Spectral density of random graphs: convergence properties and application in model fitting

Abstract

Abstract Random graph models are used to describe the complex structure of real-world networks in diverse fields of knowledge. Studying their behaviour 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) appeal to 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 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. random graphs, spectral density, model fitting, model selection, convergence.