## 3 Multi-Directional and Parallel Ant Colony Optimization (MPACO)
Although the MPACO algorithm is more successful than the ACO algorithm in static network environments, dynamic changes in network topologies might invalidate the pheromone-based route reasoning using the learning processes of MPACO. In order to overcome this issue, a new and efficient dynamic network framework (Convolution-type Multi-directional and Parallel Ant Colony Optimization (CMPACO) algorithm) is proposed. The new algorithm uses a discrete Gaussian convolution. Figure 5 illustrates how the pheromone is renewed when the collision nodes are detected using the proposed CMPACO algorithm. As shown in Figure 4(a), when a collision node is detected, the algorithm examines another collision node that shares the area within the given radius \(\alpha,\beta,\gamma\). The pheromone at each collision node is modeled as a step-based Gaussian distribution, as shown in Equation (20). Equation (20) is a discrete Gaussian distribution with mean 0 and standard deviation according to the discrete ranges(\(\alpha,\beta,\gamma\)) of the converted radius, while '\(t\)' is a parameter for controlling the concentration of diffused pheromone values. \[f(x)=\frac{t}{\sqrt{(2\pi\ast\alpha)}}exp^{-\frac{1}{2\alpha^{2}}x^{2}},\ \ \left\{ \begin{array}{ll}x=\alpha,&0<x\leq\alpha\\ x=\beta,&\alpha<x\leq\beta\ \,&0<\alpha<\beta<\gamma\\ x=\gamma,&\beta<x\end{array} \right. \tag{20}\]  
Then, pheromones in the shared regions are synthesized with the convolution of both the step-based Gaussian distributions, as shown in Figure 5b. As shown in Figure 6, the pheromone amount of A, B, and C in radii \(\alpha\), \(\beta\), and \(\gamma\) at node \(q\) are modeled using Equation (21). \[\left\{ \begin{array}{ll}\tau_{ij}\gets A,&if\ (ij)\in\pi\alpha^{2}\\ \tau_{ij}\gets B,&if\ (ij)\in\pi\beta^{2}-\pi\alpha^{2}\\ \tau_{ij}\gets C,&if\ (ij)\in\pi\gamma^{2}-\pi\beta^{2}\end{array} \right. \tag{21}\]  
Then, the pheromone convolution between both collision nodes is executed using Equations (20) and (21). Figure 7 shows the pheromone convolution using both discrete Gaussian distributions. The subsequent processes follow the processes of MPACO. While MPACO algorithm purses a node with the highest pheromone using the GPR process, topology changes in a target network might lead to mis-estimated nodes. However, the proposed CMPACO uses the discrete additions of the amount of diffusion of each pheromone when multiple collision nodes are located within a close radius,  
Figure 5: Discrete Gaussian convolution based pheromone renewal. (**a**) Detection of both Collison nodes; (**b**) Discrete Gaussian convolution based pheromone renewal. \begin{table}
\begin{tabular}{c c} \hline \hline
**Research Studies on Dynamic Networks** & **Application Fields** \\ \hline [PERSON], [PERSON] and [PERSON] [34] & Use dijkstra’s classic double bucket algorithm to find the network route \\ \hline [PERSON], [PERSON], [PERSON], [PERSON], [PERSON] and [PERSON] [35] & Intelligent system for transport fleet management \\ \hline [PERSON] and [PERSON] [36] & Route search for car navigation systems using the extended dijkstra algorithm \\ \hline [PERSON] and [PERSON] [37] & Optimal control theory to the problem of dynamic routing \\ \hline [PERSON] and [PERSON] [38] & Generate the shortest route applied ants algorithm and genetic algorithm \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Research applications on dynamic networks. as shown in Figure 7. The comparatively simple convolution-based node selection is influenced less by the dynamic changes in network topology. Then, it makes the finding of the effective route more robust against small network changes in the learning process. Section 5.2. provides the effectiveness of the proposed CMPACO algorithm with the examples of the dynamically changing networks