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where D is a diagonal matrix with diagonal entries Dii = (Ae)i = outdegree of node i, and D+ is the pseudo-inverse [Golub and van Loan, 1996], another diagonal matrix with (D+) ii ⎪ ⎪ 1 / D i i = ⎨ ⎪ ⎪⎩ 0 D i i =/ 0 D i i = 0 . (2.15) ⎧ In the context of web search, each web page corresponds to a node in G, and nodes u and v are connected with a directed edge if the page corresponding to node u links to the page corresponding to node v. Despite appearances, the setup P ̄ = AT D+ does not appear out of thin air. We’ve seen it before. From the introduction to the chapter: important web pages link to important web pages and their scores In matrix form, sj si= ∑ . j links to i total links from j s = P ̄s. We could immediately apply the ideas of section 2.2 except that they require a stochastic matrix. Thankfully, section 2.2.1 tells us how to convert P ̄ to P for PageRank. But why do we need any of these ideas? Let’s work through an example in some detail. For 3 ⎡0 1/2 0 0 2⎢⎥ 2.2 ⋅ the pagerank problem 17 0 0⎤ ⎢⎣0 0 0 0 1 0⎥⎦ thereisasingledanglingcolumn,sod=[1 0 0 0 0 0] .Theonly T 16 eigenvectorP ̄x=xisx=[0 0 0 0 1/2 1/2] .Andsoxisunique, T and α = 5/6. Using the strongly preferential PageRank model yields the but not that useful. Suppose that v = [1/6 1/6 1/6 1/6 1/6 1/6] PageRank vector T after rounding. It is the same two nodes that are the most important, but node 5 is more important than node 6. Also we learn that node 3 is the most important among the rest. Even with all of these choices, there are still details left. Should self-loops in G be retained? Should multiple-links between pages be respected? At some point, we need to end the enumeration of PageRank variants. The answers to these questions ultimately depend upon the application. x = [0.049 0.041 0.059 0.032 0.425 0.394] T ⎢⎥ ⎢0 0 0 1/3 0 0⎥ ⎢⎥ ⎢0 1/2 0 1/3 0 0⎥ thegraphG= andthematrixP ̄ =⎢ ⎥ ⎢0 0 0 0 0 0⎥ 45⎢⎥ ⎢0 0 1 1/3 0 1⎥PDF Image | ALGORITHMS FOR PAGERANK SENSITIVITY DISSERTATION
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