  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.

Probability;Dynamical Systems