  A good generalization of the Euclidean dimension to disordered systems and
non crystalline structures is commonly required to be related to large scale
geometry and it is expected to be independent of local geometrical
modifications. The spectral dimension, defined according to the low frequency
density of vibrational states, appears to be the best candidate as far as
dynamical and thermodynamical properties are concerned. In this letter we give
the rigorous analytical proof of its independence of finite scale geometry. We
show that the spectral dimension is invariant under local rescaling of
couplings and under addition of finite range couplings, or infinite range
couplings decaying faster then a characteristic power law. We also prove that
it is left unchanged by coarse graining transformations, which are the
generalization to graphs and networks of the usual decimation on regular
structures. A quite important consequence of all these properties is the
possibility of dealing with simplified geometrical models with
nearest-neighbors interactions to study the critical behavior of systems with
geometrical disorder.
