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Algorithms Substochastic matrix 2.2 ⋅ the pagerank problem 13 Theory Graph or Web graph Strongly preferential PageRank Weakly preferential PageRank Sink preferential PageRank PseudoRank PageRank Other transformations Eigensystems Linear systems Figure 2.2 – Overview of PageRank formulations. Most derivations of PageRank begin with a graph and proceed through a sub-stochastic matrix and a PageRank variant before getting to the PageRank problem. Instead, starting with the PageRank problem yields a mathematically pure look at the problem. Algorithms and implementations for PageRank often need to take advantage of the graph structure and begin with a graph or sub-stochastic matrix. The bold path illustrates the most common PageRank formulation. 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.PDF Image | ALGORITHMS FOR PAGERANK SENSITIVITY DISSERTATION
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