Fuzzy Erdos-Renyi random graphs: Spectral statistics

Within a random-matrix-theory (RMT) approach, we perform a detailed numerical study of the spectral statistics of fuzzy Erdos-Renyi (ER) random graphs. Specifically, we consider the membership values of the vertices and edges of a fuzzy ER graph model as random variables. In this way, the corresponding adjacency matrices become sparse matrices with random entries. Then, we apply standard RMT measures to characterize the eigenvalues and eigenvectors of ensembles of Hermitian and non-Hermitian adjacency matrices of both undirected and directed fuzzy graphs. We demonstrate that the average degree of the ER model serves as the scaling parameter of the spectral properties of the corresponding fuzzy setups.