Small noise limits of Markov chains and the PageRank

We recall the classical formulation of PageRank as the stationary distribution of a singularly perturbed irreducible Markov chain that is not irreducible when the perturbation parameter goes to zero. Specifically, we use the Markov chain tree theorem to derive explicit expressions for the PageRank. This analysis leads to some surprising results. These results are then extended to a much more general class of perturbations that subsume personalized PageRank. We also give examples where even simpler formulas for PageRank are possible.