# 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-

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