Complex networked systems in fields such as physics, biology, and social sciences often involve interactions that extend beyond simple pairwise ones. Hypergraphs serve as powerful modeling tools for describing and analyzing the intricate behaviors of systems with multi-body interactions. Herein, we investigate a discrete-time nonlinear averaging dynamics with three-body interactions: an underlying hypergraph, comprising triples as hyperedges, delineates the structure of these interactions, while the vertices update their states through a weighted, state-dependent average of neighboring pairs' states. This dynamics captures reinforcing group effects, such as peer pressure, and exhibits higher-order dynamical effects resulting from a complex interplay between initial states, hypergraph topology, and nonlinearity of the update. Differently from linear averaging dynamics on graphs with two-body interactions, this model does not converge to the average of the initial states but rather induces a shift. By assuming random initial states and by making some regularity and density assumptions on the hypergraph, we prove that the dynamics converges to a multiplicatively-shifted average of the initial states, with high probability. We further characterize the shift as a function of two parameters describing the initial state and interaction strength, as well as the convergence time as a function of the hypergraph structure.

翻译：在物理学、生物学和社会科学等领域的复杂网络系统中，相互作用往往超越简单的成对关系。超图作为强大的建模工具，可用于描述和分析具有多体相互作用的复杂系统行为。本文研究了一种具有三体相互作用的离散时间非线性平均动力学：以三元组作为超边的基础超图刻画了这些相互作用的结构，而顶点则通过相邻节点对状态的加权、状态依赖型平均值来更新自身状态。该动力学捕捉了强化群体效应（如同侪压力），并展现出由初始状态、超图拓扑结构及更新非线性之间复杂相互作用产生的高阶动力学效应。与具有二体相互作用的图上的线性平均动力学不同，该模型不会收敛至初始状态的平均值，而是会产生偏移。通过假设初始状态随机分布，并对超图施加一定的正则性与密度条件，我们证明该动力学以高概率收敛至初始状态的乘法偏移平均值。我们进一步将偏移量刻画为描述初始状态和相互作用强度的两个参数的函数，并将收敛时间表征为超图结构的函数。