  Random graph models are used to describe the complex structure of real-world
networks in diverse fields of knowledge. Studying their behavior and fitting
properties are still critical challenges, that in general, require model
specific techniques. An important line of research is to develop generic
methods able to fit and select the best model among a collection. Approaches
based on spectral density (i.e., distribution of the graph adjacency matrix
eigenvalues) are appealing for that purpose: they apply to different random
graph models. Also, they can benefit from the theoretical background of random
matrix theory. This work investigates the convergence properties of model
fitting procedures based on the graph spectral density and the corresponding
cumulative distribution function. We also review results on the convergence of
the spectral density for the most widely used random graph models. Moreover, we
explore through simulations the limits of these graph spectral density
convergence results, particularly in the case of the block model, where only
partial results have been established.
