Hypernetworks of complex systems

Abstract

Hypernetworks generalise the concept of a relation between two things to relations between many things. The notion of relational simplex generalises the concept of network edge to relations between many elements. Relational simplices have multi-dimensional connectivity related to hypergraphs and the Galois lattice of maximally connected sets of elements. This structure acts as a kind of backcloth for the dynamic system traffic represented by numerical mappings, where the topology of the backcloth constrains the dynamics of the traffic. Simplices provide a way of defining multilevel structure. This relates to system time measured by the formation of simplices as system events. Multilevel hyper-networks are classes of sets of relational simplices that represent the system backcloth and the traffic of systems activity it supports. Hypernetworks provide a significant generalisation of network theory, enabling the integration of relational structure, logic, and topological and analytic dynamics. They provide structures that are likely to be necessary if not sufficient for a science of complex multilevel socio-technical systems. © 2009 ICST Institute for Computer Sciences, Social Informatics and Telecommunications Engineering.