单纯形复形同步的主稳定函数
The Master Stability Function for Synchronization in Simplicial Complexes
摘要：
一个复杂系统的所有有趣和迷人的集体性质都源于其组成部分相互作用的复杂方式。物理学、生物学、社会科学和工程学中的各种系统已经被成功地建模为耦合动力系统的网络，其中图链接描述了成对的相互作用。然而，这是一个太强的限制，因为最近的研究表明，高阶多体相互作用存在于社会群体，生态系统和人类大脑中，它们实际上影响了所有这些系统的涌现动力学。在这里，我们介绍了一个一般的框架，允许研究耦合动力系统占精确的微观结构，他们的相互作用在任何可能的顺序。我们认为最一般的合奏相同的动力系统，组织在节点上的单纯复杂，并通过同步非侵入耦合功能的相互作用。单纯复形可以是任意维的，这意味着它可以同时考虑两两相互作用、三体相互作用等，在这样一个广泛的背景下，我们证明了完全同步作为不变解存在，并给出了它作为稳定态被观测到的必要条件。这将现有的结果有效的成对相互作用（即图形）的情况下，复杂系统的最一般的可能架构。此外，我们展示了如何简化的方法，具体的，但经常发生的情况下，我们验证了我们所有的理论预测在合成和现实世界的系统。鉴于所提出的方法的完全一般性，我们的结果有助于多体相互作用的动力系统的理论，并可以在非常广泛的实际情况下找到应用。
英文摘要：
All interesting and fascinating collective properties of a complex system arise from the intricate way in which its components interact. Various systems in physics, biology, social sciences and engineering have been successfully modelled as networks of coupled dynamical systems, where the graph links describe pairwise interactions. This is, however, too strong a limitation, as recent studies have revealed that higher-order many-body interactions are present in social groups, ecosystems and in the human brain, and they actually affect the emergent dynamics of all these systems. Here, we introduce a general framework that allows to study coupled dynamical systems accounting for the precise microscopic structure of their interactions at any possible order. We consider the most general ensemble of identical dynamical systems, organized on the nodes of a simplicial complex, and interacting through synchronization-non-invasive coupling function. The simplicial complex can be of any dimension, meaning that it can account, at the same time, for pairwise interactions, three-body interactions and so on. In such a broad context, we show that complete synchronization exists as an invariant solution, and we give the necessary condition for it to be observed as a stable state in terms of a Master Stability Function. This generalizes the existing results valid for pairwise interactions (i.e. graphs) to the case of complex systems with the most general possible architecture. Moreover, we show how the approach can be simplified for specific, yet frequently occurring, instances, and we verify all our theoretical predictions in synthetic and real-world systems. Given the completely general character of the method proposed, our results contribute to the theory of dynamical systems with many-body interactions and can find applications in an extremely wide range of practical cases.
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