In particular, the state of a probabilistic automaton is always a stochastic vector, since the product of any two stochastic matrices is a stochastic matrix, and the product of a stochastic vector and a stochastic matrix is again a stochastic vector.
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We use a stochastic matrix to represent the transition probabilities of this system ( rows and columns in this matrix are indexed by the possible states listed above, with the pre-transition state as the row and post-transition state as the column ).
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Once the graph is constructed, it is used to form a stochastic matrix, combined with a damping factor ( as in the " random surfer model " ), and the ranking over vertices is obtained by finding the eigenvector corresponding to eigenvalue 1 ( i . e ., the stationary distribution of the random walk on the graph ).
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