Recent studies of dynamic properties in complex systems point out the profound impact of hidden geometry features known as simplicial complexes, which enable geometrically conditioned many-body interactions. Studies of collective behaviors on the controlled-structure complexes can reveal the subtle interplay of geometry and dynamics. Here we investigate the phase synchronization (Kuramoto) dynamics under the competing interactions embedded on 1-simplex (edges) and 2-simplex (triangles) faces of a homogeneous four-dimensional simplicial complex. Its underlying network is a 1-hyperbolic graph with the assortative correlations among the node's degrees and the spectral dimension that exceeds $d_s=4$. By numerically solving the set of coupled equations for the phase oscillators associated with the network nodes, we determine the time-averaged system's order parameter to characterize the synchronization level. Our results reveal a variety of synchronization and desynchronization scenarios, including partially synchronized states and nonsymmetrical hysteresis loops, depending on the sign and strength of the pairwise interactions and the geometric frustrations promoted by couplings on triangle faces. For substantial triangle-based interactions, the frustration effects prevail, preventing the complete synchronization and the abrupt desynchronization transition disappears. These findings shed new light on the mechanisms by which the high-dimensional simplicial complexes in natural systems, such as human connectomes, can modulate their native synchronization processes.

中文翻译：

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Kuramoto 振荡器在具有竞争单纯形编码耦合的高维单纯复形上的滞后和同步过程

最近对复杂系统中动态特性的研究指出了被称为单纯复合体的隐藏几何特征的深远影响，这些特征使得几何条件下的多体相互作用成为可能。对受控结构复合体的集体行为的研究可以揭示几何学和动力学之间微妙的相互作用。在这里，我们研究了嵌入在同质四维单纯形复形的 1-单纯形（边）和 2-单纯形（三角形）面上的竞争相互作用下的相位同步（Kuramoto）动力学。它的底层网络是一个1-双曲线图，其中节点的度数和超过的谱维数之间存在分类相关性$d_{秒}=4$. 通过对与网络节点相关联的相位振荡器的耦合方程组进行数值求解，我们确定了时间平均系统的阶次参数来表征同步水平。我们的结果揭示了各种同步和去同步场景，包括部分同步状态和非对称滞后环，这取决于成对相互作用的符号和强度以及三角形面上耦合促进的几何挫折。对于基于三角形的大量交互，挫折效应占主导地位，阻止了完全同步，突然的去同步转变消失了。这些发现为人类连接组等自然系统中的高维单纯复合体的机制提供了新的思路。