Eigenvectors of isospectral graph transformations

Duarte, Pedro; Torres, Maria Joana

doi:10.1016/j.laa.2015.01.038

Cited by 14 publications

(26 citation statements)

References 8 publications

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“…We first provide some key aspects of isospectral reductions [13][14][15][16][17]19], introduced first by Bunimovich and Webb [13]. This concept will allow us to extract the polynomials P ± .…”

Section: Isospectral Reductionsmentioning

confidence: 99%

“…Contrary to eigenvectors of the symmetric matrix H, the set {v i } does not need to be pairwise orthogonal, and could even be linearly dependent or pairwise identical. Their importance stems from the fact that they can be linked [17,34] to the eigenvectors of H. Namely, every eigenvector v i ∈ R |S|×1 of R S (H, λ) with eigenvalue λ i is, up to normalization, the projection of the corresponding eigenvector V i ∈ R N ×1 of H ∈ R N ×N onto the sites S, i.e., equal to V i S ∈ R |S|×1 , where HV i = λ i V i and |S| denotes the number of elements in S.…”

Section: Isospectral Reductionsmentioning

confidence: 99%

“…Namely, by directly obtaining the polynomials P ± from an underlying symmetric Hamiltonian that features cospectral sites u and v. To this end, we combine the mathematical relations underlying the works in Ref. [11,12] with the theory of isospectral reductions [13][14][15][16][17][18][19]. Isospectral reduction is a method to reduce the size of a given Hamiltonian while keeping a large amount of information on its eigenvalues and eigenvectors.…”

Section: Introductionmentioning

confidence: 99%

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Designing pretty good state transfer via isospectral reductions

et al. 2020

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We present an algorithm to design networks that feature pretty good state transfer (PGST), which is of interest for high-fidelity transfer of information in quantum computing. Realizations of PGST networks have so far mostly relied either on very special network geometries or imposed conditions such as transcendental on-site potentials. However, it was recently shown [Eisenberg et al., arXiv:1804.01645] that PGST generally arises when a network's eigenvectors and the factors P± of its characteristic polynomial P fulfill certain conditions, where P± correspond to eigenvectors which have ±1 parity on the input and target sites. We combine this result with the so-called isospectral reduction of a network to obtain P± from a dimensionally reduced form of the Hamiltonian. Equipped with the knowledge of the factors P±, we show how a variety of setups can be equipped with PGST by proper tuning of P±. Having demonstrated a method of designing networks featuring pretty good state transfer of single site excitations, we further show how the obtained networks can be manipulated such that they allow for robust storage of qubits. We hereby rely on the concept of compact localized states, which are eigenstates of a Hamiltonian localized on a small subdomain, and whose amplitudes completely vanish outside of this domain. Such states are natural candidates for the storage of quantum information, and we show how certain Hamiltonians featuring pretty good state transfer of single site excitation can be equipped with compact localized states such that their transfer is made possible.

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Section: Isospectral Reductionsmentioning

confidence: 99%

Section: Isospectral Reductionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Designing pretty good state transfer via isospectral reductions

et al. 2020

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“…To prove this result we will first need the following theorem that relates the eigenvectors of R S (M) to the eigenvectors of M, which is found in [27]. This theorem states that under an isospectral reduction the eigenvectors of the matrix are preserved in the sense that the eigenvectors of the reduced matrix are simply the projection of the original eigenvector onto the vertices the network was reduced over.…”

Section: Eigenvector Centralitymentioning

confidence: 99%

Hidden symmetries in real and theoretical networks

Smith

Webb

2019

Physica A: Statistical Mechanics and its Applications

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Symmetries are ubiquitous in real networks and often characterize network features and functions. Here we present a generalization of network symmetry called latent symmetry, which is an extension of the standard notion of symmetry. They are defined in terms of standard symmetries in a reduced version of the network. One unique aspect of latent symmetries is that each one is associated with a size, which provides a way of discussing symmetries at multiple scales in a network. We are able to demonstrate a number of examples of networks (graphs) which contain latent symmetry, including a number of real networks. In numerical experiments, we show that latent symmetries are found more frequently in graphs built using preferential attachment, a standard model of network growth, when compared to non-network like (Erdős-Rényi) graphs. Finally we prove that if vertices in a network are latently symmetric, then they must have the same eigenvector centrality, similar to vertices which are symmetric in the standard sense. This suggests that the latent symmetries present in real-networks may serve the same structural and functional purpose standard symmetries do in these networks. We conclude from these facts and observations that latent symmetries are present in real networks and provide useful information about the network potentially beyond standard symmetries as they can appear at multiple scales.

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“…What happens to an eigenvector of a matrix (graph) as it is reduced is described by the following theorem. This theorem states that under an isospectral reduction the eigenvectors of the matrix are preserved in the sense that the eigenvectors of the reduced matrix are simply the projection of the original eigenvector onto the vertices that the network was reduced over (see Theorem 1 in [17]).…”

Section: Definition 21 (Isospectral Matrix Reduction)mentioning

confidence: 99%

Finding Hidden Structures, Hierarchies, and Cores in Networks via Isospectral Reduction

Bunimovich

Smith

Webb

2019

Applied Mathematics and Nonlinear Sciences

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The method of isospectral network reduction allows one the ability to reduce a network while preserving the network’s spectral structure. In this paper we describe a number of recent applications of the theory of isospectral reductions. This includes finding hidden structures, specifically latent symmetries, in networks, uncovering different network hierarchies, and simultaneously determining different network cores. We also specify how such reductions can be interpreted as dynamical systems and describe the type of dynamics such systems have. Additionally, we show how the recent theory of equitable decompositions can be paired with the method of isospectral reductions to decompose networks.

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Eigenvectors of isospectral graph transformations

Cited by 14 publications

References 8 publications

Designing pretty good state transfer via isospectral reductions

Designing pretty good state transfer via isospectral reductions

Hidden symmetries in real and theoretical networks

Finding Hidden Structures, Hierarchies, and Cores in Networks via Isospectral Reduction

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