Universal properties of spectral dimension

The infrared singularities of a Gaussian model on a general network are invariant under a local rescaling of the masses. This exact result leads to some interesting rigorous relations concerning diffusion and harmonic oscillations on fractals and inhomogeneous structures. We show that a generic distribution of waiting probabilities does not affect the spectral dimension in diffusive problems, neither does a change of masses in an oscillating network. In particular, we prove an exact relation between random walks and vibrational spectrum showing the possibility of noncoincidence of vibrational and usual diffusive spectral dimensions.