New structures in complex systems

Networks represent a major modelling tool in complex systems and the natural sciences. When considering systems of interacting units, networks can only model pair interactions as represented by edges between nodes. This is a severe limitation when one tries to model higher order interactions, like triple interactions etc. Some higher order interactions may be reduced to systems of pair interactions, but as we will illustrate there are, for example, triple interactions which are not reducible to pair interactions for quite deep mathematical reasons (Borromean structures). Therefore there is a need for a new kind of structure extending and encompassing networks in such a way that we can describe and model truly higher order structures. We suggest that this can be done by the concept of a Hyperstructure as introduced in [1]. Hyperstructures encompass networks and hierarchies and incorporate the phenomenon of levelwise emergence. They represent a design principle for higher order structures. It is natural to ask how hyperstructures occur in the natural sciences and complex systems and how they may be synthesized. We will discuss this, and relate it to recent work in synthetic chemistry, nuclear physics, quantum mechanical many body systems and ultracold gases. Furthermore, we will introduce the notion of hyperstructured (higher order) molecular architectures and hyperstructured (higher order) materials. We will present suggestions and conjectures on these matters.