methods to account for discrete time and word counts.
  • Performing numerical experiments to verify theoretical predictions.

Main Results:

  • Exact and approximate analytical forms for the probability distribution of fluctuations were derived.
  • The derived distributions accurately describe the deviations from mean-field predictions.
  • Numerical experiments strongly supported the analytical results.

Conclusions:

  • The Yule-Simon process exhibits inherent fluctuations in word occurrences not captured by continuous approximations.
  • Analytical methods provide accurate descriptions of these discrete fluctuations.
  • This work refines our understanding of vocabulary growth models and their statistical properties.