Best-fitting BGS HOD parameters and satellite fractions (points), as a function of magnitude threshold. The top panel shows the σ_{log M} (red) and α (purple) HOD parameters, plus the satellite fractions from our HODs (light grey) and Uchuu (dark grey). The mass parameters M_min (blue), $M_1^{\prime }$ (orange) and M₀ (green) are shown in the lower panel. The dashed lines indicate the SDSS results obtained from Dong-Páez et al. (2024), while the solid lines correspond to our approximations given by the equations included in the text. For the M_r<−22 sample, the M₀ parameter is completely unconstrained.