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Next, this matrix is a square, with all of its entries nonnegative, and the sum of the entries in a column add up to 1. Therefore, the matrix is column-stochastic, and so by a proposition in [3], 1 is an eigenvalue. In other words, equation (2) is fully justified solution-wise. By normalizing the solution, deriving our desired importance vector: [x1,x2,x3,x4] = 12, 4 , 9 , 6 . Observe 31 31 31 31 that page 3, which previously ruled by a simple majority, is now usurped by page 1 once the weighting scheme is applied. 2. Troubleshooting: There are several problems that arise when using equation (1). One issue is that a web may have dangling nodes. Another possibility is that there may not be a unique solution. In the previous example, where V1 is the eigenspace for λ = 1, the dim V1 = 1, which is desirable. However, it can only be assumed that this is universally true when we are able to travel from one page to any other page in finitely many steps. Take, for example, the web modeled as follows: With the corresponding matrix: 0 1 0 0 0 1 0 0 0 0 0 0 0 1 1/2 0 0 1 0 1/2 00000 Inthiscase,thedimV1 =2,andtherearetwodistinctsolutions: 12,12,0,0,0and[0,0,12,12,0]. Also, V1 is a subspace, and there are many additional solutions by taking linear combinations of these two solutions, and so the algorithm would fail at this stage. Assuming that our web has no dangling nodes, we then may eliminate the ambiguity by letting S be the n × n matrix where each entry in the matrix is equal to 1/n. Then S is column-stochastic, with respect to S, the dimV1 = 1, and M := (1 − m)A + mS, 0 ≤ m ≤ 1 (Originally, Google used m = 0.15 [3]). For any 0 ≤ m ≤ 1, M is column-stochastic, and for 0 ̸= m, with respect to M, the dim V1 = 1, as desired. 3PDF Image | Google PageRank algorithm powered by linear algebra
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