非齐次马尔可夫链的多重均衡与自验证网络排名
Multiple equilibria of nonhomogeneous Markov chains and self-validating web rankings
摘要：
PageRank是网页的排名，衡量给定网页在网络图上被随机冲浪者访问的频率，用于简单的网络冲浪模型。PageRank也可能影响网络冲浪者的行为，这似乎是现实的。我们在这里提出了一个简单的模型，考虑到网络排名和网上冲浪之间的相互影响。我们的排名，T-PageRank，是PageRank的非线性推广。它被定义为某些非线性迭代的极限，如果它存在的话。一个正参数T，温度，测量网络冲浪者在网络排名中的信心。我们证明了，当温度足够大时，T-PageRank是唯一的，迭代全局收敛的域上。但是当温度较低时，可能存在多个T-PageRank，这可能强烈依赖于初始排名。我们的分析使用的非线性Perron-Frobenius理论，希尔伯特射影度量和Birkhoff的遍历系数的结果。
英文摘要：
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.
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