A Sharp PageRank Algorithm with Applications to Edge Ranking and Graph Sparsification

  • Fan Chung
  • Wenbo Zhao
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6516)


We give an improved algorithm for computing personalized PageRank vectors with tight error bounds which can be as small as Ω(n − p ) for any fixed positive integer p. The improved PageRank algorithm is crucial for computing a quantitative ranking of edges in a given graph. We will use the edge ranking to examine two interrelated problems – graph sparsification and graph partitioning. We can combine the graph sparsification and the partitioning algorithms using PageRank vectors to derive an improved partitioning algorithm.


Weighted Graph Network Design Problem Graph Partitioning Sparse Graph Approximation Guarantee 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Achlioptas, D.: Database-friendly random projections. In: PODS 2001, pp. 274–281 (2001)Google Scholar
  2. 2.
    Arora, S., Kale, S.: A combinatorial, primal-dual approach to semidefinite programs. In: STOC 2007, pp. 227–236 (2007)Google Scholar
  3. 3.
    Andersen, R., Chung, F., Lang, K.: Local graph partitioning using pagerank vectors. In: FOCS 2006, pp. 475–486 (2006)Google Scholar
  4. 4.
    Andersen, R., Peres, Y.: Finding sparse cuts locally using evolving sets. In: STOC 2009, pp. 235–244 (2009)Google Scholar
  5. 5.
    Arora, S., Hazan, E., Kale, S.: \(\Theta(\sqrt{\log n})\) approximation to sparsest cut in \(\tilde{O }(n^2)\) time. In: FOCS 2004, pp. 238–247 (2004)Google Scholar
  6. 6.
    Batson, J., Spielman, D.A., Srivastava, N.: Twice-Ramanujan sparsifiers. In: STOC 2009, pp. 255–262 (2009)Google Scholar
  7. 7.
    Benczúr, A.A., Karger, D.R.: Approximating s-t minimum cuts in \(\tilde{O}(n^2)\) time. In: STOC 1996, pp. 47–55 (1996)Google Scholar
  8. 8.
    Berkhin, P.: Bookmark-coloring approach to personalized pagerank computing. Internet Mathematics 3(1), 41–62 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Brin, S., Page, L.: The anatomy of a large-scale hypertextual Web search engine. Computer Networks and ISDN Systems 30(1-7), 107–117 (1998)CrossRefGoogle Scholar
  10. 10.
    Chung, F.: Spectal Graph Theory. AMS Publication, Providence (1997)Google Scholar
  11. 11.
    Chung, F., Yau, S.-T.: Discrete Green’s Functions. Journal of Combinatorial Theory, Series A 91(1-2), 191–214 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Green, G.: An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism, Nottingham (1828)Google Scholar
  13. 13.
    Haveliwala, H.: Topic-sensitive pagerank: A context-sensitive ranking algorithm for web search. IEEE Trans. Knowl. Data Eng. 15(4), 784–796 (2003)CrossRefGoogle Scholar
  14. 14.
    Jeh, G., Widom, J.: Scaling personalized web search. In: WWW 2003, pp. 271–279 (2003)Google Scholar
  15. 15.
    Karger, D.R.: Random sampling in cut, flow, and network design problems. In: STOC 1994, pp. 648–657 (1994)Google Scholar
  16. 16.
    Karger, D.R.: Using randomized sparsification to approximate minimum cuts. In: SODA 1994, pp. 424–432 (1994)Google Scholar
  17. 17.
    Karger, D.R.: Minimum cuts in near-linear time. JACM 47(1), 46–76 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Loväsz, L.: Random walks on graphs: A survey. Combinatorics, Paul Erdös is Eighty 2, 1–46 (1993)Google Scholar
  19. 19.
    Lovász, L., Simonovits, M.: The mixing rate of Markov chains, an isoperimetric inequality, and computing the volume. In: FOCS 1990, pp. 346–354 (1990)Google Scholar
  20. 20.
    Orecchia, L., Schulman, L.J., Vazirani, U.V., Vishnoi, N.K.: On partitioning graphs via single commodity flows. In: STOC 2008, pp. 461–470 (2008)Google Scholar
  21. 21.
    Page, L., Brin, S., Motwani, R., Winograd, T.: The pagerank citation ranking: Bringing order to the web, Technical report, Stanford Digital Library Technologies Project (1998)Google Scholar
  22. 22.
    Rudelson, M., Vershynin, R.: Sampling from large matrices: An approach through geometric functional analysis. Journal of the ACM 54(4) (2007)Google Scholar
  23. 23.
    Spielman, D.A., Teng, S.-H.: Spectral partitioning works: Planar graphs and finite element meshes. In: FOCS 1996, pp. 96–105 (1996)Google Scholar
  24. 24.
    Spielman, D.A., Teng, S.-H.: Nearly-linear time algorithms for graph partitioning, graph sparsification, and solving linear systems. In: STOC 2004, pp. 81–90 (2004)Google Scholar
  25. 25.
    Spielman, D.A., Srivastava, N.: Graph sparsification by effective resistances. In: STOC 2008, pp. 563–568 (2008)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Fan Chung
    • 1
  • Wenbo Zhao
    • 1
  1. 1.University of CaliforniaSan Diego, La JollaUSA

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