  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.
