Complex systems are characterized by multiple spatial and temporal scales. A natural framework to capture their multiscale nature is that of multilayer networks, where different layers represent distinct physical processes that often regulate each other indirectly. We model these regulatory mechanisms through triadic higher-order interactions between nodes and edges. In this work, we focus on how the different timescales associated with each layer impact their reciprocal effective couplings. First, we rigorously derive a decomposition of the joint probability distribution of any dynamical process acting on such multilayer networks. By inspecting this probabilistic structure, we unravel the general principles governing how information propagates across timescales, elucidating the interplay between mutual information and causality in multiscale systems. In particular, we show that feedback interactions, i.e., those representing regulatory mechanisms from slow to fast variables, generate mutual information between layers. On the contrary, direct interactions, i.e., from fast to slow layers, can propagate this information only under certain conditions that depend solely on the structure of the underlying higher-order couplings. We introduce the mutual information matrix for multiscale observables to capture these emergent functional couplings. We apply our results to study archetypal examples of biological signaling networks and effective environmental dependencies in stochastic processes. Our framework generalizes to any dynamics on multilayer networks, paving the way for a deeper understanding of how the multiscale nature of real-world systems shapes their information content and complexity.

中文翻译：

---

具有跨时间尺度高阶交互的多层系统中的信息传播

复杂系统具有多个空间和时间尺度的特征。捕捉其多尺度性质的一个自然框架是多层网络，其中不同的层代表不同的物理过程，这些过程通常间接地相互调节。我们通过节点和边之间的三元高阶交互来模拟这些调节机制。在这项工作中，我们重点关注与每一层相关的不同时间尺度如何影响其相互有效耦合。首先，我们严格推导出作用于此类多层网络的任何动态过程的联合概率分布的分解。通过检查这种概率结构，我们揭示了控制信息如何跨时间尺度传播的一般原则，阐明了多尺度系统中互信息和因果关系之间的相互作用。特别是，我们表明反馈相互作用，即代表从慢到快变量的调节机制的反馈相互作用，在层之间产生相互信息。相反，直接交互，即从快层到慢速层，只能在某些条件下传播此信息，而这些条件仅取决于底层高阶耦合的结构。我们引入多尺度可观测量的互信息矩阵来捕获这些新兴的函数耦合。我们应用我们的结果来研究生物信号网络的典型例子和随机过程中的有效环境依赖性。我们的框架泛化到多层网络上的任何动态，为更深入地理解现实世界系统的多尺度性质如何塑造其信息内容和复杂性铺平了道路。