The emergence of synchronization in systems of coupled agents is a pivotal phenomenon in physics, biology, computer science, and neuroscience. Traditionally, interaction systems have been described as networks, where links encode information only on the pairwise influences among the nodes. Yet, in many systems, interactions among the units take place in larger groups. Recent work has shown that the presence of higher-order interactions between oscillators can significantly affect the emerging dynamics. However, these early studies have mostly considered interactions up to four oscillators at time, and analytical treatments are limited to the all-to-all setting. Here, we propose a general framework that allows us to effectively study populations of oscillators where higher-order interactions of all possible orders are considered, for any complex topology described by arbitrary hypergraphs, and for general coupling functions. To this end, we introduce a multiorder Laplacian whose spectrum determines the stability of the synchronized solution. Our framework is validated on three structures of interactions of increasing complexity. First, we study a population with all-to-all interactions at all orders, for which we can derive in a full analytical manner the Lyapunov exponents of the system, and for which we investigate the effect of including attractive and repulsive interactions. Second, we apply the multiorder Laplacian framework to synchronization on a synthetic model with heterogeneous higher-order interactions. Finally, we compare the dynamics of coupled oscillators with higher-order and pairwise couplings only, for a real dataset describing the macaque brain connectome, highlighting the importance of faithfully representing the complexity of interactions in real-world systems. Taken together, our multiorder Laplacian allows us to obtain a complete analytical characterization of the stability of synchrony in arbitrary higher-order networks, paving the way toward a general treatment of dynamical processes beyond pairwise interactions.

中文翻译：

---

多阶拉普拉斯算子，用于高阶网络中的同步

耦合代理系统中同步的出现是物理学，生物学，计算机科学和神经科学中的关键现象。传统上，交互系统已被描述为网络，其中链接仅根据节点之间的成对影响来编码信息。但是，在许多系统中，单元之间的交互是在较大的组中进行的。最近的工作表明，振荡器之间存在高阶相互作用会严重影响新兴的动力学。但是，这些早期研究大多考虑了一次最多四个振荡器的相互作用，并且分析处理仅限于所有情况。在这里，我们提出了一个通用框架，该框架使我们能够有效地研究所有可能阶次的高阶相互作用都考虑在内的振荡器群，用于任意超图描述的任何复杂拓扑，以及一般的耦合函数。为此，我们引入了一个多阶拉普拉斯算子，其频谱决定了同步解决方案的稳定性。我们的框架在日益复杂的交互的三种结构上得到了验证。首先，我们研究了具有所有阶跃的所有相互作用的种群，对此我们可以以完整的分析方式得出系统的Lyapunov指数，并为此研究包括吸引性和排斥性相互作用的影响。第二，我们将多阶Laplacian框架应用于具有异质高阶交互作用的合成模型上的同步。最后，我们仅将耦合振荡器的动力学与高阶和成对耦合进行比较，用于描述猕猴大脑连接体的真实数据集，强调了如实表示真实世界系统中交互复杂性的重要性。综上所述，我们的多阶拉普拉斯算子使我们能够获得任意高阶网络中同步稳定性的完整分析特征，从而为除成对相互作用之外的动力学过程的一般处理铺平了道路。