  Contagion processes relying on the exposure to multiple sources are prevalent
in social systems, and are effectively represented by hypergraphs. In this
Letter, we derive a mean-field model that goes beyond node- and pair-based
approximations. We reveal how the stability of the contagion-free state is
decided by either two- or three-body interactions, and how this is strictly
related to the degree of overlap between these interactions. Our findings
demonstrate the dual effect of increased overlap: it lowers the invasion
threshold, yet produces smaller outbreaks. Corroborated by numerical
simulations, our results emphasize the significance of the chosen
representation in describing a higher-order process.
