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2.3 ⋅ connections with langville and meyer’s notation 19 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. Table 2.1 – Relationship to Langville and Meyer's PageRank notation. A Rosetta stone to translate my notation for readers familiar with Langville and Meyer’s popular book Google’s PageRank and Beyond. Their symbol Our symbol ad E evT GMT H P ̄ T LA +x S PT Discussion Our initiation to PageRank was through Kamvar’s papers in which d is the dangling node vector. We always make the matrix E explicit to emphasize its rank-1 struc- ture. Here the G stands for the Google matrix. We use the application neutral M for the PageRank modified matrix. Langville and Meyer use the symbol H to suggest the hyperlink matrix without any sort of correction. We use P ̄ to suggest “P−” and that P ̄ needs an eventual correction to a stochastic matrix. Using A follows common notation in graph theory where A is the adjacency matrix; L hints at the link matrix. To keep consistent with solving linear systems (Ax = b) and eigen- value problems (Ax = λx), we use x to denote the unknown Page- Rank variables in the problem. The symbol S suggests stochastic, whereas we use P to denote a standard Markov chain transition matrix.PDF Image | ALGORITHMS FOR PAGERANK SENSITIVITY DISSERTATION
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