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Aparicio L, Bordyuh M, Blumberg AJ, Rabadan R. A Random Matrix Theory Approach to Denoise Single-Cell Data. PATTERNS 2020; 1:100035. [PMID: 33205104 PMCID: PMC7660363 DOI: 10.1016/j.patter.2020.100035] [Citation(s) in RCA: 22] [Impact Index Per Article: 5.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 01/15/2020] [Revised: 03/05/2020] [Accepted: 04/02/2020] [Indexed: 02/06/2023]
Abstract
Single-cell technologies provide the opportunity to identify new cellular states. However, a major obstacle to the identification of biological signals is noise in single-cell data. In addition, single-cell data are very sparse. We propose a new method based on random matrix theory to analyze and denoise single-cell sequencing data. The method uses the universal distributions predicted by random matrix theory for the eigenvalues and eigenvectors of random covariance/Wishart matrices to distinguish noise from signal. In addition, we explain how sparsity can cause spurious eigenvector localization, falsely identifying meaningful directions in the data. We show that roughly 95% of the information in single-cell data is compatible with the predictions of random matrix theory, about 3% is spurious signal induced by sparsity, and only the last 2% reflects true biological signal. We demonstrate the effectiveness of our approach by comparing with alternative techniques in a variety of examples with marked cell populations. Sparse random matrix theory provides a suitable framework to study single-cell biology Eigenvector localization disentangles sparsity-induced signals from biological signals 95% of the information is a random matrix, 3% sparsity-induced signal, and 2% true signal The method improves clustering and identification of cell populations
Single-cell technologies are able to capture information of a biological system cell by cell. Such a level of precision is changing the way we understand complex systems such as cancer or the immune system. However, a major challenge in studying single-cell systems and their underlying biological phenomena is their inherently noisy nature due to their complexity. Random matrix theory is a field with many applications in different branches of mathematics and physics. In the words of one of its developers, the theoretical physicist Freeman Dyson, it describes a “black box in which a large number of particles are interacting according to unknown laws.” A complex system with a large number of components (such as genes, biomolecules, or cells) interacting according to unknown laws is the epitome of systems biology. Therefore, random matrix theory looks like a suitable framework to mathematically describe the noise and complexity of gene-cell expression data coming from single-cell biology.
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Affiliation(s)
- Luis Aparicio
- Department of Systems Biology, Columbia University, New York NY 10032, USA.,Department of Biomedical Informatics, Columbia University, New York NY 10032, USA
| | - Mykola Bordyuh
- Department of Systems Biology, Columbia University, New York NY 10032, USA.,Department of Biomedical Informatics, Columbia University, New York NY 10032, USA
| | - Andrew J Blumberg
- Department of Mathematics, University of Texas, Austin, TX 78705, USA
| | - Raul Rabadan
- Department of Systems Biology, Columbia University, New York NY 10032, USA.,Department of Biomedical Informatics, Columbia University, New York NY 10032, USA
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Huang J, Landon B, Yau HT. Transition from Tracy–Widom to Gaussian fluctuations of extremal eigenvalues of sparse Erdős–Rényi graphs. ANN PROBAB 2020. [DOI: 10.1214/19-aop1378] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Martínez-Martínez CT, Méndez-Bermúdez JA. Information Entropy of Tight-Binding Random Networks with Losses and Gain: Scaling and Universality. ENTROPY 2019; 21:e21010086. [PMID: 33266802 PMCID: PMC7514196 DOI: 10.3390/e21010086] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 10/12/2018] [Revised: 01/01/2019] [Accepted: 01/15/2019] [Indexed: 11/16/2022]
Abstract
We study the localization properties of the eigenvectors, characterized by their information entropy, of tight-binding random networks with balanced losses and gain. The random network model, which is based on Erdős–Rényi (ER) graphs, is defined by three parameters: the network size N, the network connectivity α, and the losses-and-gain strength γ. Here, N and α are the standard parameters of ER graphs, while we introduce losses and gain by including complex self-loops on all vertices with the imaginary amplitude iγ with random balanced signs, thus breaking the Hermiticity of the corresponding adjacency matrices and inducing complex spectra. By the use of extensive numerical simulations, we define a scaling parameter ξ≡ξ(N,α,γ) that fixes the localization properties of the eigenvectors of our random network model; such that, when ξ<0.1 (10<ξ), the eigenvectors are localized (extended), while the localization-to-delocalization transition occurs for 0.1<ξ<10. Moreover, to extend the applicability of our findings, we demonstrate that for fixed ξ, the spectral properties (characterized by the position of the eigenvalues on the complex plane) of our network model are also universal; i.e., they do not depend on the specific values of the network parameters.
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Pérez Castillo I, Metz FL. Theory for the conditioned spectral density of noninvariant random matrices. Phys Rev E 2018; 98:020102. [PMID: 30253505 DOI: 10.1103/physreve.98.020102] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/08/2018] [Indexed: 06/08/2023]
Abstract
We develop a theoretical approach to compute the conditioned spectral density of N×N noninvariant random matrices in the limit N→∞. This large deviation observable, defined as the eigenvalue distribution conditioned to have a fixed fraction k of eigenvalues smaller than x∈R, provides the spectrum of random matrix samples that deviate atypically from the average behavior. We apply our theory to sparse random matrices and unveil strikingly different and generic properties, namely, (i) their conditioned spectral density has compact support, (ii) it does not experience any abrupt transition for k around its typical value, and (iii) its eigenvalues do not accumulate at x. Moreover, our work points towards other types of transitions in the conditioned spectral density for values of k away from its typical value. These properties follow from the weak or absent eigenvalue repulsion in sparse ensembles and they are in sharp contrast to those displayed by classic or rotationally invariant random matrices. The exactness of our theoretical findings are confirmed through numerical diagonalization of finite random matrices.
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Affiliation(s)
- Isaac Pérez Castillo
- Department of Quantum Physics and Photonics, Institute of Physics, UNAM, P.O. Box 20-364, 01000 Mexico City, Mexico and London Mathematical Laboratory, 14 Buckingham Street, London WC2N 6DF, United Kingdom
| | - Fernando L Metz
- Institute of Physics, Federal University of Rio Grande do Sul, 91501-970 Porto Alegre, Brazil; Physics Department, Federal University of Santa Maria, 97105-900 Santa Maria, Brazil; and London Mathematical Laboratory, 14 Buckingham Street, London WC2N 6DF, United Kingdom
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Gera R, Alonso L, Crawford B, House J, Mendez-Bermudez JA, Knuth T, Miller R. Identifying network structure similarity using spectral graph theory. APPLIED NETWORK SCIENCE 2018; 3:2. [PMID: 30839726 PMCID: PMC6214265 DOI: 10.1007/s41109-017-0042-3] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/27/2017] [Accepted: 06/18/2017] [Indexed: 06/02/2023]
Abstract
Most real networks are too large or they are not available for real time analysis. Therefore, in practice, decisions are made based on partial information about the ground truth network. It is of great interest to have metrics to determine if an inferred network (the partial information network) is similar to the ground truth. In this paper we develop a test for similarity between the inferred and the true network. Our research utilizes a network visualization tool, which systematically discovers a network, producing a sequence of snapshots of the network. We introduce and test our metric on the consecutive snapshots of a network, and against the ground truth. To test the scalability of our metric we use a random matrix theory approach while discovering Erdös-Rényi graphs. This scaling analysis allows us to make predictions about the performance of the discovery process.
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Affiliation(s)
- Ralucca Gera
- Department of Applied Mathematics, 1 University Avenue, Naval Postgraduate School, Monterey, 93943 CA USA
| | - L. Alonso
- Instituto de Física, Benemérita Universidad Autónoma de Puebla, Apartado Postal J-48, Puebla, 72570 Mexico
| | - Brian Crawford
- Department of Computer Science, 1 University Avenue, Naval Postgraduate School, Monterey, 93943 CA USA
| | - Jeffrey House
- Department of Operation Research, 1 University Avenue, Naval Postgraduate School, Monterey, 93943 CA USA
| | - J. A. Mendez-Bermudez
- Instituto de Física, Benemérita Universidad Autónoma de Puebla, Apartado Postal J-48, Puebla, 72570 Mexico
| | - Thomas Knuth
- Instituto de Física, Benemérita Universidad Autónoma de Puebla, Apartado Postal J-48, Puebla, 72570 Mexico
| | - Ryan Miller
- Department of Applied Mathematics, 1 University Avenue, Naval Postgraduate School, Monterey, 93943 CA USA
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Slanina F. Localization in random bipartite graphs: Numerical and empirical study. Phys Rev E 2017; 95:052149. [PMID: 28618645 DOI: 10.1103/physreve.95.052149] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/11/2016] [Indexed: 06/07/2023]
Abstract
We investigate adjacency matrices of bipartite graphs with a power-law degree distribution. Motivation for this study is twofold: first, vibrational states in granular matter and jammed sphere packings; second, graphs encoding social interaction, especially electronic commerce. We establish the position of the mobility edge and show that it strongly depends on the power in the degree distribution and on the ratio of the sizes of the two parts of the bipartite graph. At the jamming threshold, where the two parts have the same size, localization vanishes. We found that the multifractal spectrum is nontrivial in the delocalized phase, but still near the mobility edge. We also study an empirical bipartite graph, namely, the Amazon reviewer-item network. We found that in this specific graph the mobility edge disappears, and we draw a conclusion from this fact regarding earlier empirical studies of the Amazon network.
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Affiliation(s)
- František Slanina
- Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, CZ-18221 Praha, Czech Republic
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Fernandes HA, da Silva R, Caparica AA, de Felício JRD. Nonequilibrium critical dynamics of the two-dimensional Ashkin-Teller model at the Baxter line. Phys Rev E 2017; 95:042105. [PMID: 28505782 DOI: 10.1103/physreve.95.042105] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/18/2016] [Indexed: 06/07/2023]
Abstract
We investigate the short-time universal behavior of the two-dimensional Ashkin-Teller model at the Baxter line by performing time-dependent Monte Carlo simulations. First, as preparatory results, we obtain the critical parameters by searching the optimal power-law decay of the magnetization. Thus, the dynamic critical exponents θ_{m} and θ_{p}, related to the magnetic and electric order parameters, as well as the persistence exponent θ_{g}, are estimated using heat-bath Monte Carlo simulations. In addition, we estimate the dynamic exponent z and the static critical exponents β and ν for both order parameters. We propose a refined method to estimate the static exponents that considers two different averages: one that combines an internal average using several seeds with another, which is taken over temporal variations in the power laws. Moreover, we also performed the bootstrapping method for a complementary analysis. Our results show that the ratio β/ν exhibits universal behavior along the critical line corroborating the conjecture for both magnetization and polarization.
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Affiliation(s)
- H A Fernandes
- Universidade Federal de Goiás - UFG, Campus Jataí,, Jataí-GO, 78000-000, Brazil
| | - R da Silva
- Instituto de Física, Universidade Federal do Rio Grande do Sul, UFRGS, Porto Alegre - RS, 91501-970, Brazil
| | - A A Caparica
- Instituto de Física, Universidade Federal de Goiás, Goiânia-GO, 74.690-900, Brazil
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Metz FL, Pérez Castillo I. Large Deviation Function for the Number of Eigenvalues of Sparse Random Graphs Inside an Interval. PHYSICAL REVIEW LETTERS 2016; 117:104101. [PMID: 27636476 DOI: 10.1103/physrevlett.117.104101] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/01/2016] [Indexed: 06/06/2023]
Abstract
We present a general method to obtain the exact rate function Ψ_{[a,b]}(k) controlling the large deviation probability Prob[I_{N}[a,b]=kN]≍e^{-NΨ_{[a,b]}(k)} that an N×N sparse random matrix has I_{N}[a,b]=kN eigenvalues inside the interval [a,b]. The method is applied to study the eigenvalue statistics in two distinct examples: (i) the shifted index number of eigenvalues for an ensemble of Erdös-Rényi graphs and (ii) the number of eigenvalues within a bounded region of the spectrum for the Anderson model on regular random graphs. A salient feature of the rate function in both cases is that, unlike rotationally invariant random matrices, it is asymmetric with respect to its minimum. The asymmetric character depends on the disorder in a way that is compatible with the distinct eigenvalue statistics corresponding to localized and delocalized eigenstates. The results also show that the level compressibility κ_{2}/κ_{1} for the Anderson model on a regular graph satisfies 0<κ_{2}/κ_{1}<1 in the bulk regime, in contrast with the behavior found in Gaussian random matrices. Our theoretical findings are thoroughly compared to numerical diagonalization in both cases, showing a reasonable good agreement.
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Affiliation(s)
- Fernando L Metz
- Departamento de Física, Universidade Federal de Santa Maria, 97105-900 Santa Maria, Brazil
| | - Isaac Pérez Castillo
- Department of Complex Systems, Institute of Physics, UNAM, P.O. Box 20-364, 01000 México, D.F., Mexico
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Méndez-Bermúdez JA, Alcazar-López A, Martínez-Mendoza AJ, Rodrigues FA, Peron TKD. Universality in the spectral and eigenfunction properties of random networks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:032122. [PMID: 25871069 DOI: 10.1103/physreve.91.032122] [Citation(s) in RCA: 18] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/29/2014] [Indexed: 06/04/2023]
Abstract
By the use of extensive numerical simulations, we show that the nearest-neighbor energy-level spacing distribution P(s) and the entropic eigenfunction localization length of the adjacency matrices of Erdős-Rényi (ER) fully random networks are universal for fixed average degree ξ≡αN (α and N being the average network connectivity and the network size, respectively). We also demonstrate that the Brody distribution characterizes well P(s) in the transition from α=0, when the vertices in the network are isolated, to α=1, when the network is fully connected. Moreover, we explore the validity of our findings when relaxing the randomness of our network model and show that, in contrast to standard ER networks, ER networks with diagonal disorder also show universality. Finally, we also discuss the spectral and eigenfunction properties of small-world networks.
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Affiliation(s)
- J A Méndez-Bermúdez
- Instituto de Física, Benemérita Universidad Autónoma de Puebla, Apartado Postal J-48, Puebla 72570, Mexico
| | - A Alcazar-López
- Instituto de Física, Benemérita Universidad Autónoma de Puebla, Apartado Postal J-48, Puebla 72570, Mexico
| | - A J Martínez-Mendoza
- Instituto de Física, Benemérita Universidad Autónoma de Puebla, Apartado Postal J-48, Puebla 72570, Mexico and Elméleti Fizika Tanszék, Fizikai Intézet, Budapesti Műszaki és Gazdaságtudományi Egyetem, H-1521 Budapest, Hungary
| | - Francisco A Rodrigues
- Departamento de Matemática Aplicada e Estatística, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Caixa Postal 668,13560-970 São Carlos, São Paulo, Brazil
| | - Thomas K Dm Peron
- Instituto de Física de São Carlos, Universidade de São Paulo, CP 369, 13560-970, São Carlos, São Paulo, Brazil
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Martínez-Mendoza AJ, Alcazar-López A, Méndez-Bermúdez JA. Scattering and transport properties of tight-binding random networks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:012126. [PMID: 23944433 DOI: 10.1103/physreve.88.012126] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/08/2013] [Indexed: 06/02/2023]
Abstract
We study numerically scattering and transport statistical properties of tight-binding random networks characterized by the number of nodes N and the average connectivity α. We use a scattering approach to electronic transport and concentrate on the case of a small number of single-channel attached leads. We observe a smooth crossover from insulating to metallic behavior in the average scattering matrix elements <|S(mn)|(2)>, the conductance probability distribution w(T), the average conductance <T>, the shot noise power P, and the elastic enhancement factor F by varying α from small (α→0) to large (α→1) values. We also show that all these quantities are invariant for fixed ξ=αN. Moreover, we proposes a heuristic and universal relation between <|S(mn)|(2)>, <T>, and P and the disorder parameter ξ.
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Affiliation(s)
- A J Martínez-Mendoza
- Instituto de Física, Benemérita Universidad Autónoma de Puebla, Apartado Postal J-48, Puebla 72570, Mexico
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11
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Clapa VI, Kottos T, Starr FW. Localization transition of instantaneous normal modes and liquid diffusion. J Chem Phys 2012; 136:144504. [DOI: 10.1063/1.3701564] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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12
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Slanina F. Equivalence of replica and cavity methods for computing spectra of sparse random matrices. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:011118. [PMID: 21405672 DOI: 10.1103/physreve.83.011118] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/04/2010] [Revised: 12/01/2010] [Indexed: 05/30/2023]
Abstract
We show by direct calculation that the replica and cavity methods are exactly equivalent for the spectrum of an Erdős-Rényi random graph. We introduce a variational formulation based on the cavity method and use it to find approximate solutions for the density of eigenvalues. We also use this variational method for calculating spectra of sparse covariance matrices.
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Affiliation(s)
- František Slanina
- Institute of Physics, Academy of Sciences of the Czech Republic, Prague, Czech Republic.
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13
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Ding X, Jiang T. Spectral distributions of adjacency and Laplacian matrices of random graphs. ANN APPL PROBAB 2010. [DOI: 10.1214/10-aap677] [Citation(s) in RCA: 53] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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14
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Metz FL, Neri I, Bollé D. Localization transition in symmetric random matrices. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:031135. [PMID: 21230053 DOI: 10.1103/physreve.82.031135] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/02/2010] [Indexed: 05/30/2023]
Abstract
We study the behavior of the inverse participation ratio and the localization transition in infinitely large random matrices through the cavity method. Results are shown for two ensembles of random matrices: Laplacian matrices on sparse random graphs and fully connected Lévy matrices. We derive a critical line separating localized from extended states in the case of Lévy matrices. Comparison between theoretical results and diagonalization of finite random matrices is shown.
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Affiliation(s)
- F L Metz
- Instituut voor Theoretische Fysica, Katholieke Universiteit Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium
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Sade M, Kalisky T, Havlin S, Berkovits R. Localization transition on complex networks via spectral statistics. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 72:066123. [PMID: 16486026 DOI: 10.1103/physreve.72.066123] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/11/2005] [Indexed: 05/06/2023]
Abstract
The spectral statistics of complex networks are numerically studied. The features of the Anderson metal-insulator transition are found to be similar for a wide range of different networks. A metal-insulator transition as a function of the disorder can be observed for different classes of complex networks for which the average connectivity is small. The critical index of the transition corresponds to the mean field expectation. When the connectivity is higher, the amount of disorder needed to reach a certain degree of localization is proportional to the average connectivity, though a precise transition cannot be identified. The absence of a clear transition at high connectivity is probably due to the very compact structure of the highly connected networks, resulting in a small diameter even for a large number of sites.
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Affiliation(s)
- M Sade
- The Minerva Center, Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel
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Semerjian G, Cugliandolo LF. Sparse random matrices: the eigenvalue spectrum revisited. ACTA ACUST UNITED AC 2002. [DOI: 10.1088/0305-4470/35/23/303] [Citation(s) in RCA: 80] [Impact Index Per Article: 3.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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17
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Bauer M, Golinelli O. Exactly solvable model with two conductor-insulator transitions driven by impurities. PHYSICAL REVIEW LETTERS 2001; 86:2621-2624. [PMID: 11289995 DOI: 10.1103/physrevlett.86.2621] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/05/2000] [Revised: 10/26/2000] [Indexed: 05/23/2023]
Abstract
We present an exact analysis of two conductor-insulator transitions in the random graph model where low connectivity means high impurity concentration. The adjacency matrix of the random graph is used as a hopping Hamiltonian. We compute the height of the delta peak at zero energy in its spectrum exactly and describe analytically the structure and contribution of localized eigenvectors. The system is a conductor for average connectivities between 1.421 529ellipsis and 3.154 985ellipsis but an insulator in the other regimes. We explain the spectral singularity at average connectivity e = 2.718 281ellipsis and relate it to other enumerative problems in random graph theory.
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Affiliation(s)
- M Bauer
- Cea Saclay, Service de Physique Théorique, 91191 Gif-sur-Yvette, France.
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18
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Araujo M, Medina E, Aponte E. Spectral statistics and dynamics of Lévy matrices. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1999; 60:3580-8. [PMID: 11970191 DOI: 10.1103/physreve.60.3580] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/10/1998] [Revised: 05/25/1999] [Indexed: 04/18/2023]
Abstract
We study the spectral statistics and dynamics of a random matrix model where matrix elements are taken from power-law tailed distributions. Such distributions, labeled by a parameter mu, converge on the Lévy basin, giving the matrix model the label "Lévy matrix" [P. Cizeau and J. P. Bouchaud, Phys. Rev. E 50, 1810 (1994)]. Such matrices are interesting because their properties go beyond the Gaussian universality class and they model many physically relevant systems such as spin glasses with dipolar or Ruderman-Kittel-Kasuya-Yosida interactions, electronic systems with power-law decaying interactions, and the spectral behavior at the metal insulator transition. Regarding the density of states we extend previous work to reveal the sparse matrix limit as mu-->0. Furthermore, we find for 2 x 2 Lévy matrices that geometrical level repulsion is not affected by the distribution's broadness. Nevertheless, essential singularities particular to Lévy distributions for small arguments break geometrical repulsion and make it mu dependent. Level dynamics as a function of a symmetry breaking parameter gives new insight into the phases found by Cizeau and Bouchaud (CB). We map the phase diagram drawn qualitatively by CB by using the delta3 statistic. Finally we compute the conductance of each phase by using the Thouless formula, and find that the mixed phase separating conducting and insulating phases has a unique character.
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Affiliation(s)
- M Araujo
- PDVSA Intevep, Apartado 76343, Caracas 1070A, Venezuela
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19
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Prosen T, Robnik M. Energy level statistics and localization in sparsed banded random matrix ensemble. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/26/5/029] [Citation(s) in RCA: 34] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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Berkovits R, Avishai Y. Spectral statistics near the quantum percolation threshold. PHYSICAL REVIEW. B, CONDENSED MATTER 1996; 53:R16125-R16128. [PMID: 9983513 DOI: 10.1103/physrevb.53.r16125] [Citation(s) in RCA: 31] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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21
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Zharekeshev IK, Kramer B. Scaling of level statistics at the disorder-induced metal-insulator transition. PHYSICAL REVIEW. B, CONDENSED MATTER 1995; 51:17239-17242. [PMID: 9978746 DOI: 10.1103/physrevb.51.17239] [Citation(s) in RCA: 41] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Kreynin M, Shapiro B. Density of levels in a generalized matrix ensemble. PHYSICAL REVIEW LETTERS 1995; 74:4122-4124. [PMID: 10058421 DOI: 10.1103/physrevlett.74.4122] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Cizeau P, Bouchaud JP. Theory of Lévy matrices. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 50:1810-1822. [PMID: 9962183 DOI: 10.1103/physreve.50.1810] [Citation(s) in RCA: 26] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Soukoulis CM, Velgakis MJ, Economou EN. One-dimensional localization with correlated disorder. PHYSICAL REVIEW. B, CONDENSED MATTER 1994; 50:5110-5118. [PMID: 9976849 DOI: 10.1103/physrevb.50.5110] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Evangelou SN. Level-spacing function P(S) at the mobility edge. PHYSICAL REVIEW. B, CONDENSED MATTER 1994; 49:16805-16808. [PMID: 10010850 DOI: 10.1103/physrevb.49.16805] [Citation(s) in RCA: 51] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Hofstetter E, Schreiber M. Relation between energy-level statistics and phase transition and its application to the Anderson model. PHYSICAL REVIEW. B, CONDENSED MATTER 1994; 49:14726-14729. [PMID: 10010563 DOI: 10.1103/physrevb.49.14726] [Citation(s) in RCA: 61] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Abstract
We present results for the statistics of the eigenvalues in random matrix ensembles characterized by an Anderson delocalization-localization transition. The nearest-level-spacing distribution function P(S) and the number variance 〈(δN(E))2〉 are shown at the mobility edge where we obtain universal curves interpolating between Wigner-Dyson and Poisson statistics valid for delocalized and for localized eigenfunctions, respectively. We also discuss the connection of level statistics with dynamics by considering the time evolution of a quantum wavepacket in a quasirandom matrix model. The critical quantum dynamics is characterized by anomalous diffusion, described via continuous sets of multifractal exponents.
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Affiliation(s)
- S. N. Evangelou
- Department of Physics, University of Ioannina, Ioannina, 45 110, Greece
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Burleigh DC, Sibert EL. A random matrix approach to rotation–vibration mixing in H2CO and D2CO. J Chem Phys 1993. [DOI: 10.1063/1.464500] [Citation(s) in RCA: 18] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Shklovskii BI, Shapiro B, Sears BR, Lambrianides P, Shore HB. Statistics of spectra of disordered systems near the metal-insulator transition. PHYSICAL REVIEW. B, CONDENSED MATTER 1993; 47:11487-11490. [PMID: 10005290 DOI: 10.1103/physrevb.47.11487] [Citation(s) in RCA: 154] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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