## 5 Numerical Studies and Performance Analysis of MPACO and CMPACO
### Performance Analyses of CMPACO Considering Dynamic Network Topology  
In order to verify the validity of the proposed CMPACO algorithm, numerical experiments are conducted on the dynamic changes of network topology, where during the learning process, a portion of the network changes. As shown in the previous section, the topology of the network often changes due to the sudden congestion that in real applications is often caused by road construction. The tests with CMPACO are analyzed and compared with tests with the ACO algorithm and MPACO algorithm. The test conditions and assumptions follow the descriptions in Tables 6 and 7 of Section 5.1. For the comparison with the previous methods used, the route topology of the maze is set to change after every (30 and 60) iterations to describe a network with a dynamic topology change. \(\sigma\) and \(t\) in Equation (20) are modeled in this experiment with (1.4 and 5), respectively. As shown in Figure 10, a network has two topological changes during the learning processes. Figure 11 depicts the amount of final pheromone using MPACO and CMPACO algorithm in the dynamically changing maze. Figure 11 shows that the x and y axes are the node coordinates on the final network, and the level contour indicates the amount of finally formed pheromones. In the case of the MPACO algorithm (Figure 11a), the pheromone distribution is strong, primarily at the start point; however, the pheromone is low in the movement route. This observation indicates that MPACO is not effective in generating the  
Figure 11: Pheromone distribution using MPACO and CMPACO in the maze. (**a**) Final pheromone distribution using MPACO in the maze; (**b**) Final pheromone distribution using CMPACO in the maze. Figure 10: A network with two topological changes during the learning process. final route considering dynamical topology changes. On the other hand, in the case of the CMPACO algorithm (Figure 11b), a comparatively strong pheromone is formed around the movement route. Table 11 summarizes the results of applying ACO, MPACO, and CMPACO. The results show that the proposed CMPACO has better performances than other algorithms in a network environment with variable topology. The test platform parameters follow Table 8.  
Figure 12 illustrates comparisons of the cost function values using ACO, MPACO, and CMPACO algorithms in the network. Each dotted line indicates the time when the topology changes. Taking into account the case of the ACO algorithm, the route search for the shortest distance takes the longest time in all dynamic environments. In the case of the MPACO algorithm, the performance of MPACO is more unstable than that of the proposed CMPACO algorithm. The MPACO algorithm updates the pheromone around the collision node by applying the GPR whenever the route changes in a dynamic environment. Then, there is a high probability that the collision node will fail to predict the amount of pheromone in the neighboring node, due to the sudden change. It is impossible to predict a new collision node caused by a dynamic network topology change and the pheromone of a changed route. In addition, an effective route cannot be produced by predicting the pheromone distributed in the previous collision node. However, the CMPACO algorithm updates the pheromone value discretely from the collision node to the radius node. As explained previously, CMPACO's discretized pheromone convolution is influenced less by the changes of network topology. The provided numerical example shows the effectiveness of the proposed algorithm, particularly when searching a complex maze with dynamic topology changes. In order to compare algorithms' performances, a number of networks with dynamic topological changes are tested using ACO, MPACO and CMPAO. As shown in Figure 13a,b, two more networks are configured with two dynamical topologies during the learning processes.