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.

Iterative Methods for the Computation of the Perron Vector of Adjacency Matrices

The power method is commonly applied to compute the Perron vector of large adjacency matrices. Blondel et al. [SIAM Rev. 46, 2004] investigated its performance when the adjacency matrix has multiple eigenvalues of the same magnitude. It is well known that the Lanczos method typically requires fewer iterations than the power method to determine eigenvectors with the desired accuracy. However, the Lanczos method demands more computer storage, which may make it impractical to apply to very large problems. The present paper adapts the analysis by Blondel et al. to the Lanczos and restarted Lanczos methods. The restarted methods are found to yield fast convergence and to require less computer storage than the Lanczos method. Computed examples illustrate the theory presented. Applications of the Arnoldi method are also discussed.

Dynamic Mode Decomposition of Elliptical Liquid Jets in Crossflow

Dynamics of round and elliptical liquid jets in subsonic crossflow is studied using high-speed imaging technique. The experiments are performed at constant gaseous weber number and liquid-gas momentum flux ratio of 6.45 and 17.87 respectively, with orifices of different aspect ratios having an equivalent diameter of 0.43 mm. All cases are carried out inside an open loop subsonic wind tunnel with a test section of 100*100*750 mm. For each case, dynamic modes are generated directly from the snapshots using a variant of Arnoldi method known as the dynamic mode decomposition (DMD). DMD results indicate that elliptical liquid jets have more small-scaled patterns with higher frequencies compared to the case of round liquid jets. As the first attempt to investigate the dynamics of elliptical liquid jets in crossflow, present work captures the dominant spatio-temporal structures. It is also found that the orifice aspect ratio can alter the jet wavelengths remarkably. The extracted data of this work can provide beneficial information on the behaviour of elliptical liquid jets exposed to the gas crossflow in the enhanced capillary breakup regime.