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In this section we make comparisons of the dynamical SDSS data to simple toy models. We begin by detailing how each toy model is constructed. In the disk and isotropic models, hosts are permitted to have more than two satellites; in these cases, hosts are four times more likely to have n as n+1 satellites.   \begin{enumerate}  \item Isotropic model - In this case, each host may again have 2-5 satellites. Each satellite is independently assigned a random position and velocity, where the magnitude of the drawn from a uniform probability distribution from 0 km/s to 200 km/s. The model is subjected to a random rotation and oberved along the z direction, i.e. the z direction is taken to be the line of sight and the xy plane is taken to be the plane of the sky.  \item  Disk model - In this model, 2-5 satellites are placed randomly between 0 and 200 kpc from the origin on the xy plane and then randomly given a z coordinate randomly between -10 and 10 kpc. All satellites are assigned a 3D velocity of 100 km/s such that each satellite is in circular motion about the host. The model is subjected to a random rotation randomly rotated  and oberved viewed  along the z direction, i.e. the z direction is taken to be the line of sight and the xy plane is taken to be the plane of the sky. axis.  \item Corotating oppositely-aligned pairs (COP) model - Here, each host is restricted to exactly two satellites. Once the first satellite is randomly placed on the xy plane, the placing of the second satellite is restricted such that the angle between the position vectors of the two satellites is greater than $170^{\circ}$. From there, each satellite is assigned a z cooridate between -10 and 10 kpc and the system is subject to random rotation and viewed along the z axis.  \item M31 model - This model is based on the position and velocities of the 13 M31 satellites belonging to the co-rotating plane identified in \citet{Ibata_2013}. The three dimensional positions of the satellites are taken from \citet{McConnachie_2012} and the line-of-sight velocities are compiled from \citet{McConnachie_2012} and \citet{2013ApJ...768..172C}. Note that we only consider the 13 satellites exhibiting coherent rotation; the two satellites aligned with the planar structure but with couter-aligned line-of-sight velocities are considered part of the isotropic background. We assign proper motions to the satellites to place them in circular orbits around the host, and select 2-5 satellites to mock observe (this selection is random, i.e. independent of the luminosities of the true M31 satellites). The system is then randomly rotated and viewed along the z axis.  \item Isotropic model - In this case, each host may again have 2-5 satellites. Each satellite is independently assigned a random position and velocity, where the magnitude of the drawn from a uniform probability distribution from 0 km/s to 200 km/s. The model is randomly rotated and viewed along the z axis as in the previous cases.  \end{enumerate}  In each case, we rotate the hos to a random viewing angle and view the model along the z axis. A statistical sample consists of N $\sim 10^5$ hosts, and most statistical samples will consist of a mixing of the isotropic model with one of the other two models.