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2017 ; 7
(1
): 1121
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Localization of Laplacian eigenvectors on random networks
#MMPMID28442760
Hata S
; Nakao H
Sci Rep
2017[Apr]; 7
(1
): 1121
PMID28442760
show ga
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.
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