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In this paper, we analyze the limiting spectral distribution of the adjacencymatrix of a random graph ensemble, proposed by Chung and Lu, in which a givenexpected degree sequence $\overline{w}_n^{^{T}} = (w^{(n)}_1,\ldots,w^{(n)}_n)$is prescribed on the ensemble. Let $\mathbf{a}_{i,j} =1$ if there is an edgebetween the nodes $\{i,j\}$ and zero otherwise, and consider the normalizedrandom adjacency matrix of the graph ensemble: $\mathbf{A}_n$ $=$ $[\mathbf{a}_{i,j}/\sqrt{n}]_{i,j=1}^{n}$. The empirical spectral distributionof $\mathbf{A}_n$ denoted by $\mathbf{F}_n(\mathord{\cdot})$ is the empiricalmeasure putting a mass $1/n$ at each of the $n$ real eigenvalues of thesymmetric matrix $\mathbf{A}_n$. Under some technical conditions on theexpected degree sequence, we show that with probability one,$\mathbf{F}_n(\mathord{\cdot})$ converges weakly to a deterministicdistribution $F(\mathord{\cdot})$. Furthermore, we fully characterize thisdistribution by providing explicit expressions for the moments of$F(\mathord{\cdot})$. We apply our results to well-known degree distributions,such as power-law and exponential. The asymptotic expressions of the spectralmoments in each case provide significant insights about the bulk behavior ofthe eigenvalue spectrum.

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