  Several biological and social contagion phenomena, such as superspreading
events or social reinforcement, are the results of multi-body interactions, for
which hypergraphs offer a natural mathematical description. In this paper, we
develop a novel mathematical framework based on approximate master equations to
study contagions on random hypergraphs with a heterogeneous structure, both in
terms of group size (hyperedge cardinality) and of membership of nodes to
groups (hyperdegree). The characterization of the inner dynamics of groups
provides an accurate description of the contagion process, without losing the
analytical tractability. Using a contagion model where multi-body interactions
are mapped onto a nonlinear infection rate, our two main results show how large
groups are influential, in the sense that they drive both the early spread of a
contagion and its endemic state (i.e., its stationary state). First, we provide
a detailed characterization of the phase transition, which can be continuous or
discontinuous with a bistable regime, and derive analytical expressions for the
critical and tricritical points. We find that large values of the third moment
of the membership distribution suppress the emergence of a discontinuous phase
transition. Furthermore, the combination of heterogeneous group sizes and
nonlinear contagion facilitates the onset of a mesoscopic localization phase,
where contagion is sustained only by the largest groups, thereby inhibiting
bistability as well. Second, we formulate the problem of optimal seeding for
hypergraph contagion, and we compare two strategies: tuning the allocation of
seeds according to either individual node or group properties. We find that,
when the contagion is sufficiently nonlinear, groups are more effective seeds
of contagion than individual nodes.
