Approximated structured pseudospectra

Noschese, Silvia; Reichel, Lothar

doi:10.1002/nla.2082

Cited by 6 publications

(14 citation statements)

References 23 publications

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“…When the adjacency matrix A is very large, we may consider replacing the vectors

u

and

v

in (8) by the vector

e = \frac{1}{\sqrt{n}} {[1, 1, \dots, 1]}^{T}

and compute

ρ (A)

and

ρ (A + ε e e^{T})

by the (standard) Arnoldi or restarted Arnoldi methods to determine the structural robustness of the graph with adjacency matrix A . This approach was applied in Reference 21 to estimate pseudospectra of large matrices.…”

Section: Estimating and Reducing The Spectral Radiusmentioning

confidence: 99%

“…This approach takes possible uncertainty of the available edge‐weights into account. The analysis in Reference 21 leads to the following numerical method: Apply the two‐sided Arnoldi method to

A \in 𝒮

to compute the Perron root

ρ (A)

, as well as the unit right and left Perron vectors

u

and

v

, respectively, with positive entries. Project

v u^{T}

into

𝒮

, normalize the projected matrix to have unit Frobenius norm, and define

E = \frac{v u^{T} |_{𝒮}}{‖ v u^{T} |_{𝒮} ‖_{F}} .

We refer to the matrix (9) as an

𝒮

‐structured analogue of the Wilkinson perturbation. This is the worst

𝒮

‐structured perturbation for

ρ (A)

; one has, by Noschese and Pasquini, 9 (Proposition 2.3)

\frac{v^{T} E u}{v^{T} u} = \frac{| v^{T} E u |}{v^{T} u} = \frac{‖ v ‖ ‖ v u^{T} |_{𝒮} ‖_{F} ‖ u ‖}{v^{T}}

…”

Section: Estimating and Reducing The Spectral Radiusmentioning

confidence: 99%

See 1 more Smart Citation

Estimating and increasing the structural robustness of a network

Noschese

Reichel

2021

Numerical Linear Algebra App

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The capability of a network to cope with threats and survive attacks is referred to as its robustness. This article discusses one kind of robustness, commonly denoted structural robustness, which increases when the spectral radius of the adjacency matrix associated with the network decreases. We discuss computational techniques for identifying edges, whose removal may significantly reduce the spectral radius. Nonsymmetric adjacency matrices are studied with the aid of their pseudospectra. In particular, we consider nonsymmetric adjacency matrices that arise when people seek to avoid being infected by Covid‐19 by wearing facial masks of different qualities.

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“…When the adjacency matrix A is very large, we may consider replacing the vectors

u

and

v

in (8) by the vector

e = \frac{1}{\sqrt{n}} {[1, 1, \dots, 1]}^{T}

and compute

ρ (A)

and

ρ (A + ε e e^{T})

Section: Estimating and Reducing The Spectral Radiusmentioning

confidence: 99%

A \in 𝒮

to compute the Perron root

ρ (A)

, as well as the unit right and left Perron vectors

u

and

v

, respectively, with positive entries. Project

v u^{T}

into

𝒮

, normalize the projected matrix to have unit Frobenius norm, and define

E = \frac{v u^{T} |_{𝒮}}{‖ v u^{T} |_{𝒮} ‖_{F}} .

We refer to the matrix (9) as an

𝒮

‐structured analogue of the Wilkinson perturbation. This is the worst

𝒮

‐structured perturbation for

ρ (A)

; one has, by Noschese and Pasquini, 9 (Proposition 2.3)

\frac{v^{T} E u}{v^{T} u} = \frac{| v^{T} E u |}{v^{T} u} = \frac{‖ v ‖ ‖ v u^{T} |_{𝒮} ‖_{F} ‖ u ‖}{v^{T}}

…”

Section: Estimating and Reducing The Spectral Radiusmentioning

confidence: 99%

Estimating and increasing the structural robustness of a network

Noschese

Reichel

2021

Numerical Linear Algebra App

Self Cite

View full text Add to dashboard Cite

show abstract

“…, 1] T and compute ρ(A) and ρ(A+ε e e T ) by the (standard) Arnoldi or restarted Arnoldi methods to determine the structural robustness of the graph with adjacency matrix A. This approach was applied in [20] to estimate pseudospectra of large matrices.…”

Section: Computation Of the Left And Right Perron Vectors Of A Nonneg...mentioning

confidence: 99%

Estimating and increasing the structural robustness of a network

Noschese¹,

Reichel²

2021

Preprint

Self Cite

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The capability of a network to cope with threats and survive attacks is referred to as its robustness. This paper discusses one kind of robustness, commonly denoted structural robustness, which increases when the spectral radius of the adjacency matrix associated with the network decreases. We discuss computational techniques for identifying edges, whose removal may significantly reduce the spectral radius. Nonsymmetric adjacency matrices are studied with the aid of their pseudospectra. In particular, we consider nonsymmetric adjacency matrices that arise when people seek to avoid being infected by Covid-19 by wearing facial masks of different qualities.

show abstract

“…However, the computation of pseudospectra is a computationally demanding task except for very small matrices. Therefore, the development of numerical methods for the efficient computation of pseudospectra of medium-sized matrices, or partial pseudospectra of large matrices, has received considerable attention; see [3,15,18,19,25,26].…”

Section: Introductionmentioning

confidence: 99%

“…Our approach is inspired by Wilkinson's analysis of eigenvalue perturbation of a single matrix; see [24]. It generalizes an approach recently developed in [18] for the efficient computation of structured or unstructured pseudospectra of a single matrix. This paper is organized as follows.…”

Section: Introductionmentioning

confidence: 99%