Activity4.5.6

Consider the Internet with eight web pages, shown in Figure 8.

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Figure4.5.8A simple model of the Internet with eight web pages.

  1. Construct the Google matrix \(G\) for this Internet. Then use a Markov chain to find the steady-state PageRank vector \(\xvec\text{.}\)

  2. 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?

  3. Now consider the Internet with five pages, shown in Figure 9.

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    Figure4.5.9A model of the Internet with five web pages.

    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 Perron-Frobenius theorem.

  4. What do you think the PageRank vector for this Internet should be? Is any one page of a higher quality than another?

  5. Now consider the Internet with eight web pages, shown in Figure 10.

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    Figure4.5.10Another model of the Internet with eight web pages.

    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 long-term behavior of a Markov chain defined by \(G\) and why is this behavior not desirable? How is this behavior consistent with the Perron-Frobenius theorem?

in-context