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Hermitian normalized Laplacian matrix for directed networks

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Hermitian normalized Laplacian matrix for directed networks. / Yu, Guihai; Dehmer, Matthias; Emmert-Streib, Frank; Jodlbauer, Herbert.

In: Information Sciences, Vol. 495, 01.08.2019, p. 175-184.

Research output: Contribution to journalArticleScientificpeer-review

Harvard

Yu, G, Dehmer, M, Emmert-Streib, F & Jodlbauer, H 2019, 'Hermitian normalized Laplacian matrix for directed networks', Information Sciences, vol. 495, pp. 175-184. https://doi.org/10.1016/j.ins.2019.04.049

APA

Yu, G., Dehmer, M., Emmert-Streib, F., & Jodlbauer, H. (2019). Hermitian normalized Laplacian matrix for directed networks. Information Sciences, 495, 175-184. https://doi.org/10.1016/j.ins.2019.04.049

Vancouver

Author

Yu, Guihai ; Dehmer, Matthias ; Emmert-Streib, Frank ; Jodlbauer, Herbert. / Hermitian normalized Laplacian matrix for directed networks. In: Information Sciences. 2019 ; Vol. 495. pp. 175-184.

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@article{4faa5840abde4a0cbb3eb71eeecf8bd6,
title = "Hermitian normalized Laplacian matrix for directed networks",
abstract = "In this paper, we extend and generalize the spectral theory of undirected networks towards directed networks by introducing the Hermitian normalized Laplacian matrix for directed networks. In order to start, we discuss the Courant–Fischer theorem for the eigenvalues of Hermitian normalized Laplacian matrix. Based on the Courant–Fischer theorem, we obtain a similar result towards the normalized Laplacian matrix of undirected networks: for each i ∈ {1, 2,…, n}, any eigenvalue of Hermitian normalized Laplacian matrix λ i ∈ [0, 2]. Moreover, we prove some special conditions if 0, or 2 is an eigenvalue of the Hermitian normalized Laplacian matrix L(X). On top of that, we investigate the symmetry of the eigenvalues of L(X)and the edge-version for the eigenvalue interlacing result. Finally we present two expressions for the coefficients of the characteristic polynomial of the Hermitian normalized Laplacian matrix. As an outlook, we sketch some novel and intriguing problems to which our apparatus could generally be applied.",
keywords = "Characteristic polynomial, Courant–Fischer theorem, Directed networks, Eigenvalue interlacing inequality, Hermitian normalized Laplacian matrix",
author = "Guihai Yu and Matthias Dehmer and Frank Emmert-Streib and Herbert Jodlbauer",
year = "2019",
month = "8",
day = "1",
doi = "10.1016/j.ins.2019.04.049",
language = "English",
volume = "495",
pages = "175--184",
journal = "Information Sciences",
issn = "0020-0255",
publisher = "Elsevier",

}

RIS (suitable for import to EndNote) - Download

TY - JOUR

T1 - Hermitian normalized Laplacian matrix for directed networks

AU - Yu, Guihai

AU - Dehmer, Matthias

AU - Emmert-Streib, Frank

AU - Jodlbauer, Herbert

PY - 2019/8/1

Y1 - 2019/8/1

N2 - In this paper, we extend and generalize the spectral theory of undirected networks towards directed networks by introducing the Hermitian normalized Laplacian matrix for directed networks. In order to start, we discuss the Courant–Fischer theorem for the eigenvalues of Hermitian normalized Laplacian matrix. Based on the Courant–Fischer theorem, we obtain a similar result towards the normalized Laplacian matrix of undirected networks: for each i ∈ {1, 2,…, n}, any eigenvalue of Hermitian normalized Laplacian matrix λ i ∈ [0, 2]. Moreover, we prove some special conditions if 0, or 2 is an eigenvalue of the Hermitian normalized Laplacian matrix L(X). On top of that, we investigate the symmetry of the eigenvalues of L(X)and the edge-version for the eigenvalue interlacing result. Finally we present two expressions for the coefficients of the characteristic polynomial of the Hermitian normalized Laplacian matrix. As an outlook, we sketch some novel and intriguing problems to which our apparatus could generally be applied.

AB - In this paper, we extend and generalize the spectral theory of undirected networks towards directed networks by introducing the Hermitian normalized Laplacian matrix for directed networks. In order to start, we discuss the Courant–Fischer theorem for the eigenvalues of Hermitian normalized Laplacian matrix. Based on the Courant–Fischer theorem, we obtain a similar result towards the normalized Laplacian matrix of undirected networks: for each i ∈ {1, 2,…, n}, any eigenvalue of Hermitian normalized Laplacian matrix λ i ∈ [0, 2]. Moreover, we prove some special conditions if 0, or 2 is an eigenvalue of the Hermitian normalized Laplacian matrix L(X). On top of that, we investigate the symmetry of the eigenvalues of L(X)and the edge-version for the eigenvalue interlacing result. Finally we present two expressions for the coefficients of the characteristic polynomial of the Hermitian normalized Laplacian matrix. As an outlook, we sketch some novel and intriguing problems to which our apparatus could generally be applied.

KW - Characteristic polynomial

KW - Courant–Fischer theorem

KW - Directed networks

KW - Eigenvalue interlacing inequality

KW - Hermitian normalized Laplacian matrix

U2 - 10.1016/j.ins.2019.04.049

DO - 10.1016/j.ins.2019.04.049

M3 - Article

VL - 495

SP - 175

EP - 184

JO - Information Sciences

JF - Information Sciences

SN - 0020-0255

ER -