Spectral properties of random matrices for stochastic block model

We consider an extension of Erdös-Rényi graph known in literature as Stochastic Block Model (SBM). We analyze the limiting empirical distribution of the eigenvalues of the adjacency matrix of SBM. We derive a fixed point equation for the Stieltjes transform of the limiting eigenvalue empirical distribution function (e.d.f.), concentration results on both the support of the limiting e.s.f. and the extremal eigenvalues outside the support of the limiting e.d.f. Additionally, we derive analogous results for the normalized Laplacian matrix and discuss potential applications of the general results in epidemics and random walks.