Science:Math Exam Resources/Courses/MATH307/April 2013/Question Section 201 06 (b)

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MATH307 April 2013

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Question Section 201 06 (b)
Consider the same graph as in the previous question, now interpreted as an internet where the vertices represent web pages and the arrows represent links. (b) What is the stochastic matrix associated with the PageRank algorithm with damping factor α = 1/2. What happens to the eigenvalues as α tends to 0 (complete damping)?

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Solution

The stochastic matrix for a damped system is ${\begin{aligned}S&=(1-\alpha )Q+\alpha P\\&={\frac {1}{2}}Q+{\frac {1}{2}}P\\&={\begin{bmatrix}1/12&1/12&1/12&1/12&1/12&1/6\\1/3&1/12&1/12&1/12&1/12&1/6\\1/12&1/3&1/12&1/12&1/12&1/6\\1/3&1/12&1/12&1/12&1/12&1/6\\1/12&1/3&1/12&7/12&1/12&1/6\\1/12&1/12&7/12&1/12&7/12&1/6\end{bmatrix}}\end{aligned}}$ When the damping factor $\displaystyle \alpha$ tends to zero, the eigenvalues of S tend to Q. Because Q projects onto a 1 dimension subspace, there is one eigenvalue equal to 1, and the rest are zero.

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