## 3 Methodology and Proposed Innovation
### PA-Centric Characteristics of Metrics  
The selection of the proposed metrics was precisely targeted. Each one corresponds to a specific, easily acquired, and broadly available aspect of network performance, and its accountability tackles the issues that they describe. However, beyond those obvious characteristics lie the distinctive common algebraic values that they share. All the utilized primary metrics are additive over the path. Therefore, as a path is constructed, the metric output becomes the sum of the individual measurements of the constituent components, whether they are nodes or links. Moreover, all metrics are classified as minimizable. Hence, by definition, the path that individually gathers the lowest values of the aforementioned metrics would also be the most preferable. Furthermore, it can be proven that all metrics adhere to the fundamental properties of the PA framework. The quality of being monotonic and isotonic enables a metric to always lead to valid and optimal results. The following theorems are utilized to prove the inherent monotonic and isotonic functionality of the employed \(PJ\), \(L\), \(RNR^{norm}\), and \(RCR^{norm}\) metrics. **Theorem 1** (Proof of Monotonicity). : _An additive-over-the-path metric that exhibits a less than or equal to preference is strictly monotonic._Proof of Theorem 1 for \(P\), \(L\), \(RNR^{norm}\), and \(RCR^{norm}\). : By the definitions in [46], any algebraically formulated metric where \((i)\) it is additive over the path, \((ii)\) the values of its mapping function \(\omega(l)\) are always positive, and \((iii)\) its order relation \(\preccurlyeq\) corresponds to the _less than or equal to_ preference exclusively does not increase the preference of a path while prefixed by a new link. Therefore, let a link \(l_{1}\) be extended by another link \(l_{2}\). Since \(l_{1},l_{2}\in E\), \(\oplus\equiv+\) from Equations (3), (6), (10), and (14), and \(\omega(l_{1}),\omega(l_{2})\in\mathbb{R}^{+}\) from Equations (1), (4), (8), and (12), then:  
\[\omega(l_{1}\oplus l_{2})=\omega(l_{1})\oplus\omega(l_{2})\equiv\omega(l_{1}) +\omega(l_{2})>\omega(l_{1}) \tag{15}\]  
Thus, the examined metrics \(PJ\), \(L\), \(RNR^{norm}\), and \(RCR^{norm}\) are proven strictly monotonic, since the above equation satisfies the implication of strict monotonicity (\(\preceq\rightarrow\prec\)) specified by Equation (16). \[\omega(l_{1})<\omega(l_{1})+\omega(l_{2})\Leftrightarrow\omega(l_{1})\prec \omega(l_{1}\oplus l_{2}) \tag{16}\]  
**Theorem 2** (Proof of Isotonicity). : _An additive-over-the-path metric that exhibits a less than or equal to preference is strictly isotonic._  
Proof of Theorem 2 for \(P\), \(L\), \(RNR^{norm}\), and \(RCR^{norm}\). : In [46], it is also established that any algebraically formulated metric where \((i)\) it is additive over the path and \((ii)\) its order relation \(\preccurlyeq\) corresponds to the _less than or equal to_ preference exclusively preserves the preference between any two paths with the same origin, when both are suffixed by the same new link. Accordingly, let a link \(l_{3}\) extend both links \(l_{1}\) and \(l_{2}\). Since \(l_{1},l_{2},l_{3}\in E\), \(\oplus\equiv+\) from Equations (3), (6), (10), (14), and \(\omega(l_{1}),\omega(l_{2}),\omega(l_{3})\in\mathbb{R}^{+}\) from Equations (1), (4), (8), and (12), then:  
\[\left. \begin{array}{l}\omega(l_{1}\oplus l_{3})=\omega(l_{1})\oplus\omega(l_ {3})\equiv\omega(l_{1})+\omega(l_{3})\\ \omega(l_{2}\oplus l_{3})=\omega(l_{2})\oplus\omega(l_{3})\equiv\omega(l_{2}) +\omega(l_{3})\\ l_{1},l_{2}\
i\omega(l_{1})<\omega(l_{2})\xrightarrow{\omega(l_{3})>0}\omega( l_{1})+\omega(l_{3})<\omega(l_{2})+\omega(l_{3})\\ \therefore\omega(l_{1})\prec\omega(l_{2})\Rightarrow\omega(l_{1}\oplus l_{3}) \prec\omega(l_{2}\oplus l_{3})\end{array} \right\} \tag{17}\]  
which satisfies the implication of strict isotonicity (\(\preceq\rightarrow\prec\)) for \(PJ\), \(L\), \(RNR^{norm}\), and \(RCR^{norm}\), as Equation (18) notates.