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is the probability that the Markov chain starts in Sj . These vectors can be used with the Internet network matrix to show the likelihood of being on an individual page pj. It is important to be able to describe the kth step distribution for the given initial distribution vector pT (0). Using elementary probability laws, it is easy to determine that pT (1) = pT (0)S. Because a Markov chain has no memory, then pT (2) = pT (1)S, which acts as if pT (1) was the initial distribution. Continuing on in this manner with substitution reveals that: pT(k)=pT(k−1)S=pT(k−2)S2 =...=pT(0)Sk pT (k) = pT (0)Sk (II.11) Where pij in S is the probability of moving from page pi to page pj in k steps. Because the matrix S is a stochastic matrix and λ1 = 1 is the dominant eigenvalue of S, the eigensystem: πTS=πT, π≥0, πTe=1 (II.12) has a unique solution π, called the stationary distribution vector. The vector π is the dominant left eigenvector corresponding to λ1. The ith component πi represents the percentage of being on page i. With the Internet network matrix, this is useful for determining the likelihood of being on a webpage by following links, and is an essential part of the PageRank algorithm, discussed more in Section 3.2. 7PDF Image | MATHEMATICS BEHIND GOOGLE PAGERANK ALGORITHM
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