  Recent studies have shown that novel collective behaviors emerge in complex
systems due to the presence of higher-order interactions. However, how the
collective behavior of a system is influenced by the microscopic organization
of its higher-order interactions remains still unexplored. In this Letter, we
introduce a way to quantify the overlap among the hyperedges of a higher-order
network, and we show that real-world systems exhibit different levels of
hyperedge overlap. We then study models of complex contagion and
synchronization of phase oscillators, finding that hyperedge overlap plays a
universal role in determining the collective dynamics of very different
systems. Our results demostrate that the presence of higher-order interactions
alone does not guarantee abrupt transitions. Rather, explosivity and
bistability require a microscopic organization of the structure with a low
value of hyperedge overlap.
