  We study in detail relevant spectral properties of the adjacency matrix of
inhomogeneous amenable networks, and in particular those arising by negligible
additive perturbations of periodic lattices. The obtained results are deeply
connected to the systematic investigation of the Bose--Einstein condensation
for the so called Pure Hopping model describing the thermodynamics of
Bardeen--Cooper pairs of Bosons in arrays of Josephson junctions.
