Subgraph centrality and clustering in complex hyper-networks

The representation of complex systems as networks is inappropriate for the study of certain problems. We show several examples of social, biological, ecological and technological systems where the use of complex networks gives very limited information about the structure of the system. Consequently, we extend the concepts of subgraph centrality and clustering for complex networks represented by hypergraphs: complex hyper-networks. The first parameter characterizes the node participation in different sub-hypergraphs and the second one characterizes the transitivity in the hyper-network through the proportion of hyper-triangles to paths of length two. Another measure characterizing the formation of triples of mutually adjacent groups in the hyper-network is also introduced. All of these characteristics are studied in three different hyper-networks: a scientific collaboration hyper-network, an ecological competition hyper-network and the hyper-network formed by the American corporate elite in 1999.