*Article* **Green's Classifications and Evolutions of Fixed-Order Networks**

#### **Allen D. Parks**

Electromagnetic and Sensor Systems Department, Naval Surface Warfare Center Dahlgren Division, 18444 Frontage Road Suite 327, Dahlgren, VA 22448-5161, USA; allen.parks@navy.mil

Received: 31 August 2018; Accepted: 20 September 2018; Published: 25 September 2018

**Abstract:** It is shown that the set of all networks of fixed order *n* form a semigroup that is isomorphic to the semigroup *BX* of binary relations on a set *X* of cardinality *n*. Consequently, *BX* provides for Green's *L*, *R*, *H*, and *D* equivalence classifications of all networks of fixed order *n*. These classifications reveal that a fixed-order network which evolves within a Green's equivalence class maintains certain structural invariants during its evolution. The "Green's symmetry problem" is introduced and is defined as the determination of all symmetries (i.e., transformations) that produce an evolution between an initial and final network within an *L* or an *R* class such that each symmetry preserves the required structural invariants. Such symmetries are shown to be solutions to special Boolean equations specific to each class. The satisfiability and computational complexity of the "Green's symmetry problem" are discussed and it is demonstrated that such symmetries encode information about which node neighborhoods in the initial network can be joined to form node neighborhoods in the final network such that the structural invariants required by the evolution are preserved, i.e., the internal dynamics of the evolution. The notion of "propensity" is also introduced. It is a measure of the tendency of node neighborhoods to join to form new neighborhoods during a network evolution and is used to define "energy", which quantifies the complexity of the internal dynamics of a network evolution.

**Keywords:** network classification; network evolution; network symmetries; Green's symmetry problem; network invariants; network internal dynamics; symmetry ensembles; propensities; energy
