Related papers: A Superintroduction to Google Matrices for Undergr…
Eigenvalue interlacing is a versatile technique for deriving results in algebraic combinatorics. In particular, it has been successfully used for proving a number of results about the relation between the (adjacency matrix or Laplacian)…
HodgeRank generalizes ranking algorithms, e.g. Google PageRank, to rank alternatives based on real-world (often incomplete) data using graphs and discrete exterior calculus. It analyzes multipartite interactions on high-dimensional networks…
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…
Recent years have witnessed intense development of randomized methods for low-rank approximation. These methods target principal component analysis (PCA) and the calculation of truncated singular value decompositions (SVD). The present…
The largest eigenvalue of a matrix is always larger or equal than its largest diagonal entry. We show that for a large class of random Laplacian matrices, this bound is essentially tight: the largest eigenvalue is, up to lower order terms,…
The PageRank algorithm employed by Google quantifies the importance of each page by the link structure of the web. To reduce the computational burden the distributed randomized PageRank algorithms (DRPA) recently appeared in literature…
Ranking on large-scale graphs plays a fundamental role in many high-impact application domains, ranging from information retrieval, recommender systems, sports team management, biology to neuroscience and many more. PageRank, together with…
Random matrix theory allows one to deduce the eigenvalue spectrum of a large matrix given only statistical information about its elements. Such results provide insight into what factors contribute to the stability of complex dynamical…
Eigenvalue and eigenvector perturbation theory is a fundamental topic in several disciplines, including numerical linear algebra, quantum physics, and related fields. The central problem is to understand how the eigenvalues and eigenvectors…
We study the statistical properties of various directed networks using ranking of their nodes based on the dominant vectors of the Google matrix known as PageRank and CheiRank. On average PageRank orders nodes proportionally to a number of…
We describe an algorithm to compute the extremal eigenvalues and corresponding eigenvectors of a symmetric matrix by solving a sequence of Quadratic Binary Optimization problems. This algorithm is robust across many different classes of…
The eigenvalue problem plays a central role in linear algebra and its applications in control and optimization methods. In particular, many matrix decompositions rely upon computation of eigenvalue-eigenvector pairs, such as diagonal or…
In order to guarantee the downloading quality requirements of users and improve the stability of data transmission in a BitTorrent-like peer-to-peer file sharing system, this article deals with eigenproblems of addition-min algebras. First,…
In this paper new results on personalized PageRank are shown. We consider directed graphs that may contain dangling nodes. The main result presented gives an analytical characterization of all the possible values of the personalized…
The eigenvalues of matrices representing the structure of large-scale complex networks present a wide range of applications, from the analysis of dynamical processes taking place in the network to spectral techniques aiming to rank the…
Learning to rank is a supervised learning problem where the output space is the space of rankings but the supervision space is the space of relevance scores. We make theoretical contributions to the learning to rank problem both in the…
Very sparse random graphs are known to typically be singular (i.e., have singular adjacency matrix), due to the presence of "low-degree dependencies'' such as isolated vertices and pairs of degree-1 vertices with the same neighbourhood. We…
In this paper we analyze and solve eigenvalue programs, which consist of the task of minimizing a function subject to constraints on the "eigenvalues" of the decision variable. Here, by making use of the FTvN systems framework introduced by…
The aim of this note (as well as of the course itself) is to give a largely self-contained proof of two of the main results in the field of low-rank matrix recovery. This field aims for identification of low-rank matrices from only limited…
The leading eigenvalue $\lambda$ of the adjacency matrix of a graph exerts much influence on the behavior of dynamical processes on that graph. It is thus relevant to relate notions of the importance (specifically, centrality measures) of…