LAUSR

Abstract PageRank is one of the most popular measures of the importance of Web pages. The dual nature of PageRank is not yet completely understood. From its definition the PageRank of a Web page (node) is the probability to find a random walk at this node when the process has reached the steady state. From another side, PageRank of a randomly selected page is a hidden Markov process due to the random number of its in- and out-degrees. Considering the stochastic nature of PageRank we aim to study its extremal index that represents the dependence measure of extremes and plays a fundamental role in the theory of extreme values. Using the representation of PageRank as a weighted branching process and the corresponding Thorny Branching Tree, we propose a nonparametric estimation of the extremal index of PageRank by samples of moderate sizes. It is based on the representation of the reciprocal of the extremal index by a mean cluster size. The cluster implies a block of node ranks with at least one exceedance over a sufficiently high threshold. It is proposed to consider generations of successors of a root node of the PageRank branching process as blocks. Among practical advantages the extremal index determines the mean first hitting time (or the mean minimal time) to reach an influential node with a large PageRank. We consider a Max-linear model as an alternative to PageRank and compare its distribution with that one of PageRank.