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12 2 ⋅ pagerank background 2.1 matrix computation preliminaries Before getting to PageRank, we need some notation for operations involv- ing matrices and vectors. Unless otherwise noted, the following conventions hold: bold capital letter bold lower case letters lower case Greek letters subscripted capital letters subscripted lower case letters calligraphic capital letters A,G,P b,v,x α, β, γ Aij,Gij,Pij bi,vi,xi G, S for matrices, for vectors, for scalars, for matrix elements, for vector elements, and for graphs and sets, which is a variation on Householder notation used in a few recent text- books [Meyer, 2000; Trefethen and Embree, 2005]. The following symbols represent standard matrix or vector operations: AT , xT A+ e ∥⋅∥ A⊗B A●B is the transpose of a matrix or vector, is the pseudo-inverse of a matrix, is the vector of all ones of appropriate length, is the 1-norm of a matrix or vector, for the Kronecker product between matrices or vectors, and for the Hadamard, or elementwise, product between matrices or vectors. Horn and Johnson [1991] have a nice background on the Kronecker and Hadamard products. These are less well known than the standard product operations between matrices and have a few fascinating properties. Throughout this thesis, P represents a square, column stochastic matrix. Formally, column stochastic implies P ≥ 0 and eTP = eT. Taking P as ij column stochastic is contrary to the notation in probability, where P is a row stochastic matrix. The matrix P ̄ is a column sub-stochastic matrix (henceforth called a sub-stochastic matrix) where ⎧⎪0 P ̄ij =0foralli (eTP ̄)j =⎨⎪ ̄ (2.1) ⎪⎩1 Pij=/0forsomei. 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.PDF Image | ALGORITHMS FOR PAGERANK SENSITIVITY DISSERTATION
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