高维单纯复形上Kuramoto振子的滞后和同步过程
Hysteresis and synchronization processes of Kuramoto oscillators on high-dimensional simplicial complexes with the competing simplex-encoded couplings
摘要：
最近对复杂系统动力学性质的研究指出了被称为单纯形复合体的隐藏几何特征的深刻影响，这些特征使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和动力学之间微妙的相互作用。在这里，我们调查的相位同步动力学下的竞争相互作用嵌入1-单纯形（边缘）和2-单纯形（三角形）面对一个均匀的4维单纯复杂。它的底层网络是一个1-双曲图，节点的度和谱维数之间的相关性超过了$d_s=4$。我们确定了时间平均系统的序参量，以提高同步水平。在不存在更高的相互作用的情况下，随着正的成对相互作用（$K_1>0$）的增加而连续达到完全同步，并且对于负的耦合（$K_1<0$）部分同步，没有明显的滞后。类似的行为发生在度保持的随机网络中。相反，在完全随机图和简单无标度网络中，负的成对耦合不存在同步。增加基于三角形的相互作用的强度$K_2\neq 0$逐渐阻碍由成对耦合促进的同步，并且非对称磁滞回线以朝向$K_1<0$分支的突然去同步化转变打开。然而，对于大量基于三角形的相互作用，挫折效应占上风，防止完全同步，并且突然过渡消失。这些发现揭示了自然系统（如人类连接体）中的高维单纯复合物可以调节其天然同步过程的机制。
英文摘要：
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 behaviours on the controlled-structure complexes can reveal the subtle interplay of geometry and dynamics. Here, we investigate the phase synchronisation dynamics under the competing interactions embedded on 1-simplex (edges) and 2-simplex (triangles) faces of a homogeneous 4-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 $d_s=4$. We determine the time-averaged system's order parameter to characterise the synchronisation level. In the absence of higher interactions, the complete synchrony is continuously reached with the increasing positive pairwise interactions ($K_1>0$), and a partial synchronisation for the negative couplings ($K_1<0$) with no apparent hysteresis. Similar behaviour occurs in the degree-preserved randomised network. In contrast, the synchronisation is absent for the negative pairwise coupling in the entirely random graph and simple scale-free networks. Increasing the strength $K_2\neq 0$ of the triangle-based interactions gradually hinders the synchronisation promoted by pairwise couplings, and the non-symmetric hysteresis loop opens with an abrupt desynchronisation transition towards the $K_1<0$ branch. However, for substantial triangle-based interactions, the frustration effects prevail, preventing the complete synchronisation, and the abrupt 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 synchronisation processes.
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