Suppose we compute PageRank with a β of 0.7, and we introduce the additional constraint that the sum of the PageRanks of the three pages must be 3, to handle the problem that otherwise any multiple of a solution will also be a solution. Compute the PageRanks a, b, and c of the three pages A, B, and C, respectively.
To compute the PageRanks of the three pages (A, B, and C) with the given constraint, we need to follow the steps of the PageRank algorithm. Here's how you can do it:
1. Start by assigning an initial PageRank value to each page. Since we have three pages, we can give them equal initial PageRanks. Let's say a = b = c = 1.
2. Calculate the outgoing links of each page. In this case, we assume that each page has one outgoing link to the other two pages. So, page A has 2 outgoing links (to B and C), and pages B and C both have 1 outgoing link (to the other page).
3. Update the PageRank values iteratively until they converge. We can use the formula:
PR(A) = (1 - β) + β * (PR(B)/L(B) + PR(C)/L(C))
PR(B) = (1 - β) + β * (PR(A)/L(A))
PR(C) = (1 - β) + β * (PR(A)/L(A))
L(X) represents the number of outgoing links from page X.
Using the given β of 0.7, we can substitute the values into the formula:
PR(A) = (1 - 0.7) + 0.7 * (PR(B)/1 + PR(C)/1)
PR(B) = (1 - 0.7) + 0.7 * (PR(A)/2)
PR(C) = (1 - 0.7) + 0.7 * (PR(A)/2)
4. We also need to add the constraint that the sum of PageRanks of the three pages must be 3. So, we can write an equation based on the constraint:
PR(A) + PR(B) + PR(C) = 3
5. Substitute the values obtained in step 3 into the equation in step 4. We can then solve for the unknown PageRanks.
Please note that these equations need multiple iterations to converge, and we need to repeat steps 3 to 5 until the PageRanks stabilize. Let's start with the initial values and iterate to calculate the final PageRanks.