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arXiv 2003.09734nlin.AOphysics.soc-ph
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高阶网络同步的多阶拉普拉斯算子

Multiorder Laplacian for synchronization in higher-order networks

Maxime Lucas, Giulia Cencetti, Federico Battiston

备注：
Was "A multi-order Laplacian framework for the stability of higher-order synchronization"
发表信息：
Phys. Rev. Research 2, 033410 (2020)
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摘要：

传统上，交互系统被描述为网络，其中链接编码关于节点之间的成对影响的信息。然而，在许多系统中，互动发生在更大的群体中。最近的研究表明，振荡器之间的高阶相互作用可以显着影响同步。然而，这些早期的研究大多考虑的相互作用多达4个振荡器的时间，和分析处理仅限于所有对所有的设置。在这里，我们提出了一个一般的框架，使我们能够有效地研究人口的振荡器，高阶相互作用的所有可能的订单被认为是任意超图描述的任何复杂的拓扑结构，并为一般的耦合功能。在这个范围内，我们引入了一个多阶拉普拉斯算子，其谱决定了同步解的稳定性。我们的框架验证了三个结构的相互作用日益复杂。首先，我们研究了一个人口与所有的相互作用在所有订单，我们可以在一个完整的分析方式推导出系统的李雅普诺夫指数，并为我们调查的影响，包括吸引力和排斥力的相互作用。其次，我们应用多阶拉普拉斯框架同步的合成模型与异构高阶相互作用。最后，我们比较了耦合振荡器与高阶和成对耦合的动力学，对于描述猕猴大脑连接体的真实数据集，突出了忠实地表示现实世界系统中相互作用的复杂性的重要性。总之，我们的多阶拉普拉斯算子使我们能够获得一个完整的分析表征的稳定性的同步在任意高阶网络，铺平了道路的动力学过程的一般治疗超越成对的相互作用。

英文摘要：

Traditionally, interaction systems have been described as networks, where links encode information on the pairwise influences among the nodes. Yet, in many systems, interactions take place in larger groups. Recent work has shown that higher-order interactions between oscillators can significantly affect synchronization. However, these early studies have mostly considered interactions up to 4 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 scope, we introduce a multi-order 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 multi-order 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 multi-order Laplacian allows us to obtain a complete analytical characterization of the stability of synchrony in arbitrary higher-order networks, paving the way towards a general treatment of dynamical processes beyond pairwise interactions.

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