On the convergence of nonlinear averaging dynamics with three-body interactions on hypergraphs

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 multibody interactions. Herein, we investigate discrete-time dynamics with three-body interactions, described by an underlying 3-uniform hypergraph, where vertices update their states through a nonlinearly weighted average depending on their neighboring pairs' states. These dynamics capture reinforcing group effects, such as peer pressure, and exhibit 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 converge 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.