Activity 4.5.6
Consider the Internet with eight web pages, shown in Figure 4.5.8.
Construct the Google matrix \(G\) for this Internet. Then use a Markov chain to find the steadystate PageRank vector \(\xvec\text{.}\)
What does this vector tell us about the relative quality of the pages in this Internet? Which page has the highest quality and which the lowest?

Now consider the Internet with five pages, shown in Figure 4.5.9.
What happens when you begin the Markov chain with the vector \(\xvec_0=\fivevec{1}{0}{0}{0}{0}\text{?}\) Explain why this behavior is consistent with the PerronFrobenius theorem.
What do you think the PageRank vector for this Internet should be? Is any one page of a higher quality than another?

Now consider the Internet with eight web pages, shown in Figure 4.5.10.
Notice that this version of the Internet is identical to the first one that we saw in this activity, except that a single link from page 7 to page 1 has been removed. We can therefore find its Google matrix \(G\) by slightly modifying the earlier matrix.
What is the longterm behavior of a Markov chain defined by \(G\) and why is this behavior not desirable? How is this behavior consistent with the PerronFrobenius theorem?