The uniqueness of multilinear PageRank vectors

The uniqueness of multilinear PageRank vectors is discussed, and the new uniqueness condition is given. The new results are better than the one given in the work of Gleich et al. published in SIAM J Matrix Anal Appl. 2015;36;1409‐1465. Numerical examples are given to demonstrate the new theoretical results.     

Extrapolation methods for fixed‐point multilinear PageRank computations

Nonnegative tensors arise very naturally in many applications that involve large and complex data flows. Due to the relatively small requirement in terms of memory storage and number of operations per step, the (shifted) higher order power method is one of the most commonly used technique for the computation of positive Z‐eigenvectors of this type of tensors. However, unlike the matrix case, the method may fail to converge even for irreducible tensors. Moreover, when it converges, its convergence rate can be very slow. These two drawbacks often make the computation of the eigenvectors demanding or unfeasible for large problems. In this work, we consider a particular class of nonnegative tensors associated with the multilinear PageRank modification of higher order Markov chains. Based on the simplified topological ε‐algorithm in its restarted form, we introduce an extrapolation‐based acceleration of power method type algorithms, namely, the shifted fixed‐point method and the inner‐outer method. The accelerated methods show remarkably better performance, with faster convergence rates and reduced overall computational time. Extensive numerical experiments on synthetic and real‐world datasets demonstrate the advantages of the introduced extrapolation techniques.     

A Power–Arnoldi algorithm for computing PageRank

The PageRank algorithm plays an important role in modern search engine technology. It involves using the classical power method to compute the principle eigenvector of the Google matrix representing the web link graph. However, when the largest eigenvalue is not well separated from the second one, the power method may perform poorly. This happens when the damping factor is sufficiently close to 1. Therefore, it is worth developing new techniques that are more sophisticated than the power method. The approach presented here, called Power–Arnoldi, is based on a periodic combination of the power method with the thick restarted Arnoldi algorithm. The justification for this new approach is presented. Numerical tests illustrate the efficiency and convergence behaviour of the new algorithm. Copyright © 2007 John Wiley & Sons, Ltd.     

Google矩阵和它的性质

网页等级(PageRank)是一个反映网页重要性的数值.当一个网页A连向另一个网页B的时候,A就等于给网页B投了有效的一票.一个网页接受的票越多,这个网页就越重要.同时,给网页B投票的网页本身的等级也决定了该选票的重要性.Google通过每张选票本身重要性和得票多少来计算一个网页的级别(重要性).Google的核心就是计算每一个网页的等级(即PageRank).本文主要介绍Google矩阵的定义和产生,解释PageRank的一些相关概念,证明Google矩阵及其第二特征值具有的一些性质,并简要介绍这些性质的应用.     

An Arnoldi-type algorithm for computing page rank

We consider the problem of computing PageRank. The matrix involved is large and cannot be factored, and hence techniques based on matrix-vector products must be applied. A variant of the restarted refined Arnoldi method is proposed, which does not involve Ritz value computations. Numerical examples illustrate the performance and convergence behavior of the algorithm. AMS subject classification (2000) 65F15, 65C40     

项目网络中考虑支配与扩散关系的组合决策

单纯侧重项目自身属性而不考虑项目关联性以及由项目衍生而来的技术、经验/信息扩散对项目组合决策时的影响,易导致决策偏差,低估具有潜在技术先导性项目的价值。对此,引用复杂网络理论,以项目关联性的视角,将项目间支配和扩散关系分别抽象为有向加权网络,运用K-shell分解方法构建项目组合网络中基于支配关系的项目影响力模型以及技术、经验/信息在项目间扩散传播的模型。然后,基于PageRank算法,综合考虑项目间支配与扩散关系,建立了项目优先级排序决策模型。最后,通过算例分析说明了该模型与算法的可行性与有效性,为企业项目组合决策提供了有益的参考。     

Personalized PageRank Clustering: A graph clustering algorithm based on random walks

Graph clustering has been an essential part in many methods and thus its accuracy has a significant effect on many applications. In addition, exponential growth of real-world graphs such as social networks, biological networks and electrical circuits demands clustering algorithms with nearly-linear time and space complexity. In this paper we propose Personalized PageRank Clustering (PPC) that employs the inherent cluster exploratory property of random walks to reveal the clusters of a given graph. We combine random walks and modularity to precisely and efficiently reveal the clusters of a graph. PPC is a top-down algorithm so it can reveal inherent clusters of a graph more accurately than other nearly-linear approaches that are mainly bottom-up. It also gives a hierarchy of clusters that is useful in many applications. PPC has a linear time and space complexity and has been superior to most of the available clustering algorithms on many datasets. Furthermore, its top-down approach makes it a flexible solution for clustering problems with different requirements.     

Relaxation methods for solving the tensor equation arising from the higher‐order Markov chains

In this paper, we propose several relaxation algorithms for solving the tensor equation arising from the higher‐order Markov chain and the multilinear PageRank. The semi‐symmetrization technique on the original equation is also employed to modify the proposed algorithms. The convergence analysis is given for the proposed algorithms. It is shown that the new algorithms are more efficient than the existing ones by some numerical experiments when relaxation parameters are chosen suitably.     

利用网络模型测定节点影响力与重要性

介绍了2014年美国大学生数学建模竞赛C题的背景与立意,针对6篇获得Outstanding奖的论文的解题思路与方法进行了归纳与总结,指出了学生答卷中的亮点与不足,并给出了建议和改进方案。     

Interdependence of Sectors of Economic Activities for World Countries from the Reduced Google Matrix Analysis of WTO Data

We apply the recently developed reduced Google matrix algorithm for the analysis of the OECD-WTO World Network of Economic Activities. This approach allows to determine interdependencies and interactions of economy sectors of several countries, including China, Russia and the USA, properly taking into account the influence of all the other world countries and their economic activities. Within this analysis, we also obtain the sensitivity of EU countries’ economies to the petroleum activity sector. We show that this approach takes into account the multiplicity of economical interactions between countries and activity sectors, thus providing a richer analysis compared to the usual export-import analysis.