Randomized algorithm to determine the eigenvector of a stochastic matrix with application to the PageRank problem

Consideration was given to estimation of the eigenvector corresponding to the greatest eigenvalue of a stochastic matrix. There exist numerous applications of this problem arising at ranking the results of search, coordination of the multiagent system actions, network control, and data analysis. The...

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Bibliographic Details
Published in:Automation and remote control 2011-02, Vol.72 (2), p.342-352
Main Authors: Nazin, A. V., Polyak, B. T.
Format: Article
Language:English
Subjects:
CAE) and Design
Calculus of Variations and Optimal Control
Optimization
Computer-Aided Engineering (CAD
Control
Mathematics
Mathematics and Statistics
Mechanical Engineering
Mechatronics
Robotics
Systems Theory
Topical Issue
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Summary:Consideration was given to estimation of the eigenvector corresponding to the greatest eigenvalue of a stochastic matrix. There exist numerous applications of this problem arising at ranking the results of search, coordination of the multiagent system actions, network control, and data analysis. The standard technique for its solution comes to the power method with an additional regularization of the original matrix. A new randomized algorithm was proposed, and a uniform—over the entire class of the stochastic matrices of a given size—upper boundary of the convergence rate was validated. It is given by {ie342-1}, where C is an absolute constant, N is size, and n is the number of iterations. This boundary seems promising because ln( N ) is smallish even for a very great size. The algorithm relies on the mirror descent method for the problems of convex stochastic optimization. Applicability of the method to the PageRank problem of ranking the Internet pages was discussed.
ISSN:0005-1179
1608-3032
DOI:10.1134/S0005117911020111