Euler-Richardson method preconditioned by weakly stochastic matrix algebras: a potential contribution to pagerank computation
Euler-Richardson method preconditioned by weakly stochastic matrix algebras: a potential contribution to pagerank computation
Let S be a column stochastic matrix with at least one full row. Then S describes a Pagerank-like random walk since the computation of the Perron vector x of S can be tackled by solving a suitable M-matrix linear system M x = y, where M = I − τ A, A is a column stochastic matrix and τ is a positive coefficient smaller than one. The Pagerank centrality index on graphs is a relevant example where these two formulations appear. Previous investigations have shown that the Euler-Richardson (ER) method can be considered in order to approach the Pagerank computation problem by means of preconditioning strategies. In this work, it is observed indeed that the classical power method can be embedded into the ER scheme, through a suitable simple preconditioner. Therefore, a new preconditioner is proposed based on fast Householder transformations and the concept of low complexity weakly stochastic algebras, which gives rise to an effective alternative to the power method for large-scale sparse problems. Detailed mathematical reasonings for this choice are given and the convergence properties discussed. Numerical tests performed on real-world datasets are presented, showing the advantages given by the use of the proposed Householder-Richardson method.
Matrix algebras, Nonnegative matrices, Pagerank, Preconditioning
254-272
Cipolla, S.
373fdd4b-520f-485c-b36d-f75ce33d4e05
Di Fiore, C.
f43c7f86-7a2e-474a-ad41-3ff3797128bc
Tudisco, F.
3c9b5744-c949-402e-ad23-b9f053c62e37
6 February 2017
Cipolla, S.
373fdd4b-520f-485c-b36d-f75ce33d4e05
Di Fiore, C.
f43c7f86-7a2e-474a-ad41-3ff3797128bc
Tudisco, F.
3c9b5744-c949-402e-ad23-b9f053c62e37
Cipolla, S., Di Fiore, C. and Tudisco, F.
(2017)
Euler-Richardson method preconditioned by weakly stochastic matrix algebras: a potential contribution to pagerank computation.
The Electronic Journal of Linear Algebra, 32, , [20].
(doi:10.13001/1081-3810.3343).
Abstract
Let S be a column stochastic matrix with at least one full row. Then S describes a Pagerank-like random walk since the computation of the Perron vector x of S can be tackled by solving a suitable M-matrix linear system M x = y, where M = I − τ A, A is a column stochastic matrix and τ is a positive coefficient smaller than one. The Pagerank centrality index on graphs is a relevant example where these two formulations appear. Previous investigations have shown that the Euler-Richardson (ER) method can be considered in order to approach the Pagerank computation problem by means of preconditioning strategies. In this work, it is observed indeed that the classical power method can be embedded into the ER scheme, through a suitable simple preconditioner. Therefore, a new preconditioner is proposed based on fast Householder transformations and the concept of low complexity weakly stochastic algebras, which gives rise to an effective alternative to the power method for large-scale sparse problems. Detailed mathematical reasonings for this choice are given and the convergence properties discussed. Numerical tests performed on real-world datasets are presented, showing the advantages given by the use of the proposed Householder-Richardson method.
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Published date: 6 February 2017
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∗Received by the editor on June 29, 2016. Accepted for publication on April 20, 2017. Handling Editor: Dario Bini. The work has been partially supported by INdAM-GNCS and, for F.T., by the ERC grant NOLEPRO. †Department of Mathemathics, University of Rome “Tor Vergata”, Rome, Italy (stefano.cipolla87@gmail.com). ‡Department of Mathematics, University of Padua, Padua, Italy.
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© 2017, International Linear Algebra Society. All rights reserved.
Keywords:
Matrix algebras, Nonnegative matrices, Pagerank, Preconditioning
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Local EPrints ID: 485623
URI: http://eprints.soton.ac.uk/id/eprint/485623
PURE UUID: a767fafd-80dd-4f01-bab9-b5fd0e3fbc98
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Date deposited: 12 Dec 2023 17:35
Last modified: 18 Mar 2024 04:17
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Author:
S. Cipolla
Author:
C. Di Fiore
Author:
F. Tudisco
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