  Multi-body interactions can reveal higher-order dynamical effects that are
not captured by traditional two-body network models. In this work, we derive
and analyse models for consensus dynamics on hypergraphs, where nodes interact
in groups rather than in pairs. Our work reveals that multi-body dynamical
effects that go beyond rescaled pairwise interactions can only appear if the
interaction function is non-linear, regardless of the underlying multi-body
structure. As a practical application, we introduce a specific non-linear
function to model three-body consensus, which incorporates reinforcing group
effects such as peer pressure. Unlike consensus processes on networks, we find
that the resulting dynamics can cause shifts away from the average system
state. The nature of these shifts depends on a complex interplay between the
distribution of the initial states, the underlying structure and the form of
the interaction function. By considering modular hypergraphs, we discover
state-dependent, asymmetric dynamics between polarised clusters where
multi-body interactions make one cluster dominate the other.
