  In this work we compare two different random dilution of a mean field
ferromagnet: the first model is built on a Bernoulli-diluted network while the
second lives on a Poisson-diluted network. While it is known that the two
models have in the thermodynamic limit the same free energy we investigate on
the structural constraints that the two models must fulfill. We rigorously
derive for each model the set of identities for the multi-overlaps distribution
using different methods for the two dilutions: constraints in the former model
are obtained by studying the consequences of the self-averaging of the internal
energy density, while in the latter are obtained by a stochastic-stability
technique. Finally we prove that the identities emerging in the two models are
the same, showing "robustness" of the ferromagnetic properties of diluted
networks with respect to the details of dilution.
