Spectra of Adjacency and Laplacian Matrices of Inhomogeneous Erdős-Rényi Random Graphs

This paper considers inhomogeneous Erdős–Rényi random graphs [Formula: see text] on [Formula: see text] vertices in the non-sparse non-dense regime. The edge between the pair of vertices [Formula: see text] is retained with probability [Formula: see text], [Formula: see text], independently of other edges, where [Formula: see text] is a continuous function such that [Formula: see text] for all [Formula: see text]. We study the empirical distribution of both the adjacency matrix [Formula: see text] and the Laplacian matrix [Formula: see text] associated with [Formula: see text], in the limit as [Formula: see text] when [Formula: see text] and [Formula: see text]. In particular, we show that the empirical spectral distributions of [Formula: see text] and [Formula: see text], after appropriate scaling and centering, converge to deterministic limits weakly in probability. For the special case where [Formula: see text] with [Formula: see text] a continuous function, we give an explicit characterization of the limiting distributions. Furthermore, we apply our results to constrained random graphs, Chung–Lu random graphs and social networks.