  We study ensembles of sparse random block matrices generated from the
adjacency matrix of a Erd\"os-Renyi random graph with $N$ vertices of average
degree $Z$, inserting a real symmetric $d \times d$ random block at each
non-vanishing entry.
  We consider some ensembles of random block matrices with rank $r < d$ and
with maximal rank, $r=d$. The spectral moments of the sparse random block
matrix are evaluated for $N \to \infty$, $d$ finite or infinite, and several
probability distributions for the blocks (e.g. fixed trace, bounded trace and
Gaussian).
  Because of the concentration of the probability measure in the $d \to \infty$
limit, the spectral moments are independent of the probability measure of the
blocks (with mild assumptions of isotropy, smoothness and sub-gaussian tails).
  The Effective Medium Approximation is the limiting spectral density of the
sparse random block ensembles with finite rank. Analogous classes of
universality hold for the Laplacian sparse block ensemble. The same limiting
distributions are obtained using random regular graphs instead of Erd\"os-Renyi
graphs.
