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 hyper-graphs 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 hypernetworks 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.