The random surfer's favourite web page

Question

A tiny web has three pages, A, B, C, linked as follows. Page A links only to B. Page B links to A and C. Page C links only to A. A surfer clicks a uniformly random outgoing link on each page and keeps clicking forever. In the long run, what fraction of time is spent on each page? (This fraction is exactly the page's "PageRank.")

Accepted Answer

Write the transition probabilities from the link structure: - From A: go to B with probability 1. - From B: go to A or C, each with probability 	frac12. - From C: go to A with probability 1. Let (\piA, \piB, \piC) be the long-run fractions. Each page's share equals the total inflow from pages that link to it: \piA = \piB\cdot	frac12 + \piC\cdot 1, \qquad \piB = \piA \cdot 1, \qquad \piC = \piB \cdot 	frac12. Let \piB = x. Then \piA = x (from the second equation) and \piC = 	frac12 x (third equation). Check the first: \piA = 	frac12 x + 	frac12 x = x. Consistent. Normalize so the fractio…

The random surfer's favourite web page

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