## 4 Abstract Concept Modeling with RANs
### RANs Proof of Hypothesis and Complexity  
At the beginning of this Section 4 it was hypothesized that nodes in the newly created layer symbolically represent abstract concepts of the five classes (Class-1, Class-2, Class-3, Class-4, and Class-5) of Toy-data. This hypothesis can be proven through classification operation using the RAN model generated with Toy-data. The classification experiment setup consists of 30 iterations of an experiment. Each experiment consist of 9 Research Design (RD)(see Table A3 in Appendix A.2), where, in every RD a 10-fold cross-validation procedure was applied. To carry out the evaluation operation _True-labels_, and _Test-labels_ are determined via Abstract Concept Labeling (ACL) operation of RANs (see Appendix A.3 for ACL's description). Further, these labels were used to form a multi-class confusion matrix for the 3 classes of IRIS data and with the aid of this confusion matrix, 4 metrics (i.e., Precision, Recall, F1-Score, and Accuracy) were calculated.  
Figure 5: Plot of Similarity Translation Function with respect to varying input values in range \([0,1]\).  
```
Input: Vector \([a_{1},\,a_{2},\,\ldots,\,a_{n_{a}}]\) as input at layer \(l\). Output: New activation vector \([A_{1},\,A_{2},\,\ldots,\,A_{n_{A}}]\) in layer \(l+1\). for Each node \(A_{j}\) in layer \(l+1\)do  Calculate Normalized Euclidean Distance: \(d_{j}=\frac{\sqrt{\sum_{i=1}^{n_{a}}(W_{j}-a_{i})^{2}}}{\sqrt{n_{a}}}\)  Transform \(d_{j}\) through STF Equation (3): \(A_{j}=f(d_{j}^{2})\) endfor  Where:  i = [1, 2,..., \(n_{a}\)].  j = [1, 2,..., \(n_{A}\)]. \(W_{j,i}\) is ILW see Equation (1).
```  
**Algorithm 1** Upwards Activation Propagation algorithm  
Multi-class Receiver Operating Characteristics (ROC) curves were also plotted for the five classes to support the classification experiment with Toy-data. The binary labels corresponding to the True-labels (obtained via ACL operation) were obtained using the method node-wise binary transformation of input True-label (see Appendix A.5). Further, the confidence scores for the binary vectors were calculated using the node-wise confidence-score calculation method (described in Appendix A.5).  
Table 2 not only shows the RAN's comparison with the other 5 classifiers but also that RAN indeed performed well in the classification process with a performance of 99% (ca.) for all classification metrics. Figure 6 shows the ROC-AUC analysis of RANs model with Toy-data, in this graph one can see that an average AUC for all the five classes is 99% (ca.). These results show the ability of RAN's modeling to identify the abstract concept where the three nodes (\(N_{1}\), \(N_{2}\), \(N_{3}\  
In RAN's algorithm the four operations have different complexities: (1) the concept identification process is expressed as \(O(f(n))\) where \(f(n)\) is the complexity of the concept identifier (or clustering algorithm); (1) the concept creation has complexity of \(O(k)\) where \(k\) is the number of clusters; (3) the inter layer learning also has complexity of \(O(k)\) because it is an assignment operation and is equal to number of identified cluster centers; (4) the upward activation operation has the complexity of \(O(n)\) when \(n\) is the number data instances. The overall complexity of the RAN's modeling for creating a single layer is expressed by Equation (4).  
\[T(n)=O(max\left\{O(f(n)),O(n)\right\}) \tag{4}\]  
\[f(n)=O(n^{(k+2/p)}) \tag{5}\]  
where: \(k\) is number of clusters; p is number of features.  
The time complexity of the K-means algorithm is given by Equation (5) and when K-means is chosen to be the concept identifier is, therefore, the \(T(n)=O(n^{(k+2/p)})\). Table 3 lists the time complexities of all the algorithms used in this article including the RAN's time complexities with both K-means and Affinity Propagation algorithms. In Table 3 the complexities of K-means and Affinity Propagation algorithms, in fact, are the complexities of the RAN's modeling because their complexities are greater than \(O(n)\).