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

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

Chapter Six
The Sensitivity of PageRank

Psychologists say that a person’s sensitivities give insights into the personality. They say sensitivity to name-calling might indicate a maligned childhood. Sensitivity to injury, a pampered, spoiled upbringing; a short fuse with the boss, anger toward parents, and so on. It seems the same is true for the PageRank model. The sensitivities of the PageRank model reveal quite a bit about the popularity scores it produces. For example, when α gets very close to 1 (its upperbound), it seems to really get PageRank’s goat. In this chapter, we explain exactly how PageRank reacts to changes like this.

In fact, the sensitivity of the PageRank vector can be analyzed by examining each parameter of the Google matrix G separately. In Chapter 5, we emphasized G’s dependence on three specific parameters: the scaling parameter α, the hyperlink matrix H, and the personalization vector vT. We discuss the effect of each of these on the PageRank vector in turn in this chapter.


6.1 SENSITIVITY WITH RESPECT TO α

In this section, we use the derivative to show the effect of changes in α on πT. The derivative is a classical tool for answering questions of sensitivity. The derivative of πT with respect to α, written dπT (α)/dα, tells how much the elements in the PageRank vector πT vary when α varies slightly. If element j of dπT (α)/dα, denoted j(α)/dα, is large in magnitude, then we can conclude that as α increases slightly, πj (the PageRank for page Pj) is very sensitive to small changes in α. The signs of the derivatives also give important information; if j(α)/dα > 0, then small increases in α imply that the PageRank for Pj will increase. And similarly, j(α)/dα < 0 implies the PageRank decreases. It is important to remember that dπT (α)/dα is only an approximation of how elements in πT change when α changes, and does not describe exactly how they change. Nevertheless, analyzing this derivative can reveal important information about how changes in α affect πT.

Even though the parameter α is usually set to .85, it can theoretically vary between 0 <α< 1. Of course, G depends on α, and so, G(α) = αS + (1 − α)evT. The question about how sensitive πT (α) is to changes in α can be answered precisely if the derivative dπT (α)/dα, which gives the rate of change of πT (α) with respect to small changes in α, can be evaluated. But before attempting to differentiate we should be sure that this derivative is well defined. The distribution πT (α) is a left-hand eigenvector for G(α), but eigenvector components need not be differentiable (or even continuous) functions of the entries of G(α) [127, p. 497], so the existence of dπT (α)/dα is not a slam dunk. The following theorem provides what is needed. (We have postponed all proofs in this chapter until the last section, section 6.5.)

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