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combinations of unit eigenvectors of the ergodic classes extended to all states with 0 probability.1 PageRank values are also the stationary distribution probabilities for the modified Markov chain, namely the PageRank vector x is the stationary dis- tribution vector. For this reason, the PageRank vector is a probability distri- bution vector and has the natural normalization 1 A special case is also not difficult tosee.Whenvi ≥ 0thenMcan be symmetrically permuted and partitioned so that M = [ M1 ], M12 M2 where the spectral radius of M1 < 1 and M2 is stochastic and irre- ducible. The rows and columns in M2 correspond to all states reach- able from the support of v. Solu- tionsto[M1 ][x1]=[x1]are M12 M2 x2 x2 unique because ρ(M1) < 1 implies that x1 = 0 and also stochasticity and irreducibility of M2 implies that x2 is unique. x ≥0,eTx=1. i The discussion thus far is the eigenvector definition of PageRank: Mx = x and eT x = 1. (2.3) As a probability distribution, the PageRank vector is also the solution of the linear system (I − αP)x = (1 − α)v, (2.4) which follows from eT x = 1 and Mx = αPx + (1 − α)v = x. This system is non-singular for all α < 1, and (I − αP) is an M-matrix. We could hardly ask to be luckier! Note that there is no difficulty with non-negative v for the linear system. Both representations yield quite a bit of flexibility in working with the problem. We summarize this section with the following definition. Problem1(PageRank). GivenacolumnstochasticP,0≤α<1,andadistri- bution vector v, set M = αP + (1 − α)veT . Solving PageRank is computing or approximating the unique vector x in Mx = x and eT x = 1 or (I − αP)x = (1 − α)v. (2.5) All the relevant pieces of the PageRank problem are present in this statement. Anything that calls itself PageRank will compute a vector that satisfies this property for some α, P, and v. Hence, this problem is the core PageRank problem at the heart of figure 2.2. 2.2.1 PageRank variations An alternative starting point for PageRank is to begin with a sub-stochastic matrix P ̄ . As discussed in the next section, sub-stochastic matrices often arise from random walk or influence propagation definitions on graphs. For such P ̄, there are two established formulations of the PageRank problem: strongly 2.2 ⋅ the pagerank problem 17PDF Image | Instagram Cheat Sheet
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