  We devise an analytical method to deal with a class of nonlinear
Schr\"odinger lattices with random potential and subquadratic power
nonlinearity. An iteration algorithm is proposed based on multinomial theorem,
using Diophantine equations and a mapping procedure onto a Cayley graph. Based
on this algorithm, we were able to obtain several hard results pertaining to
asymptotic spreading of the nonlinear field beyond a perturbation theory
approach. In particular, we show that the spreading process is subdiffusive and
has complex microscopic organization involving both long-time trapping
phenomena on finite clusters and long-distance jumps along the lattice
consistent with L\'evy flights. The origin of the flights is associated with
the occurrence of degenerate states in the system; the latter are found to be a
characteristic of the subquadratic model. The limit of quadratic power
nonlinearity is also discussed and shown to result in a delocalization border,
above which the field can spread to long distances on a stochastic process and
below which it is Anderson localized similarly to a linear field.
