explore other SAMs in future work, but do not expect our conclusions to change.

N-body Simulation {#Sec:Mill}
-----------------

The G13 model is implemented in the Millennium-WMAP7 N-body simulation [@Guo:2013a; @Gonzalez:2014; @Lacey:2016]. This simulation has similar specifications to the original Millennium simulation of [@Springel:2005] but uses a WMAP7 cosmology (ie. $\Omega_{\rm \Lambda0}$ = 0.728, $\Omega_{\rm m0}$= $ \Omega_{\rm dm0}$+$\Omega_{\rm b0}$ = 0.272, $\Omega_{\rm b0}$ = 0.0455, $\sigma_8$ = 0.81, $n_{\rm s}$ = 0.967, $h$ = 0.704.). The simulation uses $2160^3$ particles in a periodic box of comoving volume $(\,500\, h^{-1}\, {\rm~Mpc})^3$ corresponding to a particle mass of $ 9.31\times10^8 h^{-1} \, {\rm M_{ \odot}}$ and a softening value of $5\, h^{-1}\, {\rm~kpc}$. There are 61 simulation snapshots output between $z=50$ and $z=0$.

Halo merger trees are constructed from the simulation outputs. Haloes are identified using the friends-of-friends ([FoF]{}) group finding algorithm [@Davis:1985] at each snapshot of the simulation, using a minimum of 20 particles per halo (equivalent to a mass of $ 1.86\times10^{10} h^{-1} \, {\rm M_{ \odot}}$). [SUBFIND]{} is then run on these groups to identify subhaloes [@Springel:2001]. Merger trees are constructed by linking a subhalo in one snapshot to a single descendant subhalo in the subsequent output, i.e., a 