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2.3 Markov Chains A row-stochastic matrix is a non-negative n × n matrix where each row sums to 1. It is important here for its use in Markov chains. A Markov chain is a stochastic process that satisfies the Markov property, a memoryless property where the probability of future behavior is independent of past behavior (Meyer, 2000). The Internet network matrix can also be defined by using the transition probability : Sij = P(Xt = pj|Xt−1 = pi) (II.7) where each entry can be described as the odds that a web surfer would follow a link to page pj, given that they were on page pi. Since: S = [Sij], then S is a transition matrix. With a transition matrix S we can then find a probability distribution vector p, a non-negative vector where: (II.8) T∑ p =(p1 p2 ... pn)suchthat pk =1 In a Markov chain, the kth step probability distribution vector is: pT(k) = (p1(k) p2(k) ... pn(k)), k = 1,2,..., where pj(k) = P(Xk = Sj) Where pj (k) is the probability of being on page pj on the kth step, and the intitial distribution vector: pT (0) = (p1(0) p2(0) ... pn(0)), where pj (0) = P (X0 = Sj ) (II.10) 6 k (II.9)PDF Image | MATHEMATICS BEHIND GOOGLE PAGERANK ALGORITHM
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