Schützenberger Symmetries in Network Structures

Abstract

It is known that the set of all networks of fixed order form a semigroup. This fact provides for the Green's L, R, H and D equivalence classifications of all such networks. These classifications reveal certain structural invariants common to all networks within a Green's equivalence class and enables the computation of the associated invariant preserving symmetries that transform a network into another network within a Green's equivalence class. Here, the notion of Schützenberger symmetries in network structures is introduced. These are computable symmetries which transform any network within an H-equivalence class into another network within that class in a manner that preserves the associated structural invariants. Useful applications of Schützenberger symmetries include enabling the classification and analysis of biological network data, identifying important relationships in social networks, and understanding the consequences of link reconfiguration in communication and sensor networks.