Natural and manmade complex systems are comprised of different elementary units, being either system components or diverse subsystems as in the case of networked systems. These units interact with each other in a possibly nonlinear way, which results in a complex dynamics that is generally dissipative and nonstationary. One of the challenges in the modeling of such systems is the identification of not only pairwise but, more importantly, higher-order interactions, together with their directions and strengths from measured multivariate time series. Here, we propose a novel data-driven approach for characterizing interactions of different orders. Our approach is based on solving a set of linear equations constructed from Kramers-Moyal coefficients derived from statistical moments of $\mathcal{N}$-dimensional multivariate time series. We demonstrate the substantial potential for applications by a data-driven reconstruction of interactions in various multidimensional and networked dynamical systems.

中文翻译：

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揭示高维复杂系统中的高阶交互：数据驱动的方法

自然和人造的复杂系统由不同的基本单元组成，这些基本单元要么是系统组件，要么是不同的子系统，就像网络系统一样。这些单元以可能非线性的方式相互作用，从而产生通常是耗散且非平稳的复杂动态。此类系统建模的挑战之一不仅是识别成对的交互作用，更重要的是识别高阶交互作用，以及测量的多元时间序列中它们的方向和强度。在这里，我们提出了一种新颖的数据驱动方法来表征不同订单的相互作用。我们的方法基于求解一组由 Kramers-Moyal 系数构造的线性方程，这些系数源自以下的统计矩$\mathcal{氮}$维多元时间序列。我们通过数据驱动的各种多维和网络动力系统中的相互作用重建来展示应用的巨大潜力。