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Fitness Fluctuations and Correlation Time Scaling in the Barycentric Bak-Sneppen Model

We consider the barycentric version of the Bak-Sneppen model, a one-dimensional self-organized critical model that describes generalized Keynesian beauty contests with a local interaction rule. We numerically investigate the power spectral density of the fitness variable and correlation time. Through data collapse for both variables, we estimate the critical exponents. For global and local fitness variables, the power spectral density exhibits $1/f^α$ with $0< α< 2$, indicative of long-range correlations. We also investigate the cover time, defined as the duration required for the extinction or mutation of species across the entire system in the critical state of the barycentric BS model. Using finite-size scaling and extreme value theory, we analyze the statistical properties of the cover time. Our results show power-law scaling with system size for the mean, variance, mode, and peak probability. Furthermore, the cumulative probability distribution exhibits data collapse, and the associated scaling function is well described by the generalized extreme value density, closely approximating the Gumbel family.