Google's Pagerank and Beyond: The Science of Search Engine Rankings

By Amy N. Langville; Carl D. Meyer | Go to book overview

Chapter Nine
Accelerating the Computation of PageRank

People have a natural fascination with speed. Look around; articles abound on Nascar and the world’s fastest couple—Marion Jones and Tim Montgomery—speedboat racing and speed dating, fast food and the Concorde jet. So the interest in speeding up the computation of PageRank seems natural, but actually it’s essential because the PageRank computation by the standard power method takes days to converge. And the Web is growing rapidly, so days could turn into weeks if new methods aren’t discovered.

Because the classical power method is known for its slow convergence, researchers immediately looked to other solution methods. However, the size and sparsity of the web matrix create limitations on the solution methods and have caused the predominance of the power method. This restriction to the power method has forced new research on the often criticized power method and has resulted in numerous improvements to the vanilla-flavored power method that are tailored to the PageRank problem. Since 1998, the resurgence in work on the power method has brought exciting, innovative twists to the old, unadorned workhorse. As each iteration of the power method on a web-sized matrix is so expensive, reducing the number of iterations by a handful can save hours of computation. Some of the most valuable contributions have come from researchers at Stanford who have discovered several methods for accelerating the power method. There are really just two ways to reduce the work involved in any iterative method: either reduce the work per iteration or reduce the total number of iterations. These goals are often at odds with one another. That is, reducing the number of iterations usually comes at the expense of a slight increase in the work per iteration, and vice versa. As long as this overhead is minimal, the proposed acceleration is considered beneficial. In this chapter, we review three of the most successful methods for reducing the work associated with the PageRank vector.


9.1 AN ADAPTIVE POWER METHOD

The goal of the PageRank game is to compute πT, the stationary vector of G, or technically, the power iterates π(k)T such that ‖π(k)Tπ(k−1)T1 < τ, where τ is some acceptable convergence criterion. Suppose, for the moment, that we magically know πT from the start. We’d, of course, be done, problem solved. But, out of curiosity, let’s run the power method to see how far the iterates π(k)T are from the final answer πT. We want to know what kind of progress the power method is making throughout the iteration history. There are several ways to do this. You can take a macroscopic view and look at how far π(k)T, the current iterate, is from πT, the magical final answer, by computing ‖π(k)TπT1. By using the norm, the individual errors in each component are lumped into a single scalar which gives the aggregated error. The standard power method takes the macroscopic view at each iteration, using a convergence test that looks at an aggregated er-

-89-

Notes for this page

Add a new note
If you are trying to select text to create highlights or citations, remember that you must now click or tap on the first word, and then click or tap on the last word.
One moment ...
Default project is now your active project.
Project items
Notes
Cite this page

Cited page

Style
Citations are available only to our active members.
Buy instant access to cite pages or passages in MLA 8, MLA 7, APA and Chicago citation styles.

(Einhorn, 1992, p. 25)

(Einhorn 25)

(Einhorn 25)

1. Lois J. Einhorn, Abraham Lincoln, the Orator: Penetrating the Lincoln Legend (Westport, CT: Greenwood Press, 1992), 25, http://www.questia.com/read/27419298.

Note: primary sources have slightly different requirements for citation. Please see these guidelines for more information.

Cited page

Bookmark this page
Google's Pagerank and Beyond: The Science of Search Engine Rankings
Table of contents

Table of contents

Settings

Settings

Typeface
Text size Smaller Larger Reset View mode
Search within

Search within this book

Look up

Look up a word

  • Dictionary
  • Thesaurus
Please submit a word or phrase above.
Print this page

Print this page

Why can't I print more than one page at a time?

Help
Full screen
Items saved from this book
  • Bookmarks
  • Highlights & Notes
  • Citations
/ 224

matching results for page

    Questia reader help

    How to highlight and cite specific passages

    1. Click or tap the first word you want to select.
    2. Click or tap the last word you want to select, and you’ll see everything in between get selected.
    3. You’ll then get a menu of options like creating a highlight or a citation from that passage of text.

    OK, got it!

    Cited passage

    Style
    Citations are available only to our active members.
    Buy instant access to cite pages or passages in MLA 8, MLA 7, APA and Chicago citation styles.

    "Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn, 1992, p. 25).

    "Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn 25)

    "Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn 25)

    "Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences."1

    1. Lois J. Einhorn, Abraham Lincoln, the Orator: Penetrating the Lincoln Legend (Westport, CT: Greenwood Press, 1992), 25, http://www.questia.com/read/27419298.

    Cited passage

    Thanks for trying Questia!

    Please continue trying out our research tools, but please note, full functionality is available only to our active members.

    Your work will be lost once you leave this Web page.

    Buy instant access to save your work.

    Already a member? Log in now.

    Search by... Author
    Show... All Results Primary Sources Peer-reviewed

    Oops!

    An unknown error has occurred. Please click the button below to reload the page. If the problem persists, please try again in a little while.