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22 2 ⋅ pagerank background 2.3 connections with langville and meyer’s notation Langville and Meyer [2006a] established a different set of notation for the PageRank problem. There are many relationships between our notations, but we prefer to separate PageRank from a web-ranking context. The biggest difference between our notations is the column vs. row orien- tation of the matrices. Langville and Meyer use row stochastic matrices and then write the PageRank equations as +T (αS + (1 − α)E) = +T . Such notation closely follows standard probability and Markov chain theory, although most of that literature also utilizes row vectors instead of the column vector, which would make it +(αS + (1 − α)E) = +. Instead, our notation is designed to avoid unnecessary transpose symbols and retain column vectors. Thus we write (αP + (1 − α)veT )x = x. Table 2.1 summarizes the symbol relationships between our symbols. 2.4 algorithms So far, we have seen how PageRank is formulated as the linear equation or the eigensystem (I − αP)x = (1 − α)v Mx = x. Both of these formulations correspond to extremely well studied problems: solving linear systems and computing eigenvectors, respectively [Golub and van Loan, 1996]. So why do we need to write about algorithms for PageRank? The property that makes PageRank an interesting problem is that the matrices are H U G E! Recent reports about the size of the web establish that there are over one trillion (1,000,000,000,000) pages [Alpert and Hajaj, 2008]— although many are duplicates—and at least one search engine has crawled over 180 billion pages [Cuil, 2009]. Algorithms to compute PageRank, then,PDF Image | Instagram Cheat Sheet
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