Embracing n-ary Relations in Network Science

Most network scientists restrict their attention to relations between pairs of things, even though most complex systems have structures and dynamics determined by n-ary relation where n is greater than two. Various examples are given to illustrate this. The basic mathematical structures allowing more than two vertices have existed for more than half a century, including hypergraphs and simplicial complexes. To these can be added hypernetworks which, like multiplex networks, allow many relations to be defined on the vertices. Furthermore, hypersimplices provide an essential formalism for representing multilevel part-whole and taxonomic structures for integrating the dynamics of systems between levels. Graphs, hypergraphs, networks, simplicial complex, multiplex network and hypernetworks form a coherent whole from which, for any particular application, the scientist can select the most suitable.