Abstract
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.
Highlights
Licensee MDPI, Basel, Switzerland.With the rapid development of the Internet, web search engines become very popular for information retrieval
The PageRank model was developed by the Google team to rank the importance of Web pages based on the frequency of page visits recorded by a random user who keeps browsing the World Wide Web with an equal
Probability of choosing the hyperlinks on each page. Speaking, it requires the computation of the stationary distribution of this Markov random walk on the linking structure of pages, the values within the distribution represent the frequency of visits to each Web page
Summary
Licensee MDPI, Basel, Switzerland.With the rapid development of the Internet, web search engines become very popular for information retrieval. Web pages matching the search query, it is necessary to rank higher the most important pages to make this tool practical. Fur this purpose, the PageRank model was developed by the Google team to rank the importance of Web pages based on the frequency of page visits recorded by a random user who keeps browsing the World Wide Web with an equal. Probability of choosing the hyperlinks on each page Speaking, it requires the computation of the stationary distribution of this Markov random walk on the linking structure of pages, the values within the distribution represent the frequency of visits to each Web page. The transition probability matrix P ∈ Rn×n of this random walk process is defined as
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