Toward a Theory of Markov Influence Systems and their Renormalization
Toward a Theory of Markov Influence Systems and their Renormalization
复制标题
马尔可夫影响系统理论及其重整化
DOI：
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发表时间：
2018
期刊：
影响因子：
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通讯作者：
B. Chazelle
中科院分区：
文献类型：
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作者：
B. Chazelle
非线性马尔可夫链是物理学、生物学和社会科学中常用的概率模型。在“马尔可夫影响系统”（MIS）中，链的转移概率作为当前状态分布的函数而变化。这项工作介绍了一个重整化框架，用于分析MIS的动态。它分为两个独立的部分：首先，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动态情况。然后，我们使用这个框架进行分歧分析的几个重要的MIS家庭。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。我们还给出了一个“超迟钝”混合的例子，在超指数时间内达到平稳分布，这是任何马尔可夫链都无法实现的时间尺度。
Nonlinear Markov chains are probabilistic models commonly used in physics, biology, and the social sciences. In "Markov influence systems" (MIS), the transition probabilities of the chains change as a function of the current state distribution. This work introduces a renormalization framework for analyzing the dynamics of MIS. It comes in two independent parts: first, we generalize the standard classification of Markov chain states to the dynamic case by showing how to parse graph sequences. We then 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 also give an example of "hyper-torpid" mixing, where a stationary distribution is reached in super-exponential time, a timescale that cannot be achieved by any Markov chain.