  We investigate consensus dynamics on temporal hypergraphs that encode network
systems with time-dependent, multi-way interactions. We compare this dynamics
with that on appropriate projections of this higher-order network
representation that flatten the temporal, the multi-way component, or both. For
linear average consensus dynamics, we find that the convergence of a randomly
switching time-varying system with multi-way interactions is slower than the
convergence of the corresponding system with pairwise interactions, which in
turn exhibits a slower convergence rate than a consensus dynamics on the
corresponding static network. We then consider a nonlinear consensus dynamics
model in the temporal setting. Here we find that in addition to an effect on
the convergence speed, the final consensus value of the temporal system can
differ strongly from the consensus on the aggregated, static hypergraph. In
particular we observe a first-mover advantage in the consensus formation
process: If there is a local majority opinion in the hyperedges that are active
early on, the majority in these first-mover groups has a higher influence on
the final consensus value - a behaviour that is not observable in this form in
projections of the temporal hypergraph.
