  We prove that the Aizenman-Contucci relations, well known for fully connected
spin glasses, hold in diluted spin glasses as well. We also prove more general
constraints in the same spirit for multi-overlaps, systematically confirming
and expanding previous results. The strategy we employ makes no use of
self-averaging, and allows us to generate hierarchically all such relations
within the framework of Random Multi-Overlap Structures. The basic idea is to
study, for these structures, the consequences of the closely related concepts
of stochastic stability, quasi-stationarity under random shifts, factorization
of the trial free energy. The very simple technique allows us to prove also the
phase transition for the overlap: it remains strictly positive (in average)
below the critical temperature if a suitable external field is first applied
and then removed in the thermodynamic limit. We also deduce, from a cavity
approach, the general form of the constraints on the distribution of
multi-overlaps found within Quasi-Stationary Random Multi-Overlap Structures.
