###### Activity 4.5.7

The following Sage cell will generate the Markov chain for the modified Google matrix \(G\) if you simply enter the original Google matrix \(G\) in the appropriate line.

Consider the original Internet with three pages shown in Figure 4.5.7 and find the PageRank vector \(\xvec\) using the modified Google matrix in the Sage cell above. How does this modified PageRank vector compare to the vector we found using the original Google matrix \(G\text{?}\)

Find the modified PageRank vector for the Internet shown in Figure 4.5.9. Explain why this vector seems to be the correct one.

Find the modified PageRank vector for the Internet shown in Figure 4.5.10. Explain why this modified PageRank vector fixes the problem that appeared with the original PageRank vector.