[Paper Review] Turnover Rate of Popularity Charts in Neutral Models

Tim Evans, Andrea Giometto|arXiv (Cornell University)|May 20, 2011

Opinion Dynamics and Social Influence4 citations

TL;DR

This paper investigates the turnover rate of popularity charts in neutral cultural transmission models, comparing the Wright-Fisher and Moran models. It finds that Bentley et al.'s proposed formula $ z = 2\sqrt{\mu}y $ is an inaccurate approximation in the Wright-Fisher model and invalid in the Moran model, with significant dependence on population size $ N $, challenging its use for parameter estimation from real data.

ABSTRACT

It has been shown recently that in many different cultural phenomena the turnover rate on the most popular artefacts in a population exhibit some regularities. A very simple expression for this turnover rate has been proposed by Bentley et al. and its validity in two simple models for copying and innovation is investigated in this paper. It is found that Bentley's formula is an approximation of the real behaviour of the turnover rate in the Wright-Fisher model, while it is not valid in the Moran model.

Motivation & Objective

• To evaluate the validity of Bentley et al.'s formula $ z = 2\sqrt{\mu}y $ for popularity chart turnover in neutral models.
• To compare the turnover rate behavior in the Wright-Fisher and Moran models, which differ in their update mechanisms.
• To determine whether the turnover rate depends on population size $ N $, innovation rate $ \mu $, and chart size $ y $, and to assess the functional form of this dependence.
• To provide a more accurate empirical fit for turnover rate that can be used to infer model parameters from real-world popularity data.

Proposed method

• Simulates the Wright-Fisher and Moran models with $ N $ individuals, where each individual copies an existing artefact with probability $ 1-\mu $ or innovates with probability $ \mu $.
• Computes the turnover rate $ z $ as the sum of artefacts exiting and entering the top $ y $ chart at each time step, using a consistent definition across simulations.
• Performs ensemble averaging over $ E $ simulations with $ \tau $ burn-in steps to reach steady state, followed by $ T $ steps for turnover measurement.
• Uses logarithmic fitting of $ \ln z $ on $ \ln \mu $, $ \ln y $, and $ \ln N $ to estimate power-law exponents and test functional dependence.
• Employs a sequential simulation strategy where each run starts from the final state of the previous one to improve computational efficiency.
• Validates steady-state convergence using the second moment $ \langle k(k-1) \rangle / [N(N-1)] $, confirming equilibrium is reached.

Experimental results

Research questions

• RQ1 Does Bentley et al.'s formula $ z = 2\sqrt{\mu}y $ accurately describe the turnover rate in the Wright-Fisher model?
• RQ2 How does the turnover rate in the Moran model compare to the Wright-Fisher model, and does it follow a simple power law in $ \mu $, $ y $, and $ N $?
• RQ3 Is the turnover rate independent of population size $ N $, as implied by Bentley et al.'s formula, or is there a significant $ N $-dependence?
• RQ4 Can a more accurate functional form for the turnover rate be derived from simulation data that accounts for $ \mu $, $ y $, and $ N $, especially in the $ y \ll N $ regime?
• RQ5 Is the turnover rate in the Moran model better described by a function of the ratio $ y/N $, rather than individual dependencies on $ y $ and $ N $?

Key findings

• The formula $ z = 2\sqrt{\mu}y $ proposed by Bentley et al. is statistically excluded by the authors' simulations in the Wright-Fisher model, with deviations of 10% in the exponents of $ \mu $ and $ y $.
• The turnover rate in the Wright-Fisher model shows significant dependence on population size $ N $, contradicting the assumption of $ N $-independence in Bentley's formula.
• A more accurate fit for the Wright-Fisher model is found in the regime $ y \ll N $, which reproduces simulation results within 6% and is recommended for real data analysis.
• The turnover rate in the Moran model does not follow a simple power law in $ \mu $, $ y $, and $ N $, and instead appears to depend on the ratio $ y/N $, suggesting a different underlying mechanism.
• The two models exhibit fundamentally different behaviors: the Wright-Fisher model allows for a functional fit to $ \mu $, $ y $, and $ N $, while the Moran model does not, indicating a lack of universal scaling.
• The authors identify a conflict between two fits for the Wright-Fisher model: one for $ y \ll N $ and one for larger $ y $, suggesting the $ N $-dependence of $ z $ may not be a simple power law, requiring further study.