Multiple equilibria of nonhomogeneous Markov chains and self-validating web rankings

PageRank is a ranking of the web pages that measures how often a given web page is visited by a random surfer on the web graph, for a simple model of web surfing. It seems realistic that PageRank may also have an influence on the behavior of web surfers. We propose here a simple model taking into account the mutual influence between web ranking and web surfing. Our ranking, the T-PageRank, is a nonlinear generalization of the PageRank. It is defined as the limit, if it exists, of some nonlinear iterates. A positive parameter T, the temperature, measures the confidence of the web surfer in the web ranking. We prove that, when the temperature is large enough, the T-PageRank is unique and the iterates converge globally on the domain. But when the temperature is small, there may be several T-PageRanks, that may strongly depend on the initial ranking. Our analysis uses results of nonlinear Perron-Frobenius theory, Hilbert projective metric and Birkhoff's coefficient of ergodicity.