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、生物和社会科学等领域的复杂网络系统中，经常涉及超越简单的成对相互作用的情况。超图是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。本文研究三体相互作用下的离散时间非线性平均动态：一个基础的超图，包含三元组作为超边，描述这些相互作用的结构，而顶点通过周围成对状态的加权状态依赖平均值更新其状态。这种动态捕捉到增强的群体效应，如同辈压力，并表现出由初始状态、超图拓扑和更新的非线性之间复杂相互作用引起的高阶动态效应。与拥有两体相互作用的图的线性平均动态不同，该模型不会收敛到初始状态的平均值，而是会引起平移。通过假设随机初始状态，并对超图进行一些正则性和密度假设，我们证明了该动态以高概率收敛到初始状态的乘法平移平均值。我们进一步将移位表征为两个参数的函数，这两个参数描述了初始状态和相互作用强度，以及超图结构的收敛时间的函数。