  Higher-order networks have emerged as a powerful framework to model complex
systems and their collective behavior. Going beyond pairwise interactions, they
encode structured relations among arbitrary numbers of units through
representations such as simplicial complexes and hypergraphs. So far, the
choice between simplicial complexes and hypergraphs has often been motivated by
technical convenience. Here, using synchronization as an example, we
demonstrate that the effects of higher-order interactions are highly
representation-dependent. In particular, higher-order interactions typically
enhance synchronization in hypergraphs but have the opposite effect in
simplicial complexes. We provide theoretical insight by linking the
synchronizability of different hypergraph structures to (generalized) degree
heterogeneity and cross-order degree correlation, which in turn influence a
wide range of dynamical processes from contagion to diffusion. Our findings
reveal the hidden impact of higher-order representations on collective
dynamics, highlighting the importance of choosing appropriate representations
when studying systems with nonpairwise interactions.
