Universality in the spectral and eigenfunction properties of random networks

Méndez-Bermúdez, J. A.; Alcazar-Lopez, A.; Martínez‐Mendoza, A. J.; Rodrigues, Francisco A.; Peron, Thomas

doi:10.1103/physreve.91.032122

Cited by 42 publications

(65 citation statements)

References 97 publications

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“…This random network ensemble has already been studied in Refs. [3,36,46] and is constructed as follows. Starting with the standard ER network, we add self-edges and further consider all edges to have random strengths.…”

Section: Resultsmentioning

confidence: 99%

Normal mode analysis of spectra of random networks

Vargas

Fossión

Méndez-Bermúdez

2020

Physica A: Statistical Mechanics and its Applications

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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]

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Section: Resultsmentioning

confidence: 99%

Normal mode analysis of spectra of random networks

Vargas

Fossión

Méndez-Bermúdez

2020

Physica A: Statistical Mechanics and its Applications

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“…Therefore, the latter quantity is a good measure of eigenvector localization/delocalization. In fact, this quantity has already been used to characterize quantitatively the complexity and localization properties of the eigenvectors of the adjacency matrices of several random network models (see examples in [16,17,20,21,22] and references therein). Below we use exact numerical diagonalization to compute the eigenvectors Ψ k and eigenvalues λ k (k = 1 .…”

Section: Eigenvector Properties Scaling and Universalitymentioning

confidence: 99%

“…It is important to stress that the nearest-neighbor energy-level spacing distribution P (s) [23] is already a well accepted quantity to measure the degree of chaos or disorder in complex systems and has been extensively used to characterize spectral properties of complex networks (see examples in [16,21,22] and references therein). However, the use of P (r) is more convenient here since it does not require the process known in RMT as spectral unfolding [23], whose implementation for spectra with kinks as those in Figs.…”

Section: Spectral Propertiesmentioning

confidence: 99%

Spectral and localization properties of random bipartite graphs

Martinez-Martinez

Méndez-Bermúdez

Moreno

et al. 2019

Chaos, Solitons & Fractals: X

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Bipartite graphs are often found to represent the connectivity between the components of many systems such as ecosystems. A bipartite graph is a set of n nodes that is decomposed into two disjoint subsets, having m and n − m vertices each, such that there are no adjacent vertices within the same set. The connectivity between both sets, which is the relevant quantity in terms of connections, can be quantified by a parameter α ∈ [0, 1] that equals the ratio of existent adjacent pairs over the total number of possible adjacent pairs. Here, we study the spectral and localization properties of such random bipartite graphs. Specifically, within a Random Matrix Theory (RMT) approach, we identify a scaling parameter ξ ≡ ξ(n, m, α) that fixes the localization properties of the eigenvectors of the adjacency matrices of random bipartite graphs. We also show that, when ξ < 1/10 (ξ > 10) the eigenvectors are localized (extended), whereas the localization-to-delocalization transition occurs in the interval 1/10 < ξ < 10. Finally, given the potential applications of our findings, we round off the study by demonstrating that for fixed ξ, the spectral properties of our graph model are also universal.

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“…For random regular graphs with uniform edges, in which all eigenvectors are delocalized [27][28][29], we show that σ 2 (λ) = 0 for any λ. On the other hand, for random graph models with localized eigenvectors [22][23][24]30,31], the prefactor σ 2 (λ) exhibits a maximum for a certain λ, while it vanishes for |λ| → 0. These results indicate that the linear scaling of the variance is a consequence of the uncorrelated nature of the eigenvalues in the localized regions of the spectrum.…”

Section: Introductionmentioning

confidence: 99%

“…Although the eigenvalue distribution of random graphs has been computed using different techniques [19], the statistical properties of the index have not been addressed so far. Several random graph models typically contain localized eigenvectors at finite sectors of the spectrum [20][21][22][23], usually corresponding to extreme eigenvalues, where the nearest-level spacing distribution follows a Poisson law [23,24]. In these regions, neighboring eigenvalues are free to be arbitrarily close to each other, which should heavily influence the index fluctuations.…”

Section: Introductionmentioning

confidence: 99%

Index statistical properties of sparse random graphs

Metz

Stariolo

2015

Phys. Rev. E

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Using the replica method, we develop an analytical approach to compute the characteristic function for the probability P N (K,λ) that a large N × N adjacency matrix of sparse random graphs has K eigenvalues below a threshold λ. The method allows to determine, in principle, all moments of P N (K,λ), from which the typical sample-to-sample fluctuations can be fully characterized. For random graph models with localized eigenvectors, we show that the index variance scales linearly with N 1 for |λ| > 0, with a model-dependent prefactor that can be exactly calculated. Explicit results are discussed for Erdös-Rényi and regular random graphs, both exhibiting a prefactor with a nonmonotonic behavior as a function of λ. These results contrast with rotationally invariant random matrices, where the index variance scales only as ln N , with an universal prefactor that is independent of λ. Numerical diagonalization results confirm the exactness of our approach and, in addition, strongly support the Gaussian nature of the index fluctuations.

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Universality in the spectral and eigenfunction properties of random networks

Cited by 42 publications

References 97 publications

Normal mode analysis of spectra of random networks

Normal mode analysis of spectra of random networks

Spectral and localization properties of random bipartite graphs

Index statistical properties of sparse random graphs

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