Abstract Network scientists have shown that there is great value in studying pairwise interactions between components a system. From linear algebra point of view, this involves defining and evaluating functions the associated adjacency matrix. Recent work indicates are further benefits from accounting directly for higher order interactions, notably through hypergraph representation where an edg...

A powerful vertex-partitioning algorithm is developed and applied for vertex partitioning of graphs of chemical and spectroscopic interest. The codes developed on the basis of these algorithms are tested and compared for performance with other methods based on the Morgan algorithm and the principal eigenvector algorithm based on the Givens-Householder method. The newly developed algorithm and c...

The angular synchronization problem is to obtain an accurate estimation (up to a constant additive phase) for a set of unknown angles θ1, . . . , θn from m noisy measurements of their offsets θi − θj mod 2π. Of particular interest is angle recovery in the presence of many outlier measurements that are uniformly distributed in [0, 2π) and carry no information on the true offsets. We introduce an...

This paper aims to address two fundamental challenges arising in eigenvector estimation and inference for a low-rank matrix from noisy observations: 1) how estimate an unknown when the eigen-gap (i.e. spacing between associated eigenvalue rest of spectrum) is particularly small; 2) perform on linear functionals eigenvector—a sort “fine-grained” statistical reasoning that goes far beyond usual <...

The spectral properties of the adjacency matrix, in particular its largest eigenvalue and the associated principal eigenvector, dominate many structural and dynamical properties of complex networks. Here we focus on the localization properties of the principal eigenvector in real networks. We show that in most cases it is either localized on the star defined by the node with largest degree (hub...

We address the homogenization of an eigenvalue problem for the neutron transport equation in a periodic heterogeneous domain, modeling the criticality study of nuclear reactor cores. We prove that the neutron flux, corresponding to the first and unique positive eigenvector, can be factorized in the product of two terms, up to a remainder which goes strongly to zero with the period. One term is ...

This paper establishes converses to the well-known result: for any vector ũ such that the sine of the angle sin θ(u, ũ) = O( ), we have ρ(ũ) def = ũ∗Aũ ũ∗ũ = λ+O( ), where λ is an eigenvalue and u is the corresponding eigenvector of a Hermitian matrix A, and “∗” denotes complex conjugate transpose. It shows that if ρ(ũ) is close to A’s largest eigenvalue, then ũ is close to the corresponding ei...

The least-squares method can be used to approximate an eigenvector for a matrix when only an approximation is known for the corresponding eigenvalue. In this paper, this technique is analyzed and error estimates are established proving that if the error in the eigenvalue is sufficiently small, then the error in the approximate eigenvector produced by the least-squares method is also small. Also...

In this note we prove a conjecture from [DFJMN] on the asymptotics of the composition of n quantum vertex operators for the quantum affine algebra Uq(ŝl2), as n goes to ∞. For this purpose we define and study the leading eigenvalue and eigenvector of the product of two components of the quantum vertex operator. This eigenvector and the corresponding eigenvalue were recently computed by M.Jimbo....

The problem of partitioning a graph such that the number of edges incident to vertices in diierent partitions is minimized, arises in many contexts. Some examples include its recursive application for minimizing ll-in in matrix factorizations and load-balancing for parallel algorithms. Spectral graph partitioning algorithms partition a graph using the eigenvector associated with the second smal...