luminosity. Purity is defined to be the percentage of systems detected in the mock SDSS catalog, using the C4 algorithm, that match a known dark matter halo in the HV simulation. Clearly, the purity of the C4 catalog remains high over nearly 2 orders of magnitude in luminosity and is a direct result of searching for clustering in a high–dimensional space where projection effects are rare.

In Figure \[completeness\], I also present the completeness of the C4 catalog, as a function of the dark halo mass (${\rm M_{200}}$), and demonstrate that the C4 catalog remains $>90\%$ complete for systems with ${\rm M_{200}} > 10^{14} {\rm M_{\odot}}$. The accuracy of these completeness measurements greatly benefits from the large sample sizes available from the HV simulation. However, for the lower mass systems, there may be some incompletenesses in the original dark matter halo catalog because of the mass resolution of the HV simulations. Furthermore, the purity and completeness of the C4 catalog is only a weak function of the input parameters, [*e.g.*]{}, the FDR threshold, and thus robust against the exact parameter choices. In summary, using these mock SDSS catalogs, we have addressed the first two challenges given in Section \[algorithms\], [*i.e.*]{}, 