Moment-Based Spectral Analysis of Large-Scale Generalized Random Graphs

This paper investigates the spectra of the adjacency matrix and Laplacian matrix for an artificial complex network model—the generalized random graph. We deduce explicit expressions for the first four asymptotic spectral moments of the adjacency matrix and the Laplacian matrix associated with a generalized random graph with independent and identically distributed vertex weights. An estimate for the upper bound of the spectral radius is obtained on the basis of the fourth asymptotic spectral moment of the adjacency matrix, as well as the Laplacian matrix. These expressions are applied to study the behavior of a viral infection in a generalized random graph. On the basis of our results, we can design generalized random graphs with good antivirus ability when facing an initial virus infection. Numerical simulations agree with our analytical predictions.