Ranking algorithms in random graphs

In this talk I will give an overview of the line of work on mathematical properties of Google PageRank algorithm. We will first discuss computational aspects, and sensitivity to changes in the network. Next, we will zoom in on the remarkable property that the distribution of PageRank in scale-free networks follows a power law with the same exponent as in-degree. We will see how this can be explained by a probabilistic model, based on a stochastic fixed point equation. The main result is the distribution of a family of rankings, which includes Google's PageRank, on a directed configuration model (DCM). The result states that the rank of a randomly chosen node in the graph converges in distribution to a finite random variable that can be written as a linear combination of i.i.d. copies of the endogenous solution to a stochastic fixed point equation. For the first time in the literature, this result establishes a limiting behavior for a complete PageRank distribution. This provides a very accurate approximation for the PageRank distribution on the DCM but also on a complete English Wikipedia graph. The essence of the proof is in coupling of the DCM with a specially constructed tree. The main result is obtained by showing that the ranking in the graph converges with any given precision before the coupling breaks. (joint work with Mariana Olvera-Cravioto and Ningyuan Chen)