**11. Example**

Let *E* → *F* be a Green's *R* evolution in *NV*, *V* = {1, 2} (or equivalently in *B*2), where (in *B*2)

$$E = \begin{bmatrix} \mathbf{0} & \mathbf{0} \\ \mathbf{1} & \mathbf{0} \end{bmatrix}, F = \begin{bmatrix} \mathbf{0} & \mathbf{0} \\ \mathbf{1} & \mathbf{1} \end{bmatrix},$$

with *Ic*(*E*) = {2} = *Ic*(*F*) (note that this evolution corresponds to the *τ* → *λ* Green's *R* evolution in [3]). Theorem 3 guarantees the existence of at least one *A* such that

$$EA = \begin{bmatrix} \begin{array}{cc} 0 & 0\\ 1 & 0 \end{array} \end{bmatrix} \circ \begin{bmatrix} \ a\_{11} & a\_{12} \\\ a\_{21} & a\_{22} \end{bmatrix} = \begin{bmatrix} 0 & 0\\ 1 & 1 \end{bmatrix} = F.$$

The disjunctive normal form logical expression (1) for this equation yields the following system of equations

$$(0 \land a\_{11}) \lor (0 \land a\_{21}) = 0 \ (0 \land a\_{12}) \lor (0 \land a\_{22}) = 0$$
 
$$(1 \land a\_{11}) \lor (0 \land a\_{21}) = 1 \ (1 \land a\_{12}) \lor (0 \land a\_{22}) = 1$$

which can be used to solve the associated Green's symmetry problem.

For the two equations in the second row of this system to be satisfied requires the assignment *a*11 = 1 = *a*12. By inspection it is seen that the complete system is satisfied when, in addition to these assignments, *a*21 and *a*22 each assume both values from the set {0, <sup>1</sup>}. Thus,

$$G\_{\ast 1} = G\_{\ast 2} = \left\{ \left[ \begin{array}{c} 1 \\ 0 \end{array} \right], \left[ \begin{array}{c} 1 \\ 1 \end{array} \right] \right\}$$

so that *γ* = |*<sup>G</sup>*∗<sup>1</sup>||*<sup>G</sup>*∗<sup>2</sup>| = 2·2 = 4 = |*I*R| and the evolution *E* → *F* is 4-satisfiable. The associated symmetry ensemble is the set

$$\mathcal{E}\_{\overline{\mathcal{R}}} = \left\{ \begin{bmatrix} 1 & 1 \\ 0 & 0 \end{bmatrix}, \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}, \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}, \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix} \right\} \equiv \{A^{(1)}, A^{(2)}, A^{(3)}, A^{(4)}\}.$$

To calculate the computational complexity of this Green's symmetry problem, refer to Section 8.1 and observe that *M*1 = ∅, *M*2 = {1}, and *Q*(1) = {1} = *Q*(2). Application of Theorem 4 yields CR = 2<sup>2</sup>−|*<sup>M</sup>*1| + 2<sup>2</sup>−|*<sup>M</sup>*1| = 22 + 22 = 8, i.e., four combinations of value assignments must be checked for each *j* since, according to Lemma 4, *aij* values cannot be assigned when *Fij* = 0 because *Mi* = ∅.

The propensity and propensity energy for the ensemble are

$$
\overline{A} = \begin{bmatrix} 1 & 1 \\ \end{bmatrix}
$$

and ER = 3, respectively, and the energies of evolution are <sup>E</sup>R*A*(1) = 2, <sup>E</sup>R*A*(2) = 2 = <sup>E</sup>R*A*(3), and <sup>E</sup>R*A*(4) = 3. Please note that this validates Lemma 6. These energies also indicate that *A*(1) produces the least energy of evolution in the sense that the evolution involves simpler internal dynamical activity than evolutions produced by the other symmetries in the ensemble.

To illustrate this further, first observe that *<sup>I</sup>*(*<sup>E</sup>*∗<sup>1</sup>) = {2}, *<sup>I</sup>*(*<sup>E</sup>*∗<sup>2</sup>) = ∅, and *<sup>I</sup>*(*<sup>F</sup>*∗<sup>1</sup>) = {2} = *<sup>I</sup>*(*<sup>F</sup>*∗<sup>2</sup>) (here the *j*th column vector is set directly equal to the nodes in the in-neighborhood of node *j*). It is also easily determined that the motions of the dynamics for: *A*(1) are *Ψ*1 = {1} = *Ψ*2; *A*(2) are *Ψ*1 = {1} and *Ψ*2 = {1, <sup>2</sup>}; *A*(3) are *Ψ*1 = {1, 2} and *Ψ*2 = {1}; and *A*(4) are *Ψ*1 = {1, 2} = *Ψ*2. By inspection it is found that each of these motions satisfies (3). Using *A*(4) as an example, it is seen that (3) yields the correct set theoretic relationship *<sup>I</sup>*(*<sup>E</sup>*∗<sup>1</sup>) ∪ *<sup>I</sup>*(*<sup>E</sup>*∗<sup>2</sup>) ⊆ *<sup>I</sup>*(*<sup>F</sup>*∗<sup>1</sup>) ∪ *<sup>I</sup>*(*<sup>F</sup>*∗<sup>2</sup>) or {2} ∪ ∅ ⊆ {2} ∪ {2} or {2} ⊆ {2} for both *Ψ*1 and *Ψ*2. Also note that the internal dynamics for *A*(1) are simpler than those for the other symmetries in the ensemble, in the sense that both of the *A*(1) motions are singleton sets, whereas at least one of the motions for the other symmetries is a doubleton set. This is consistent with the fact mentioned above that *A*(1) produces the least energy of evolution.

Now consider the energies of motion for each ensemble symmetry. They are easily calculated from the theory and are found to be:

$$\begin{aligned} \mathfrak{E}\_{\mathcal{R}}\left[A^{(1)};\Psi\_{1}\right] &= 1 = \mathfrak{E}\_{\mathcal{R}}\left[A^{(1)};\Psi\_{2}\right]; \\\\ \mathfrak{E}\_{\mathcal{R}}\left[A^{(2)};\Psi\_{1}\right] &= 1, \; \mathfrak{E}\_{\mathcal{R}}\left[A^{(2)};\Psi\_{2}\right] = 1; \\\\ \mathfrak{E}\_{\mathcal{R}}\left[A^{(3)};\Psi\_{1}\right] &= 1, \; \mathfrak{E}\_{\mathcal{R}}\left[A^{(3)};\Psi\_{2}\right] = 1; \end{aligned}$$

and

$$\mathfrak{E}\_{\mathbb{Z}}\left[A^{(4)};\Psi\_{1}\right] = 1 = \mathfrak{E}\_{\mathbb{Z}}\left[A^{(4)};\Psi\_{2}\right].$$

Thus, the motions associated with an *A*(1) evolution are the least energetic since

$$\mathfrak{e}\_{\mathcal{R}}\left[A^{(1)};\Psi\_{\rangle}\right] \le \mathfrak{e}\_{\mathcal{R}}\left[A^{(k)};\Psi\_{\rangle}\right], \ k = 2,3,4; j = 1,2... $$

This is also consistent with the fact that an *A*(1) induced evolution is the least energetic and involves the least complex internal dynamics.

Finally, observe that these results validate Theorem 6. In particular,

$$\mathfrak{e}\_{\mathcal{R}}\left[A^{(1)};\Psi\_{1}\right] + \mathfrak{e}\_{\mathcal{R}}\left[A^{(1)};\Psi\_{2}\right] = 2 = \mathfrak{e}\_{\mathcal{R}}\left[A^{(1)}\right];$$

$$\mathfrak{e}\_{\mathcal{R}}\left[A^{(2)};\Psi\_{1}\right] + \mathfrak{e}\_{\mathcal{R}}\left[A^{(2)};\Psi\_{2}\right] = 2 = \mathfrak{e}\_{\mathcal{R}}\left[A^{(2)}\right];$$

$$\mathfrak{e}\_{\mathcal{R}}\left[A^{(3)};\Psi\_{1}\right] + \mathfrak{e}\_{\mathcal{R}}\left[A^{(3)};\Psi\_{2}\right] = 2 = \mathfrak{e}\_{\mathcal{R}}\left[A^{(3)}\right];$$

and

$$\mathfrak{E}\_{\mathcal{R}}\left[A^{(4)};\mathcal{Y}\_1\right] + \mathfrak{E}\_{\mathcal{R}}\left[A^{(4)};\mathcal{Y}\_2\right] = \mathfrak{J} = \mathfrak{E}\_{\mathcal{R}}\left[A^{(4)}\right].$$

#### **12. Concluding Remarks**

The research documented in [3] was inspired by earlier research performed by Konieczny [6] and Plemmons et al. [7]. This paper has reviewed the results developed in [3], i.e., that the set of all networks on a fixed number of nodes can be classified using the Green's equivalence relations of semigroup theory and that all networks within a Green's equivalence class have a common structural invariant (neighborhoods or poset relationships between node sets generated by neighborhoods). By extension, it was deduced in this paper from these results that if a network evolves from an initial network configuration to a final network configuration such that both the initial and final networks are in the same Green's equivalence class, then the structural invariants for the class are preserved by the evolution. In addition, the Green's symmetry problem was also defined in this paper. This problem is to determine by computation all symmetries which produce a network evolution within a Green's *R* or a Green's *L* equivalence class (i.e., a symmetry ensemble). These symmetries were shown to be solutions to special Boolean equations whose form is dictated by semigroup theory. Each such symmetry encodes information about the internal dynamics of the associated evolution and an ensemble associated with an evolution was used to define propensities and energies which quantify aspects of the internal dynamics of the evolution. However, it should be noted that a practical limitation exists for solving the Green's symmetry problem. This occurs because the cardinality of symmetry ensembles associated with large real networks can be quite large, thereby requiring the use of considerable computational resources to solve such problems (see future research suggestions below).

In conclusion, it is believed that the results of this paper are new and not in general use (perhaps having the closest resemblance to these results are the applications of Green's relations to social networks [10] and automata theory, e.g., [11]). However, the results of this paper are important and should be of general interest to network science researchers and those working in areas of applied network theory. In addition to applications similar to those mentioned in Section 1 (actor identification in social networks and communication network reconfiguration), contemporary areas of frontier research, such as identifying emerging scientific disciplines, e.g., [12], analyzing brain connectivity, e.g., [13–15], and finding symmetries in engineering processes [16], could also benefit from the results of this paper.

Before closing it is worthwhile to mention several directions for related future research. First, because of the computational resources required to solve the Green's symmetry problem, it would be useful to investigate how sampling and statistics can be used to obtain symmetry sub-ensembles that effectively yield the same information about propensities and energies as the associated full ensemble. A second research area involves understanding symmetries and their computation for network evolutions occurring within Green's *H* and *D* equivalence classes. A third and potentially very interesting research area concerns determining the relationships (if any) between the theory developed in this paper and the relatively new theory of persistence that is used to analyze large data sets, e.g., [17].

**Funding:** This research was supported by a Grant from the Naval Surface Warfare Center Dahlgren Division's In-house Laboratory Independent Research program.

**Conflicts of Interest:** The author declares no conflict of interest. The funder had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, and in the decision to publish the results.
