1-Let's consider a web with 5 pages which respectively contain a link to each other page [1]. Suppose, further, that one of the pages contains a link to another page which itself contains no link. The web therefore contains a total of 6 pages. Using the simplified version of PageRank, calculate the PageRank of each page with the convention that the sum of the PageRanks must be 1. Justify your answer by providing the matrices and vectors used in your calculation. However, you do not have to do the calculation by hand: you are allowed to use a program for the calculations.

What can you say about the case where there are not 5 pages, but 50 pages in the clique?

Tips. Page X has links to all other pages except itself. Page Y has no link. All other pages have 4 links.

2-Use the same configuration as in question 1. We have 5 pages which respectively contain a link to every other page and where page X contains a link to page Y which contains no link.

Suppose a user searches for a certain keyword that is only found in page X. Calculate the authority and hub coefficients for each page. Justify your answer by providing the matrix and the vectors used for your calculations. Again, you don't have to do the math by hand. What does the HITS algorithm "recommend" to the user?

Tips. Check that your matrix A is correct: Ai,j has the value 1 if and only if page j has a link to page i. Thus, the sum of the components of a column gives the number of links present on the corresponding page. If a page has no links, then the corresponding column should only contain zeros.