In order to simplify the notation we will use the capital $M$ to denote dark matter halo masses and the lower case $m$ to denote galaxy stellar masses.

The simulation and halo catalogs {#s:simulation}
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High-resolution dissipationless N-body simulations have shown that distinct halos[^1] contain subhalos which orbit within the potential of their host halo. These subhalos were distinct halos in the past, and entered the larger halo via merging during the process of hierarchical assembly. We will refer to the galaxy at the center of a distinct halo as a central galaxy, and the galaxies within subhalos as “satellites”, and we will use the term ‘halo’ to refer to the distinct halo for central galaxies and to the subhalo in which the galaxy originally formed for satellite galaxies.

[*A*b initio]{} models of galaxy formation predict that the stellar mass of a galaxy is tightly correlated with the depth of the potential well of the halo in which it formed. For distinct halos, the relevant mass is the virial mass at the time of observation. Subhalos, however, lose mass while orbiting in a larger system as their outer regions are tidally stripped. Stars are centrally concentrated and more tightly bound 