## 2 Materials and Methods
### Analyzing NCMs for Decision Support
#### 2.2.2 Dynamic Analysis of NCMs
The system nonlinearly transforms the weighted input to each node \(C_{i}\), as follows:  
\[C_{i}(t_{n+1})=f\Big{[}\sum_{k=1}^{N}e_{ki}(t_{n})C_{k}(t_{n})\Big{]} \tag{7}\]  
The summation term \(\sum_{k=1}^{N}e_{ki}C_{k}\) is known as the activation of the concept node. The nonlinear function, \(f\), can be a simple thresholding operation with a threshold value \(T\), where the output, \(C_{j}\), is given by the following:  
\[C_{j}=\left\{ \begin{array}{l}O\;if\;activation_{j}\;\leq T\\ 1\;if\;activation_{j}\;\geq T\\ I\;if\;C_{j}\;is\;not\;an\;integer\end{array} \right. \tag{8}\]  
A thresholding function such as the one shown above results in binary concept values and is used in our study. **Definition 10** ([29]). : _A fixed point is a dynamical system's equilibrium state that is represented by a single state vector. Consider the NCM with \(C_{1}\), \(C_{2}\)... \(C_{n}\) as its nodes. For example, let us start the dynamical system by turning on \(C_{1}\). Assume the NCM settles with \(C_{1}\) and \(C_{n}\) on, i.e., the state vector remains (1, 0,.., 1); this neutrosophic state vector (1, 0,.., 0, 1) is referred to as the fixed point._  
**Definition 11** ([29]). : _If the NCM settles with a neutrosophic state vector repeating in the pattern \(A_{1}\)\(\to\)\(A_{2}\)\(\rightarrow\)\(\ldots\)\(\ldots\)\(A_{i}\)\(\to\)\(A_{1}\), this equilibrium is termed the NCM limit cycle._  
NCMs work in the following way. The activation values of the nodes represent different concepts in the problem domain. These activation values are set according to the current state of the problem domain. The value of each node depends on the information available about the concept that it represents. The NCM nodes are then allowed to interact with one another. According to [47], this interaction lasts until one of the following:  
1. The NCM stabilizes to a stable state (the fixed-point attractor), with certain ideas \"on\" and others not. 2. The NCM repeatedly cycles through the same set of output states (limit cycle). 3. The NCM shows unstable behavior (chaotic attractor), shifting states instead of stabilizing as in (1) and (2) above. Simple threshold NCMs quickly reach stable limit cycles or fixed points [44]. These limit cycles reveal \"hidden patterns\" within the causal network of the NCM. Consider the following neutrosophic digraph (Figure 7). The neutrosophic adjacency matrix N(E) for the directed graph (Figure 7) is shown below (Figure 8). Let us assume that we take the state vector \(C_{1}=(1\ 0\ 0\ 0\ 0\ 0\ 0)\). We will now examine the effect of \(C_{1}\) on N(E). \(C_{1}\)N(E) = \((0\ 1-1\ 1\ 1\ 0\ 0)\rightarrow(1\ 1\ 0\ 1\ 1\ 0\ 0)=C_{2}\)  
\(C_{2}\)N(E) = \((1+2,\ I,\ -1+1,\ 1\ 0\ 0)\rightarrow(1\ 1\ 0\ 1\ 1\ 0\ 0)=C_{3}=C_{2}\)  
Thus, \(C_{2}=(1\ 1\ 0\ 1\ 0\ 0)\) is a fixed point, indicating that the activation or ON state of concept 1 positively influences concepts 4 and 5. However, other factors, such as concept 2, remain indeterminate in their relationship to concept 1. Let us note some important rules that apply when using the usual operators in neutrosophic matrices. An element of a neutrosophic matrix can take the form of \(a+bI\), where \(a\) and \(b\) are real numbers and \(I\) represents the indeterminacy factor. Factor \(I\) then satisfies the following properties [25]: \(I^{2}=I,\ I+I+\ldots+I=nI,\ 0\times I=0\) and if \(k\in\ \mathrm{K}\) (the ring of real numbers), then \(k\times I=kI\).