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2.2 ⋅ the pagerank problem 15 stochastic matrix. The matrix P ̄ is a column sub-stochastic matrix (henceforth called a sub-stochastic matrix) where ⎧ ⎪0 P ̄ij =0foralli (eTP ̄)j =⎨ ⎪⎩1 P ̄ij=/0forsomei. (2.1) We use two definitions consistently in the remainder of the text, except where explicitly noted. One common exception is that e may also denote an error vector when we discuss approximations to exact solutions. 2.2 the pagerank problem Discussing PageRank from both a theoretical and a practical view is hard. There are many slight variants of the PageRank problem, and this section enumerates three of them after introducing the core PageRank problem. The distinctions among the variants are important, although this core formulation of the PageRank problem masks them. Hiding the distinctions is mathemati- cally advantageous as most properties of the PageRank problem are preserved for all variants and thus it simplifies analysis. Figure 2.2 provides guidance for the discussion of the next few sections. This section introduces the mathematical PageRank formulations. The sub- sections describe the variations. Without further ado, what distinguishes PageRank? PageRank begins with any idea that defines a stochastic matrix P. Ideally, the importance of items is proportional to the dominant eigenvector Px = x of the stochastic matrix, but this may not be unique. (We’ll see that x is not unique for the “importance” model given in the introduction.) Given any stochastic matrix P, PageRank modifies it to produce a new problem with a unique answer. These modifications, then, define PageRank and not the starting stochastic matrix. In a slightly hyperbolic sense, PageRank is a technique to take any cockamamied idea and fix it. The data for the PageRank problem are P a column stochastic matrix that defines the transitions of a Markov chain; α ateleportationparameterordampingparameter,0≤α<1;andPDF Image | Instagram Cheat Sheet
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