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The Ramanujan Journal

, Volume 23, Issue 1–3, pp 371–396 | Cite as

Bentley’s conjecture on popularity toplist turnover under random copying

  • Kimmo Eriksson
  • Fredrik Jansson
  • Jonas Sjöstrand
Article

Abstract

Bentley et al. studied the turnover rate in popularity toplists in a ‘random copying’ model of cultural evolution. Based on simulations of a model with population size N, list length and invention rate μ, they conjectured a remarkably simple formula for the turnover rate: \(\ell \sqrt{\mu}\). Here we study an overlapping generations version of the random copying model, which can be interpreted as a random walk on the integer partitions of the population size. In this model we show that the conjectured formula, after a slight correction, holds asymptotically.

Keywords

Toplists Random walk Integer partitions Moran model Popularity distribution 

Mathematics Subject Classification (2000)

60G50 05A17 

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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Kimmo Eriksson
    • 1
    • 2
  • Fredrik Jansson
    • 1
    • 2
  • Jonas Sjöstrand
    • 1
    • 2
  1. 1.School of Communication, Culture and CommunicationMälardalen UniversityVästeråsSweden
  2. 2.Centre for the Study of Cultural Evolution at Stockholm UniversityStockholmSweden

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