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16 2 ⋅ pagerank background Algorithms Graph or Web graph Substochastic matrix PseudoRank PageRank Eigensystems Linear systems Strongly preferential PageRank Weakly preferential PageRank Sink preferential PageRank Theory Other transformations 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. v a teleportation distribution, where v ≥ 0 and eT v = 1, also known as i the preference vector. To fix the scheme, PageRank modifies it so that it does something predictable from v. The parameter α controls the trade-off between P and v. Transitions of the PageRank process are given by a modified Markov matrix M=α P +(1−α)veT =M(α,P,v). (2.2) follow transitions reset Subsequently, we will omit the explicit dependence on the parameters when they are clear from context. Interpreted as a Markov chain, PageRank is a process that follows transitions in the original process P with probability α or resets according to a known distribution over the states with probability (1 − α). In contrast with P, the dominant eigenvector Mx = x is always unique. This x is the PageRank vector (with a slight detail addressed below). Uniqueness of the eigenvector is trivial when vi > 0 and follows from the Perron-Frobenius theorem because α < 1 implies that Mi j > 0. A detail often swept under the rug is what happens when vi ≥ 0. Without a completely positive v, M is no longer irreducible and the simple theorems for a unique eigenvector do not apply. That said, the eigenvector is still unique because M has only a single ergodic class over the set of states reachable from the support of v. Berman and Plemmons [1994, theorem 3.23] justifies this statement with the more general result that all unit eigenvectors of a stochastic matrix are convexPDF Image | Instagram Cheat Sheet
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