## Asymptotic behaviour of ranking algorithms in directed random networks

There is a vast empirical research on the behaviour of ranking algorithms, e.g. Google PageRank, in scale-free networks. In this talk, we address this problem by analytical probabilistic methods. In particular, it is well-known that the PageRank in scale-free networks follows a power law with the same exponent as in-degree. Recent probabilistic analysis has provided an explanation for this phenomenon by obtaining a natural approximation for PageRank based on stochastic fixed-point equations. For these equations, explicit solutions can be constructed on weighted branching trees, and their tail behavior can be described in great detail.

In this talk we present a model for generating directed random graphs with prescribed degree distributions where we can prove that the PageRank of a randomly chosen node does indeed converge to the solution of the corresponding fixed-point equation as the number of nodes in the graph grows to infinity. The proof of this result is based on classical random graph coupling techniques combined with the now extensive literature on the behavior of branching recursions on trees.