Information Retrieval

, Volume 15, Issue 6, pp 503–526 | Cite as

Lumping algorithms for computing Google’s PageRank and its derivative, with attention to unreferenced nodes

  • Qing Yu
  • Zhengke Miao
  • Gang Wu
  • Yimin WeiEmail author


In this paper, we introduce five type nodes for lumping the Web matrix, and give a unified presentation of some popular lumping methods for PageRank. We show that the PageRank problem can be reduced to solving the PageRank corresponding to the strongly non-dangling and referenced nodes, and the full PageRank vector can be easily derived by some recursion formulations. Our new lumping strategy can reduce the original PageRank problem to a much smaller one, and it is much cheaper than the recursively reordering scheme. Furthermore, we discuss sensitivity of the PageRank vector, and present a lumping algorithm for computing its first order derivative. Numerical experiments show that the new algorithms are favorable when the matrix is large and the damping factor is high.


Google PageRank Web information retrieval Dangling nodes Unreferenced nodes 



We would like to express our sincere thanks to Professor Justin Zobel and two anonymous reviewers for their invaluable suggestions that make us greatly improve the representation of this paper. Meanwhile, we are grateful to Dr. David Gleich and Professor Tim Davis for data files of the Web matrices, and to Dr. Amy N. Langville for providing us MATLAB codes of the recursively reordering algorithm (Langville and Meyer 2006a). Moreover, we thank Dr. Xing-hua Shi for MATLAB codes of Algorithms 1 and 2. Finally, Gang Wu and Qing Yu would like to thank School of Mathematical Sciences of Xuzhou Normal University for the use of their facilities during the development of this project. Zhengke Miao is supported by the National Natural Science Foundation of China under grants 10871166 and 11171288. Gang Wu is supported by the National Science Foundation of China under grants 10901132 and 11171289, the Qing-Lan Project of Jiangsu Province, and the 333 Project of Jiangsu Province. Yimin Wei is supported by the National Natural Science Foundation of China under grant 10871051, Doctoral Program of the Ministry of Education under grant 20090071110003, 973 Program Project under grant 2010CB327900, Shanghai Education Committee under Dawn Project 08SG01 and Shanghai Science & Technology Committee.


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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of Arts and SciencesXuzhou Higher Normal SchoolXuzhouPeople’s Republic of China
  2. 2.School of Mathematical SciencesXuzhou Normal UniversityXuzhouPeople’s Republic of China
  3. 3.School of Mathematical Sciences and Shanghai Key Laboratory of Contemporary Applied MathematicsFudan UniversityShanghaiPeople’s Republic of China

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