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I have been stuck on a homework problem for days.

Construct a strongly connected graph in which the basic PageRank computation does not converge.

I tried everything and still cannot find a solution. My latest attempt is to brute force generate all possible 4 and 5 node graphs and to solve for their eigenvectors and look for cases where none of the eigenvectors are all the same sign. Not working so far so I would greatly appreciate some hint. I'm beginning to think no such graph exists....

If I can find a Matrix that will continually oscillate a vector with entries (1/n, 1/n, 1/n..., where is the number of dimensions) I believe that is the solution. But how can I find such a Matrix? I tried a 2D or 3D rotation matrix but that doesn't work as it gives a matrix with negative entries.

Alp Uzman
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mchangun
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1 Answers1

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Maybe you should try with

  • an ergodic Markov chain which is not regular (and therefore MUST be cyclic)
  • no damping factor (aka $\alpha=1$)

A directed cycle should work.

Alp Uzman
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asdf
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