2015/12/11 by Preciado, Victor M., Rahimian, M. Amin
#05C80 #60B20 #Applications (stat.AP) #FOS: Computer and information sciences #FOS: Mathematics #FOS: Physical sciences #Physics and Society (physics.soc-ph) #Probability (math.PR) #Social and Information Networks (cs.SI) #Statistics Theory (math.ST)
paper · doi:10.48550/arxiv.1512.03489
In this paper, we analyze the limiting spectral distribution of the adjacency matrix of a random graph ensemble, proposed by Chung and Lu, in which a given expected degree sequence wn^T = (w(n)1,…,w(n)n) is prescribed on the ensemble. Let ai,j =1 if there is an edge between the nodes \i,j\ and zero otherwise, and consider the normalized random adjacency matrix of the graph ensemble: An = [ai,j/√(n)]i,j=1n. The empirical spectral distribution of An denoted by Fn(\mathord⋅) 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(\mathord⋅) converges weakly to a deterministic distribution F(\mathord⋅). Furthermore, we fully characterize this distribution by providing explicit expressions for the moments of F(\mathord⋅). We apply our results to well-known degree distributions, such as power-law and exponential. The asymptotic expressions of the spectral moments in each case provide significant insights about the bulk behavior of the eigenvalue spectrum.