Localization of Laplacian eigenvectors on random networks

Abstract In large random networks, each eigenvector of the Laplacian matrix tends to localize on a subset of network nodes having similar numbers of edges, namely, the components of each Laplacian eigenvector take relatively large values only on a particular subset of nodes whose degrees are close....

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Autores principales:	Shigefumi Hata, Hiroya Nakao
Formato:	article
Lenguaje:	EN
Publicado:	Nature Portfolio 2017
Materias:	Medicine R Science Q
Acceso en línea:	https://doaj.org/article/3b61cac4c4914fafa2a343c00f7453b7
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spelling	oai:doaj.org-article:3b61cac4c4914fafa2a343c00f7453b72021-12-02T16:06:39ZLocalization of Laplacian eigenvectors on random networks10.1038/s41598-017-01010-02045-2322https://doaj.org/article/3b61cac4c4914fafa2a343c00f7453b72017-04-01T00:00:00Zhttps://doi.org/10.1038/s41598-017-01010-0https://doaj.org/toc/2045-2322Abstract In large random networks, each eigenvector of the Laplacian matrix tends to localize on a subset of network nodes having similar numbers of edges, namely, the components of each Laplacian eigenvector take relatively large values only on a particular subset of nodes whose degrees are close. Although this localization property has significant consequences for dynamical processes on random networks, a clear theoretical explanation has not yet been established. Here we analyze the origin of localization of Laplacian eigenvectors on random networks by using a perturbation theory. We clarify how heterogeneity in the node degrees leads to the eigenvector localization and that there exists a clear degree-eigenvalue correspondence, that is, the characteristic degrees of the localized nodes essentially determine the eigenvalues. We show that this theory can account for the localization properties of Laplacian eigenvectors on several classes of random networks, and argue that this localization should occur generally in networks with degree heterogeneity.Shigefumi HataHiroya NakaoNature PortfolioarticleMedicineRScienceQENScientific Reports, Vol 7, Iss 1, Pp 1-11 (2017)
institution	DOAJ
collection	DOAJ
language	EN
topic	Medicine R Science Q
spellingShingle	Medicine R Science Q Shigefumi Hata Hiroya Nakao Localization of Laplacian eigenvectors on random networks
description	Abstract In large random networks, each eigenvector of the Laplacian matrix tends to localize on a subset of network nodes having similar numbers of edges, namely, the components of each Laplacian eigenvector take relatively large values only on a particular subset of nodes whose degrees are close. Although this localization property has significant consequences for dynamical processes on random networks, a clear theoretical explanation has not yet been established. Here we analyze the origin of localization of Laplacian eigenvectors on random networks by using a perturbation theory. We clarify how heterogeneity in the node degrees leads to the eigenvector localization and that there exists a clear degree-eigenvalue correspondence, that is, the characteristic degrees of the localized nodes essentially determine the eigenvalues. We show that this theory can account for the localization properties of Laplacian eigenvectors on several classes of random networks, and argue that this localization should occur generally in networks with degree heterogeneity.
format	article
author	Shigefumi Hata Hiroya Nakao
author_facet	Shigefumi Hata Hiroya Nakao
author_sort	Shigefumi Hata
title	Localization of Laplacian eigenvectors on random networks
title_short	Localization of Laplacian eigenvectors on random networks
title_full	Localization of Laplacian eigenvectors on random networks
title_fullStr	Localization of Laplacian eigenvectors on random networks
title_full_unstemmed	Localization of Laplacian eigenvectors on random networks
title_sort	localization of laplacian eigenvectors on random networks
publisher	Nature Portfolio
publishDate	2017
url	https://doaj.org/article/3b61cac4c4914fafa2a343c00f7453b7
work_keys_str_mv	AT shigefumihata localizationoflaplacianeigenvectorsonrandomnetworks AT hiroyanakao localizationoflaplacianeigenvectorsonrandomnetworks
_version_	1718384897347813376

Localization of Laplacian eigenvectors on random networks

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