Preciado, VM and Rahimian, MA
(2017)
Moment-based spectral analysis of random graphs with given expected degrees.
IEEE Transactions on Network Science and Engineering, 4 (4).
215 - 228.
Abstract
We analyze the eigenvalues of a random graph ensemble, proposed by Chung and Lu, in which a given sequence of expected degrees, denoted by wn = (w(n) 1 ; . . .; 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 An = nrn p a(n) i;j n i;j=1, where rn = 1= Pn i=1 w(n) i and a(n) i;j = 1 if there is an edge between the nodes fi; jg, 0 otherwise. The empirical spectral distribution of An, denoted by Fn, is the empirical measure putting a mass 1=n at each of the n real eigenvalues of the symmetric matrix An. Under some technical conditions on the expected degree sequence, we show that with probability one Fn converges weakly to a deterministic distribution F as n→∞. 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.
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