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