On rank statistics of PageRank and MarkovRank

An important statistic in analyzing some (finite) network data, called \emph{PageRank}, and a related new statistic, which we call \emph{MarkovRank}, are studied in this paper. The PageRank was originally developed by the cofounders of \emph{Google}, Sergey Brin and Larry Page, to optimize the ranking of websites for their search engine outcomes, and it is computed using an iterative algorithm, based on the idea that nodes with a larger number of incoming edges are more important. The aim of this paper is to analyze the common features and some significant differences between the PageRank and the new Rank. A common merit of the two Ranks is that both statistics can be easily computed by either the mathematical computation or the iterative algorithm. According to the analysis of some examples, these two statistics seem to return somewhat different values, but the resulting rank statistics of both statistics are not far away from each other. One of the differences is that only MarkovRank has the property that its rank statistic does not depend on any tuning parameter, and it is determined only through the adjacency matrix for given network data. Moreover, it is also shown that the rank statistic of MarkovRank coincides with that of the stationary distribution vector of the corresponding Markov chain (with finite state space) whenever the Markov chain is regular. Thus, MarkovRank may have a potential to play a similar role to the well established PageRank from a practical point of view, not only commonly with light computational tasks, but also with some new theoretical validity.