  We propose a class of mean-field models for the isostatic transition of
systems of soft spheres, in which the contact network is modeled as a random
graph and each contact is associated to $d$ degrees of freedom. We study such
models in the hypostatic, isostatic, and hyperstatic regimes. The density of
states is evaluated by both the cavity method and exact diagonalization of the
dynamical matrix. We show that the model correctly reproduces the main features
of the density of states of real packings and, moreover, it predicts the
presence of localized modes near the lower band edge. Finally, the behavior of
the density of states $D(\omega)\sim\omega^\alpha$ for $\omega\to 0$ in the
hyperstatic regime is studied. We find that the model predicts a nontrivial
dependence of $\alpha$ on the details of the coordination distribution.
