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Intermittent Synchronization in Finite-State Random Networks Under Markov Perturbations
Communications in Mathematical Physics ( IF 2.4 ) Pub Date : 2021-05-08 , DOI: 10.1007/s00220-021-04104-z
Arno Berger , Hong Qian , Shirou Wang , Yingfei Yi

By introducing extrinsic noise as well as intrinsic uncertainty into a network with stochastic events, this paper studies the dynamics of the resulting Markov random network and characterizes a novel phenomenon of intermittent synchronization and desynchronization that is due to an interplay of the two forms of randomness in the system. On a finite state space and in discrete time, the network allows for unperturbed (or “deterministic”) randomness that represents the extrinsic noise but also for small intrinsic uncertainties modelled by a Markov perturbation. It is shown that if the deterministic random network is synchronized (resp., uniformly synchronized), then for almost all realizations of its extrinsic noise the stochastic trajectories of the perturbed network synchronize along almost all (resp., along all) time sequences after a certain time, with high probability. That is, both the probability of synchronization and the proportion of time spent in synchrony are arbitrarily close to one. Under smooth Markov perturbations, high-probability synchronization and low-probability desynchronization occur intermittently in time. If the perturbation is \(C^m\) (\(m \ge 1\)) in \(\varepsilon \), where \(\varepsilon \) is a perturbation parameter, then the relative frequencies of synchronization with probability \(1-O(\varepsilon ^{\ell })\) and of desynchronization with probability \(O(\varepsilon ^{\ell })\) can both be precisely described for \(1\le \ell \le m\) via an asymptotic expansion of the invariant distribution. Existence and uniqueness of invariant distributions are established, as well as their convergence as \(\varepsilon \rightarrow 0\). An explicit asymptotic expansion is derived. Ergodicity of the extrinsic noise dynamics is seen to be crucial for the characterization of (de)synchronization sets and their respective relative frequencies. An example of a smooth Markov perturbation of a synchronized probabilistic Boolean network is provided to illustrate the intermittency between high-probability synchronization and low-probability desynchronization.



中文翻译:

马尔可夫扰动下有限状态随机网络的间歇同步

通过将外部噪声以及固有不确定性引入具有随机事件的网络中,本文研究了由此产生的马尔可夫随机网络的动力学并描述了由于系统中两种形式的随机性相互影响而导致的间歇性同步和去同步的新现象。在有限状态空间和离散时间中,网络允许代表外部噪声的不受干扰(或“确定性”)随机性,也允许由马尔可夫扰动建模的较小的固有不确定性。结果表明,如果确定性随机网络是同步的(分别是均匀同步的),那么对于其外在噪声的几乎所有实现,被扰动网络的随机轨迹将在经过一定时间后沿几乎所有(分别沿着所有)时间序列同步。一定时间,很有可能。即,同步的概率和花费在同步上的时间的比例都任意地接近一。在平稳的马尔可夫扰动下,高概率同步和低概率去同步会间歇性地发生。如果扰动是\(\ varepsilon \)中的\(C ^ m \)\(m \ ge 1 \)),其中\(\ varepsilon \)是一个摄动参数,然后以概率\(1-O (\ varepsilon ^ {\ ELL})\) ,并用概率去同步化的\(O(\ varepsilon ^ {\ ELL})\)既可用于精确地描述\(1 \文件\ ELL \文件米\)经由不变分布的渐近展开。建立了不变分布的存在性和唯一性,以及它们的收敛性为\(\ varepsilon \ rightarrow 0 \)。导出一个显式渐近展开。外来噪声动力学的遍历性被认为对于(去)同步集及其各自的相对频率的表征至关重要。提供了一个同步概率布尔网络的平稳马尔可夫扰动的示例,以说明高概率同步与低概率不同步之间的间歇性。

更新日期:2021-05-08
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