马尔可夫影响系统及其重整化理论
Toward a Theory of Markov Influence Systems and their Renormalization
摘要：
我们引入马尔可夫影响系统（MIS）的概念，并分析其动力学。MIS模拟图中的随机游走，其边缘和转移概率作为电流分布的函数内生地变化。本文由两个独立的部分组成：第一部分，我们通过展示如何“解析”图序列，将马尔可夫链状态的标准分类推广到时变情况;第二部分，我们使用这个框架对几个重要的MIS族进行分支分析。我们表明，在一般情况下，这些系统可以是混沌的，但不可约MIS几乎总是渐近周期的。我们给出了一个“超迟钝”混合的例子，在超指数时间内达到一个平稳分布，这是一个超越任何马尔可夫链的时间尺度。
英文摘要：
We introduce the concept of a Markov influence system (MIS) and analyze its dynamics. An MIS models a random walk in a graph whose edges and transition probabilities change endogenously as a function of the current distribution. This article consists of two independent parts: in the first one, we generalize the standard classification of Markov chain states to the time-varying case by showing how to "parse" graph sequences; in the second part, we use this framework to carry out the bifurcation analysis of a few important MIS families. We show that, in general, these systems can be chaotic but that irreducible MIS are almost always asymptotically periodic. We give an example of "hyper-torpid" mixing, where a stationary distribution is reached in super-exponential time, a timescale beyond the reach of any Markov chain.
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