Question 10.5.4: Consider a simple random walk on the graph in Figure 1. What...

Linear Algebra and Its Applications

Consider a simple random walk on the graph in Figure 1. What is the probability that a walker starting at state 1 visits state 4 before visiting state 7?

5.1

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Changing state 4 and state 7 to absorbing states and then computing the absorption probabilities starting at state 1 will answer this question. Begin by reordering the states as 4, 7, 1, 2, 3, 5, 6 and rewrite states 4 and 7 as absorbing states:

\text { The resulting transition matrix is }\left[\begin{array}{l|l}I & S \\\hline O & Q\end{array}\right] \text {, with }

and

so

M=(I-Q)^{-1}=

and

Since the first column of A corresponds to state 1 and the rows correspond to states 4 and 7, respectively, the probability of visiting 4 before visiting 7 is 3/5.

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