  We show that the quantifier elimination result for the Shelah-Spencer almost
sure theories of sparse random graphs $G(n,n^{-\alpha})$ given by Laskowski in
$[7]$ extends to their various analogues. The analogues will be obtained as
theories of generic structures of certain classes of finite structures with a
notion of strong substructure induced by rank functions and we will call the
generics Baldwin-Shi hypergraphs. In the process we give a method of
constructing extensions whose `relative rank' is negative but arbitrarily small
in context. We give a necessary and sufficient condition for the theory of a
Baldwin-Shi hypergraph to have atomic models. We further show that for certain
well behaved classes of theories of Baldwin-Shi hypergraphs, the existentially
closed models and the atomic models correspond.
