高阶动态网络的降维
Dimension Reduction of Higher-Order Dynamical Networks
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中文总结 AI 辅助
本文针对仅含高阶相互作用的网络动力学系统，提出一种基于有效高阶相互作用强度（$\beta_{\triangle}$）的一维降维方法，经多模型验证可高精度捕捉系统有效稳态与相变。
中文摘要 AI 辅助
低维约化是研究复杂网络上高维动力学的有用框架，但现有多数方法局限于成对相互作用。本文针对仅含高阶相互作用的网络动力学系统，提出一种一维约化方法，该方法通过有效高阶相互作用强度（$\beta_{\triangle}$）构建，与底层网络的三角相互作用及动力学系统的有效状态相关联。本文为该降维方法建立了理论框架，并在三个仅含高阶相互作用的动力学模型中验证了其有效性。研究发现，降维精度主要由节点状态的同质性决定，即状态值偏差极小。合成网络与真实网络的数值结果表明，该约化模型能以良好精度捕捉完整系统的有效稳态与相变。
英文摘要
Low-dimensional reductions provide a useful framework for studying high-dimensional dynamics on complex networks, but most existing approaches are restricted to pairwise interactions. Here, we develop a one-dimensional reduction for dynamical systems on networks with purely higher-order interactions. The reduction is formulated through an effective higher-order interaction strength ($β_Δ$), associated with the triangular interactions of the underlying network and the dynamical system's effective state. We present a theoretical framework for the dimension-reduction approach and validate it across three dynamical models with exclusively higher-order interactions. We find that the reduction accuracy is mainly determined by the homogeneity of node states, i.e., the deviations in state values become very small. Numerical results on synthetic and real networks show that the reduced model captures the effective steady states and transitions of the full system with good accuracy.
