Moment-Based Spectral Analysis of Random Graphs with Given Expected Degrees

We analyze the eigenvalues of a random graph ensemble, proposed by Chung and Lu, in which a given sequence of expected degrees, denoted by (W) over bar (n) = (w(1)((n)), ..., w(n)((n))), is prescribed on the n nodes of a random graph. We focus on the eigenvalues of the normalized (random) adjacency matrix of the graph ensemble, defined as A(n) = root n rho n[a(i,j)((n))](i,j=1)(n), where rho n = 1/Sigma(n)(i=1) w(i)((n)) and a(i,j)((n)) = 1 if there is an edge between the nodes {i, j), 0 otherwise. The empirical spectral distribution of A(n), denoted by F-n(.), is the empirical measure putting a mass 1/n at each of the n real eigenvalues of the symmetric matrix A(n). Under some technical conditions on the expected degree sequence, we show that with probability one F-n(.) converges weakly to a deterministic distribution F(.) as n -> infinity. Furthermore, we fully characterize this deterministic distribution by providing explicit closed-form expressions for the moments of F(.). We illustrate our results with two well-known degree distributions, namely, the power-law and the exponential degree distributions. Based on our results, we provide significant insights about the bulk behavior of the eigenvalue spectrum; in particular, we analyze the quasi-triangular spectral distribution of power-law networks.