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：物理学、生物学和社会科学等领域的复杂网络系统往往涉及超越简单成对相互作用的交互。超图作为描述和分析具有多体相互作用系统复杂行为的强大建模工具，本文研究了一种离散时间下含三体相互作用的非线性平均动力学：底层超图（以三元组作为超边）勾勒出这些相互作用的拓扑结构，而顶点通过相邻顶点对状态的加权、状态依赖性平均值来更新自身状态。该动力学能够捕捉群体效应（如同伴压力）等强化机制，并呈现出由初始状态、超图拓扑结构及更新非线性之间复杂相互作用所导致的高阶动力学效应。与仅含二体相互作用的图上的线性平均动力学不同，该模型不会收敛到初始状态的平均值，而是引起一种偏移。通过假设随机初始状态并对超图施加一定的正则性和密度假设，我们证明该动力学以高概率收敛到初始状态的乘法偏移平均值。我们进一步将这种偏移表征为描述初始状态和交互强度的两个参数的函数，同时将收敛时间表征为超图结构的函数。