**1. Introduction**

Symmetry is a principle which has served as a guide for the spectacular advances that have been made in modern science, especially physics. For example, the continuous translational symmetry of ordinary space and time guarantees the invariance of the laws of physics under such translations. Thus, any mathematical expression describing a physical system, whether subatomic or macroscopic, must be invariant under space and time translations.

Group theory is the mathematical language used to describe symmetry and its associated invariant properties (recall that an abstract group is a set *S* of elements together with a law of composition " ◦ " such that for *x*, *y*, *z* ∈ *S* (*i*) *x* ◦ *y* ∈ *S*; (*ii*) *x* ◦ (*y* ◦ *z*) = (*x* ◦ *y*) ◦ *z*; (*iii*) there is an identity element *e* ∈ *S* such that *x* ◦ *e* = *e* ◦ *x* = *x*; and (*iv*) for *x* ∈ *S* there is an inverse *x*<sup>−</sup><sup>1</sup> ∈ *S* such that *x* ◦ *x*<sup>−</sup><sup>1</sup> = *x*<sup>−</sup><sup>1</sup> ◦ *x* = *e*). As a simple example, the set *S* of 0◦, 90◦, 180◦, and 270◦ rotations in the plane of a square about its fixed center under "composition of rotations" form a symmetry group for the square (0◦ is the identity element and the inverse of an *X*◦ ∈ *S* rotation is a 360◦ − *X*◦ rotation). Each of these rotations is a symmetry which brings the square into coincidence with itself, i.e., they preserve the invariant shape of the square. A much more complicated example are the so called gauge symmetries of the standard model of physics which classify and describe three fundamental forces of nature (i.e., the electromagnetic, weak, and strong forces) in terms of groups (specifically, the unitary group *U*(1) of degree 1 and the special unitary groups *SU*(2) and *SU*(3) of degree 2 and 3, respectively).

In recent years, the notion of generalized symmetry has been introduced to further describe graph symmetry [1,2]. The generalized symmetries of a graph are a generalization of the notion of the automorphism group of a graph and are derived from the application of Green's equivalence relations to the endomorphism monoid of the graph (the automorphism group is a subgroup of the graph's endomorphism monoid). Since these symmetries and invariant properties are strictly associated with a single graph, they do not address properties that remain fixed when the connection topology of the graph changes.

An important problem in network theory is identifying those properties of networks that remain fixed (invariant) as the network's connection topology changes with time. It was shown in [3] that the set of all networks (i.e., all connection topologies) on a fixed number of nodes also forms a semigroup. There it was also shown that the application of Green's equivalence relations to this semigroup partitions the associated set of networks into equivalence classes, each of which contains many fixed node number networks with various connection topologies, such that all networks within each class share some identifiable invariant connectivity property. If the connection topology of a network changes such that its initial and final configurations are in the same equivalence class, then the initial and final configurations share a common invariant property. It follows that, in this context, Green's equivalence classifications can be useful for identifying invariant properties of networks which evolve within an equivalence class. Such connectivity invariants can be used, for example, to identify important actors in evolving social networks and to select communication network reconfigurations that will retain a desired connectivity between specific node sets.

Transformations between networks within an equivalence class which preserve the associated invariant connectivity properties are called "Green's symmetries". Here, in addition to reviewing the Green's classification of networks [3], the "Green's symmetry problem" is introduced and defined. This problem is to determine (by calculation) the ensemble set of all the Green's symmetries which evolve an initial network configuration into a final configuration within a fixed Green's *R* equivalence class or within a fixed Green's *L* equivalence class. As discussed below, each such symmetry encodes information about the internal dynamics of the evolution, i.e., how node neighborhoods in the initial network configuration are joined to form node neighborhoods in the final configuration such that the invariant properties are preserved.

Since the cardinality of such ensembles can be large, the statistical notion of propensity is introduced. This quantity provides measures of the overall tendency of node neighborhoods in an initial network configuration to associate and form node neighborhoods in the final network configuration. Propensities are used to define "propensity energies", which quantify the overall complexity of the internal dynamics of a network evolution, and "energies of evolution", which quantify the complexity of internal dynamical activity for an evolution produced by a specific ensemble symmetry.

The objective of this paper is to motivate the application of Green's symmetry principles to network science by demonstrating how Green's equivalence relations can be applied to: classify networks; identify associated structural invariants; determine symmetries that preserve these invariants; and define associated measures that quantify aspects of the internal dynamics of network evolutions. The remainder of this paper is organized as follows: To make this paper reasonably self-contained, the relevant definitions and terminology from semigroup theory are summarized in the next section (for additional depth and clarification the reader is invited to consult such standard references as [4,5]). The semigroup *BX* of all binary relations on a finite set *X* and the semigroup *Bn* of *n* × *n* Boolean matrices are defined and shown to be isomorphic to one another in Section 3. The semigroup of networks *NV* on a fixed set *V* of nodes is introduced and is shown to be isomorphic to *BV* in Section 4. This isomorphism provides for the Green's equivalence classifications of *NV* given in Section 5. Green's evolutions of networks and their associated invariant properties are discussed in Section 6. The "Green's symmetry problem" is defined in Section 7 and its satisfiability and computational complexity are discussed in Section 8. The information encoded in symmetries

as internal dynamics is detailed in Section 9. Symmetry "ensembles" and their "propensities" and "energies" are introduced in Section 10. A simple example illustrating aspects of the theory is presented in Section 11. Concluding remarks comprise the final section of this paper.
