A Preconditioned Variant of the Refined Arnoldi Method for Computing PageRank Eigenvectors

The PageRank model computes the stationary distribution of a Markov random walk on the linking structure of a network, and it uses the values within to represent the importance or centrality of each node. This model is first proposed by Google for ranking web pages, then it is widely applied as a centrality measure for networks arising in various fields such as in chemistry, bioinformatics, neuroscience and social networks. For example, it can measure the node centralities of the gene-gene annotation network to evaluate the relevance of each gene with a certain disease. The networks in some fields including bioinformatics are undirected, thus the corresponding adjacency matrices are symmetry. Mathematically, the PageRank model can be stated as finding the unit positive eigenvector corresponding to the largest eigenvalue of a transition matrix built upon the linking structure. With rapid development of science and technology, the networks in real applications become larger and larger, thus the PageRank model always desires numerical algorithms with reduced algorithmic or memory complexity. In this paper, we propose a novel preconditioning approach for solving the PageRank model. This approach transforms the original PageRank eigen-problem into a new one that is more amenable to solve. We then present a preconditioned version of the refined Arnoldi method for solving this model. We demonstrate theoretically that the preconditioned Arnoldi method has higher execution efficiency and parallelism than the refined Arnoldi method. In plenty of numerical experiments, this preconditioned method exhibits noticeably faster convergence speed over its standard counterpart, especially for difficult cases with large damping factors. Besides, this superiority maintains when this technique is applied to other variants of the refined Arnoldi method. Overall, the proposed technique can give the PageRank model a faster solving process, and this will possibly improve the efficiency of researches, engineering projects and services where this model is applied.

On the spectral radii and principal eigenvectors of uniform hypergraphs

Let [Formula: see text] be a connected [Formula: see text]-uniform hypergraph. The unique positive eigenvector [Formula: see text] with [Formula: see text] corresponding to spectral radius [Formula: see text] is called the principal eigenvector of [Formula: see text]. In this paper, we present some lower bounds for the spectral radius [Formula: see text] and investigate the bounds of entries of the principal eigenvector of [Formula: see text].

The local spectral radius of a nonnegative orbit of compact linear operators

We consider orbits of compact linear operators in a real Banach space which are nonnegative with respect to the partial ordering induced by a given cone. The main result shows that under a mild additional assumption the local spectral radius of a nonnegative orbit is an eigenvalue of the operator with a positive eigenvector.

Combinatorial vs. Algebraic Characterizations of Completely Pseudo-Regular Codes

Given a simple connected graph $\Gamma$ and a subset of its vertices $C$, the pseudo-distance-regularity around $C$ generalizes, for not necessarily regular graphs, the notion of completely regular code. We then say that $C$ is a completely pseudo-regular code. Up to now, most of the characterizations of pseudo-distance-regularity has been derived from a combinatorial definition. In this paper we propose an algebraic (Terwilliger-like) approach to this notion, showing its equivalence with the combinatorial one. This allows us to give new proofs of known results, and also to obtain new characterizations which do not depend on the so-called $C$-spectrum of $\Gamma$, but only on the positive eigenvector of its adjacency matrix. Along the way, we also obtain some new results relating the local spectra of a vertex set and its antipodal. As a consequence of our study, we obtain a new characterization of a completely regular code $C$, in terms of the number of walks in $\Gamma$ with an endvertex in $C$.

Positive root vectors

As a generalisation of the well-known result of Perron and Frobenius, it was shown by Rothblum [13] and independently by Richman and Schneider [12] that every nonzero matrix with non-negative entries has a basis of the root space corresponding to the maximal eigenvalue, represented by root vectors with non-negative entries. Krein and Rutman [9] showed that a positive compact nonquasinilpotent operator on a Banach lattice has a positive eigenvector corresponding to its spectral radius. As an extension of both results, we give sufficient conditions on such an operator in order that its spectral subspace corresponding to its spectral radius has a basis made exclusively of positive root vectors.

A dynamical systems proof of the Krein–Rutman Theorem and an extension of the Perron Theorem

SynopsisIn this paper we establish Perron and Krein–Rutman-like theorems for an operator mapping a cone into the interior of the cone, by considering the discrete dynamical system for the induced operator on the projective space (= sphere). Existence of a positive eigenvector reduces to showing that the ω-limit set of the induced operator consists of a single equilibrium. A special feature of our approach is that the convexity of the cone is needed only for establishing the non-emptiness of the w-limit set. This allows us in finite dimensions to establish an abstract Perron Theorem for non-convex cones.