Normal mode analysis of spectra of random networks

Abstract

Several spectral fluctuation measures of random matrix theory (RMT) have been applied in the study of spectral properties of networks. However, the calculation of those statistics requires performing an unfolding procedure, which may not be an easy task. In this work, network spectra are interpreted as time series, and we show how their short and long-range correlations can be characterized without implementing any previous unfolding. In particular, we consider three different representations of Erdős-Rényi (ER) random networks: standard ER networks, ER networks with random-weighted self-edges, and fully random-weighted ER networks. In each case, we apply singular value decomposition (SVD) such that the spectra are decomposed in trend and fluctuation normal modes. We obtain that the fluctuation modes exhibit a clear crossover between the Poisson and the Gaussian orthogonal ensemble statistics when increasing the average degree of ER networks. Moreover, by using the trend modes, we perform a data-adaptive unfolding to calculate, for comparison purposes, traditional fluctuation measures such as the nearest neighbor spacing distribution, number variance Σ 2 , as well as ∆3 and δn statistics. The thorough comparison of RMT short and long-range correlation measures make us identify the SVD method as a robust tool for characterizing random network spectra. arXiv:1911.05231v1 [cond-mat.dis-nn]

Computational and analytical studies of the Randić index in Erdös–Rényi models

“…This statement is in accordance with the results reported in [29,30], where the spectral and transport properties of Erdös-Rényi graphs where shown to be universal for the scaling parameter np, see also [31,32,33]. Additionally, from our previous experience, see e.g., [29,30,31,32,33], we expect that other quantities related to R(G) will also be scaled with ξ. Indeed, we validate this conjecture by analyzing the energy E(n, p) of the Erdös-Rényi graphs G(n, p) defined as [34,35]…”

Structural and spectral properties of generative models for synthetic multilayer air transportation networks

“…The probability distribution functions of s and r in the Poisson limit, that we will use below as a reference, are as follows [ 58 ]: respectively. It is important to stress that P ( s ) is already a well accepted quantity to measure the degree of disorder in complex networks, however, the use of P ( r ) is relatively recent; see an example in [ 59 ]. The entropic eigenfunction localization length of the eigenfunction Ψ n is given as [ 53 ]: where S n is the Shannon entropy of Ψ n , defined as .…”

Non-uniform random graphs on the plane: A scaling study

“…We would like to mention that in contrast to the Shannon entropy which is a well accepted quantity to measure the degree of disorder in complex networks, the use of the ratio of consecutive eigenvalue spacings is relative recent in graph studies; see for example [25][26][27][28].…”

“…Here, we will follow a recently introduced approach under which the adjacency matrices of random graphs are represented by RMT ensembles; see the application of this approach on Erdös-Rényi graphs [26,29], RGGs and random rectangular graphs [30], β-skeleton graphs [31], multiplex and multilayer networks [32], and bipartite graphs [27]. Consequently, we define the elements of the adjacency matrix A of our random graph model as…”