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Multiplex Decomposition of Non-Markovian Dynamics and the Hidden Layer Reconstruction Problem

机译：非马洛维亚动力学和隐藏层重建问题的多路复用分解

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Elements composing complex systems usually interact in several different ways, and as such, the interaction architecture is well modeled by a network with multiple layers—a multiplex network—where the system’s complex dynamics is often the result of several intertwined processes taking place at different levels. However, only in a few cases can such multilayered architecture be empirically observed, as one usually only has experimental access to such structure from an aggregated projection. A fundamental challenge is thus to determine whether the hidden underlying architecture of complex systems is better modeled as a single interaction layer or if it results from the aggregation and interplay of multiple layers. Assuming a prior of intralayer Markovian diffusion, here we show that by using local information provided by a random walker navigating the aggregated network, it is possible to determine, in a robust manner, whether these dynamics can be more accurately represented by a single layer or if they are better explained by a (hidden) multiplex structure. In the latter case, we also provide Bayesian methods to estimate the most probable number of hidden layers and the model parameters, thereby fully reconstructing its architecture. The whole methodology enables us to decipher the underlying multiplex architecture of complex systems by exploiting the non-Markovian signatures on the statistics of a single random walk on the aggregated network. In fact, the mathematical formalism presented here extends above and beyond detection of physical layers in networked complex systems, as it provides a principled solution for the optimal decomposition and projection of complex, non-Markovian dynamics into a Markov switching combination of diffusive modes. We validate the proposed methodology with numerical simulations of both (i)?random walks navigating hidden multiplex networks (thereby reconstructing the true hidden architecture) and (ii)?Markovian and non-Markovian continuous stochastic processes (thereby reconstructing an effective multiplex decomposition where each layer accounts for a different diffusive mode). We also state and prove two existence theorems guaranteeing that an exact reconstruction of the dynamics in terms of these hidden jump-Markov models is always possible for arbitrary finite-order Markovian and fully non-Markovian processes. Finally, we showcase the applicability of the method to experimental recordings from (i)?the mobility dynamics of human players in an online multiplayer game and (ii)?the dynamics of RNA polymerases at the single-molecule level.
机译：构成复杂系统的元素通常以几种不同的方式交互，因此，交互架构由具有多个层的网络建模良好 - 多层网络 - 系统的复杂动态通常是在不同级别进行多个交织过程的结果。然而，只有在少数情况下，可以经验上观察到这种多层架构，因为通常只能从聚合投影中对这种结构进行实验性访问。因此，基本挑战是确定复杂系统的隐藏的基础架构是否更好地建模为单个交互层，或者如果它来自多个层的聚合和相互作用。假设intoralAlay Markovian扩散之前，这里我们示出了通过使用由航管步行者提供的本地信息导航聚合网络，可以以稳健的方式确定这些动态是否可以更准确地由单层表示或者如果它们更好地由（隐藏）多路复用结构解释。在后一种情况下，我们还提供贝叶斯的方法来估计最可能的隐藏层数和模型参数，从而完全重建其架构。整个方法使我们能够通过利用在聚合网络上单个随机散步的统计数据上的非马洛维亚签名来破译复杂系统的底层多路复用体系结构。实际上，这里呈现的数学形式主义在网络化复杂系统中的物理层的检测方案上方扩展，因为它为复合的非马车动态的最佳分解和投影提供了一个原则性的解决方案，进入扩散模式的马尔可夫切换组合。我们用两者（i）的数值模拟验证了所提出的方法？随机行走导航隐藏的多路复用网络（从而重建真实的隐藏架构）和（ii）？马尔可维亚和非马尔可夫持续随机过程（从而重建有效的多路复用分解图层占用不同的扩散模式）。我们还陈述并证明了两个存在定理，保证了在这些隐藏的Jump-Markov模型方面精确地重建动态，始终可以进行任意有限的马尔可维亚和完全非马洛维亚流程。最后，我们展示了来自（i）的实验记录的方法的适用性？在线多人游戏中人类参与者的移动性动态和（ii）？单分子水平下RNA聚合酶的动态。

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