A multi-class approach for ranking graph nodes: models and experiments with incomplete data

After the phenomenal success of the PageRank algorithm, many researchers have extended the PageRank approach to ranking graphs with richer structures beside the simple linkage structure. In some scenarios we have to deal with multi-parameters data where each node has additional features and there are relationships between such features. This paper stems from the need of a systematic approach when dealing with multi-parameter data. We propose models and ranking algorithms which can be used with little adjustments for a large variety of networks (bibliographic data, patent data, twitter and social data, healthcare data). In this paper we focus on several aspects which have not been addressed in the literature: (1) we propose different models for ranking multi-parameters data and a class of numerical algorithms for efficiently computing the ranking score of such models, (2) by analyzing the stability and convergence properties of the numerical schemes we tune a fast and stable technique for the ranking problem, (3) we consider the issue of the robustness of our models when data are incomplete. The comparison of the rank on the incomplete data with the rank on the full structure shows that our models compute consistent rankings whose correlation is up to 60% when just 10% of the links of the attributes are maintained suggesting the suitability of our model also when the data are incomplete

Estimating an Eigenvector by the Power Method with a Random Start

This paper addresses the problem of approximating an eigenvector belonging to the largest eigenvalue of a symmetric positive definite matrix by the power method. We assume that the starting vector is randomly chosen with uniform distribution over the unit sphere. This paper provides lower and upper as well as asymptotic bounds on the randomized error in the Lp sense, p ∈[1,+ ∞]. We prove that it is impossible to achieve sharp bounds that are independent of the ratio between the two largest eigenvalues. This should be contrasted to the problem of approximating the largest eigenvalue, for which Kuczyński and Woźniakowski [SIAM J. Matrix Anal. Appl., 13 (1992), pp. 1094--1122] proved that it is possible to bound the randomized error at the kth step with a quantity that depends only on k and on the size of the matrix. We prove that the rate of convergence depends on the ratio of the two largest eigenvalues, on their multiplicities, and on the particular norm. The rate of convergence is at most linear in the ratio of the two largest eigenvalues

Compression of unitary rank--structured matrices to CMV-like shape with an application to polynomial rootfinding

This paper is concerned with the reduction of a unitary matrix U to CMV-like shape. A Lanczos--type algorithm is presented which carries out the reduction by computing the block tridiagonal form of the Hermitian part of U, i.e., of the matrix U+U^H. By elaborating on the Lanczos approach we also propose an alternative algorithm using elementary matrices which is numerically stable. If U is rank--structured then the same property holds for its Hermitian part and, therefore, the block tridiagonalization process can be performed using the rank--structured matrix technology with reduced complexity. Our interest in the CMV-like reduction is motivated by the unitary and almost unitary eigenvalue problem. In this respect, finally, we discuss the application of the CMV-like reduction for the design of fast companion eigensolvers based on the customary QR iteration

Block Tridiagonal Reduction of Perturbed Normal and Rank Structured Matrices

It is well known that if a matrix $A\in\mathbb C^{n\times n}$ solves the matrix equation $f(A,A^H)=0$, where $f(x, y)$ is a linear bivariate polynomial, then $A$ is normal; $A$ and $A^H$ can be simultaneously reduced in a finite number of operations to tridiagonal form by a unitary congruence and, moreover, the spectrum of $A$ is located on a straight line in the complex plane. In this paper we present some generalizations of these properties for almost normal matrices which satisfy certain quadratic matrix equations arising in the study of structured eigenvalue problems for perturbed Hermitian and unitary matrices.Comment: 13 pages, 3 figure

A CMV--based eigensolver for companion matrices

In this paper we present a novel matrix method for polynomial rootfinding. By exploiting the properties of the QR eigenvalue algorithm applied to a suitable CMV-like form of a companion matrix we design a fast and computationally simple structured QR iteration.Comment: 14 pages, 4 figure

Counting Fiedler pencils with repetitions

We introduce a new notation based on diagrams to deal with Fiedler pencils with repetitions (FPR), and use it to solve several counting problems. In particular, we give explicit recurrences to count the number of FPRs of a given degree d, the number of symmetric, palindromic and antipalindromic ones (where the latter two structures are intended in the sense of [5]). We relate these structures to the presence of symmetries in the associated diagrams

An implicit multishift QRQR-algorithm for Hermitian plus low rank matrices

Hermitian plus possibly non-Hermitian low rank matrices can be efficiently reduced into Hessenberg form. The resulting Hessenberg matrix can still be written as the sum of a Hermitian plus low rank matrix. In this paper we develop a new implicit multishift $QR$-algorithm for Hessenberg matrices, which are the sum of a Hermitian plus a possibly non-Hermitian low rank correction.The proposed algorithm exploits both the symmetry and low rank structure to obtain a $QR$-step involving only \mO{n} floating point operations instead of the standard \mO{n^2} operations needed for performing a $QR$-step on a Hessenberg matrix. The algorithm is based on a suitable\mO{n} representation of the Hessenberg matrix. The low rank parts present in both the Hermitian and low rank part of the sum are compactly stored by a sequence of Givens transformations and few vectors.Due to the new representation, we cannot apply classical deflation techniques for Hessenberg matrices. A new, efficient technique is developed to overcome this problem.Some numerical experiments based on matrices arising in applications are performed.The experiments illustrate effectiveness and accuracy of both the $QR$-algorithm and the newly developed deflation technique. <br /