  The quantum efficiency in the transfer of an initial excitation in disordered
finite networks, modeled by the $k$-body embedded Gaussian ensembles of random
matrices, is studied for bosons and fermions. The influence of the presence or
absence of time-reversal symmetry and centrosymmetry/centrohermiticity are
addressed. For bosons and fermions, the best efficiencies of the realizations
of the ensemble are dramatically enhanced when centrosymmetry
(centrohermiticity) is imposed. For few bosons distributed in two
single-particle levels this permits perfect state transfer for almost all
realizations when one-particle interactions are considered. For fermionic
systems the enhancement is found to be maximal for cases when all but one
single particle levels are occupied.
