\label{sec:preprocessing_v_distance}
We next explore how the above group-preprocessed fractions vary with the current distance of the satellites from their host. Figure \ref{fig:infall.fraction_v_distance} (top and bottom) show the same preprocessed fractions as in Figure \ref{fig:infall.fraction_v_mass} (top), but as a function of \(d / {R_{\rm vir}}\) at \(z = 0\), similar to Figure \ref{fig:infall.time_v_distance}. Again, we compute these quanities in bins of \(d / {R_{\rm vir}}\) for each MW/M31 halo, though we also show the dependence on \(d\) (using the median \({R_{\rm vir}}\) across the MW/M31 halos) along the top axis. At any distance, lower-mass satellites are more likely to have been preprocessed. Moreover, at nearly all satellite masses, those closer to the host center are more likely to have been preprocessed, with a nearly \(2 \times\) increase from \(d / {R_{\rm vir}}= 1\) to 0.1 for low-mass satellites. Most likely, this gradient arises because satellites that fell in as part of a group remained bound to that (more massive) group for some time after infall, so they experienced more efficient dynamical friction that dragged them to the center of the MW/M31 halo more rapidly.
Thus, given that faint and ultra-faint satellite galaxies are observable only at small distances within the MW halo, most likely about half of them were satellites in another group before / during falling into the MW halo.