Intermittent Synchronization in Finite-State Random Networks Under Markov Perturbations

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 Cm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^m$$\end{document} (m≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m \ge 1$$\end{document}) in ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document}, where ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} is a perturbation parameter, then the relative frequencies of synchronization with probability 1-O(εℓ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1-O(\varepsilon ^{\ell })$$\end{document} and of desynchronization with probability O(εℓ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\varepsilon ^{\ell })$$\end{document} can both be precisely described for 1≤ℓ≤m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le \ell \le m$$\end{document} via an asymptotic expansion of the invariant distribution. Existence and uniqueness of invariant distributions are established, as well as their convergence as ε→0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \rightarrow 0$$\end{document}. 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.