  Going beyond networks, to include higher-order interactions of arbitrary
sizes, is a major step to better describe complex systems. In the resulting
hypergraph representation, tools to identify structures and central nodes are
scarce. We consider the decomposition of a hypergraph in hyper-cores, subsets
of nodes connected by at least a certain number of hyperedges of at least a
certain size. We show that this provides a fingerprint for data described by
hypergraphs and suggests a novel notion of centrality, the hyper-coreness. We
assess the role of hyper-cores and nodes with large hyper-coreness in
higher-order dynamical processes: such nodes have large spreading power and
spreading processes are localized in central hyper-cores. Additionally, in the
emergence of social conventions very few committed individuals with high
hyper-coreness can rapidly overturn a majority convention. Our work opens
multiple research avenues, from comparing empirical data to model validation
and study of temporally varying hypergraphs.
