  The aim of the paper is to address the behavior in large population of
diffusions interacting on a random, possibly diluted and inhomogeneous graph.
This is the natural continuation of a previous work, where the homogeneous
Erd\H os-R\'enyi case was considered. The class of graphs we consider includes
disordered $W$-random graphs, with possibly unbounded graphons. The main result
concerns a quenched convergence (that is true for almost every realization of
the random graph) of the empirical measure of the system towards the solution
of a nonlinear Fokker-Planck PDE with spatial extension, also appearing in
different contexts, especially in neuroscience. The convergence of the spatial
profile associated to the diffusions is also considered, and one proves that
the limit is described in terms of a nonlinear integro-differential equation
which matches the neural field equation in certain particular cases.
