  Hypergraphs naturally represent higher-order interactions, which persistently
appear from social interactions to neural networks and other natural systems.
Although their importance is well recognized, a theoretical framework to
describe general dynamical processes on hypergraphs is not available yet. In
this paper, we bridge this gap and derive expressions for the stability of
dynamical systems defined on an arbitrary hypergraph. The framework allows us
to reveal that, near the fixed point, the relevant structure is the
graph-projection of the hypergraph and that it is possible to identify the role
of each structural order for a given process. We also analytically solve two
dynamics of general interest, namely, social contagion and diffusion processes,
and show that the stability conditions can be decoupled in structural and
dynamical components. Our results show that in social contagion processes, only
pairwise interactions play a role in the stability of the absorbing state,
while for the diffusion dynamics, the order of the interactions plays a
differential role. Ours is the first attempt to provide a general framework for
further exploration of dynamical processes on hypergraphs.
