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  1. Article ; Online: Localization and Universality of Eigenvectors in Directed Random Graphs.

    Metz, Fernando Lucas / Neri, Izaak

    Physical review letters

    2021  Volume 126, Issue 4, Page(s) 40604

    Abstract: Although the spectral properties of random graphs have been a long-standing focus of network theory, the properties of right eigenvectors of directed graphs have so far eluded an exact analytic treatment. We present a general theory for the statistics of ...

    Abstract Although the spectral properties of random graphs have been a long-standing focus of network theory, the properties of right eigenvectors of directed graphs have so far eluded an exact analytic treatment. We present a general theory for the statistics of the right eigenvector components in directed random graphs with a prescribed degree distribution and with randomly weighted links. We obtain exact analytic expressions for the inverse participation ratio and show that right eigenvectors of directed random graphs with a small average degree are localized. Remarkably, if the fourth moment of the degree distribution is finite, then the critical mean degree of the localization transition is independent of the degree fluctuations, which is different from localization in undirected graphs that is governed by degree fluctuations. We also show that in the high connectivity limit the distribution of the right eigenvector components is solely determined by the degree distribution. For delocalized eigenvectors, we recover in this limit the universal results from standard random matrix theory that are independent of the degree distribution, while for localized eigenvectors the eigenvector distribution depends on the degree distribution.
    Language English
    Publishing date 2021-02-11
    Publishing country United States
    Document type Journal Article
    ZDB-ID 208853-8
    ISSN 1079-7114 ; 0031-9007
    ISSN (online) 1079-7114
    ISSN 0031-9007
    DOI 10.1103/PhysRevLett.126.040604
    Database MEDical Literature Analysis and Retrieval System OnLINE

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  2. Article ; Online: Publisher's Note: Eigenvalue Outliers of Non-Hermitian Random Matrices with a Local Tree Structure [Phys. Rev. Lett. 117, 224101 (2016)].

    Neri, Izaak / Metz, Fernando Lucas

    Physical review letters

    2017  Volume 118, Issue 1, Page(s) 19901

    Abstract: This corrects the article DOI: 10.1103/PhysRevLett.117.224101. ...

    Abstract This corrects the article DOI: 10.1103/PhysRevLett.117.224101.
    Language English
    Publishing date 2017-01-05
    Publishing country United States
    Document type Published Erratum
    ZDB-ID 208853-8
    ISSN 1079-7114 ; 0031-9007
    ISSN (online) 1079-7114
    ISSN 0031-9007
    DOI 10.1103/PhysRevLett.118.019901
    Database MEDical Literature Analysis and Retrieval System OnLINE

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  3. Book ; Online: Linear stability analysis for large dynamical systems on directed random graphs

    Neri, Izaak / Metz, Fernando Lucas

    2019  

    Abstract: We present a linear stability analysis of stationary states (or fixed points) in large dynamical systems defined on random directed graphs with a prescribed distribution of indegrees and outdegrees. We obtain two remarkable results for such dynamical ... ...

    Abstract We present a linear stability analysis of stationary states (or fixed points) in large dynamical systems defined on random directed graphs with a prescribed distribution of indegrees and outdegrees. We obtain two remarkable results for such dynamical systems: First, infinitely large systems on directed graphs can be stable even when the degree distribution has unbounded support; this result is surprising since their counterparts on nondirected graphs are unstable when system size is large enough. Second, we show that the phase transition between the stable and unstable phase is universal in the sense that it depends only on a few parameters, such as, the mean degree and a degree correlation coefficient. In addition, in the unstable regime we characterize the nature of the destabilizing mode, which also exhibits universal features. These results follow from an exact theory for the leading eigenvalue of infinitely large graphs that are locally tree-like and oriented, as well as, for the right and left eigenvectors associated with the leading eigenvalue. We corroborate analytical results for infinitely large graphs with numerical experiments on random graphs of finite size. We discuss how the presented theory can be extended to graphs with diagonal disorder and to graphs that contain nondirected links. Finally, we discuss the influence of small cycles and how they can destabilize large dynamical systems when they induce strong enough feedback loops.

    Comment: 35 pages, 8 figures, a few typo's have been corrected in the new version
    Keywords Condensed Matter - Statistical Mechanics ; Condensed Matter - Disordered Systems and Neural Networks ; Computer Science - Social and Information Networks ; Physics - Physics and Society ; Quantitative Biology - Populations and Evolution
    Subject code 515
    Publishing date 2019-08-19
    Publishing country us
    Document type Book ; Online
    Database BASE - Bielefeld Academic Search Engine (life sciences selection)

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  4. Article ; Online: Eigenvalue Outliers of Non-Hermitian Random Matrices with a Local Tree Structure.

    Neri, Izaak / Metz, Fernando Lucas

    Physical review letters

    2016  Volume 117, Issue 22, Page(s) 224101

    Abstract: Spectra of sparse non-Hermitian random matrices determine the dynamics of complex processes on graphs. Eigenvalue outliers in the spectrum are of particular interest, since they determine the stationary state and the stability of dynamical processes. We ... ...

    Abstract Spectra of sparse non-Hermitian random matrices determine the dynamics of complex processes on graphs. Eigenvalue outliers in the spectrum are of particular interest, since they determine the stationary state and the stability of dynamical processes. We present a general and exact theory for the eigenvalue outliers of random matrices with a local tree structure. For adjacency and Laplacian matrices of oriented random graphs, we derive analytical expressions for the eigenvalue outliers, the first moments of the distribution of eigenvector elements associated with an outlier, the support of the spectral density, and the spectral gap. We show that these spectral observables obey universal expressions, which hold for a broad class of oriented random matrices.
    Language English
    Publishing date 2016-11-25
    Publishing country United States
    Document type Journal Article
    ZDB-ID 208853-8
    ISSN 1079-7114 ; 0031-9007
    ISSN (online) 1079-7114
    ISSN 0031-9007
    DOI 10.1103/PhysRevLett.117.224101
    Database MEDical Literature Analysis and Retrieval System OnLINE

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    In stock of ZB MED Bonn / Germany

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