## 3 Experiments and Results
### Pheromone Distribution Analysis
Subsequently, a decline is observed, but only when \(\rho=0.01\). Under \(\rho=0.001\), both mean and standard deviation continue to rise over time. This discrepancy can be attributed to the more efficient exploratory activity of ants in this setup, resulting in fewer ants being present in the network at any given moment to counteract ongoing pheromone evaporation, particularly with the higher evaporation rate of \(\rho=0.01\). The third plot illustrates the evolving number of ants in the network over time, starting at 100 and peaking at 200, indicating the entry of groups 1 and 2. As time progresses, ants exit more rapidly under \(\rho=0.001\) and slower under \(\rho=0.01\). The plots for \(g=5\) and \(n_{g}=40\) in Figure 5 follow a similar trend. Both the mean and the standard deviation exhibit an initial increase during the simulation, followed by a decline, but only when \(\rho=0.01\). Under \(\rho=0.001\), both metrics reach stabilization. As ants gradually exit the environment, their ability to counteract pheromone evaporation is more effective with a low evaporation rate (\(\rho=0.001\)), resulting in a higher degree of pheromone heterogeneity. Conversely, when the evaporation rate is higher (\(\rho=0.01\)), the remaining ants in the network struggle to counteract pheromone evaporation, leading to a decrease in the mean and heterogeneity. Concerning the number of ants in the environment, it initiatesat 40, with the fluctuations in the plot representing the intermittent entry of the five groups. Over time, ants exit at varying rates depending on the value of \(\rho\). In Figures 6 and 7, the plots for the configurations \(g=10\) and \(N_{g}=20\), as well as \(g=20\) and \(n_{g}=10\), exhibit a trend similar to previous configurations. For both configurations, both the mean and the standard deviation show an initial increase during the simulation, followed by a decline, but only under \(\rho=0.01\). Under \(\rho=0.001\), both metrics in both cases reach a stabilization. In other words, as ants gradually exit the environment their ability to counteract pheromone evaporation is more effective with a low evaporation rate (\(\rho=0.001\)), resulting in a prolonged period of higher pheromone heterogeneity. Conversely, when the evaporation rate is higher (\(\rho=0.01\)) the remaining ants in the network struggle to counteract pheromone evaporation, leading to a decrease in mean and standard deviation. Regarding the number of ants in the environment, they initiate at 20 and 10, respectively, with the fluctuations in the plot representing the intermittent entry of the other groups (10 and 20, respectively). For \(g=10\), it is clear that over time ants exit more rapidly when \(\rho=0.01\) than when \(\rho=0.001\), suggesting that the ants increasingly benefit from the remaining pheromones on the edges as time unfolds. In contrast, for \(g=20\) this pattern holds true but only until around 11,000 ticks, beyond which the two trends reverse. In Figure 8, for the configuration \(g=50\) and \(n_{g}=4\), both the mean and standard deviation show a continuous increase during the simulation when \(\rho=0.001\). However, with \(\rho=0.01\), the trend in the standard deviation, after an initial phase of increase, starts to decrease, followed by stabilization and a slight subsequent increase. This is in contrast to the mean, which experiences a more pronounced decrease. Furthermore, in this configuration a high degree of heterogeneity is sustained for a more extended period. These behaviors are closely linked to the evolving number of ants in the network, as observed in the third plot. Initially (for \(0<T<5000\)), groups of four ants engage in exploration, swiftly exiting with no noticeable peaks in the curve. Concurrently, other groups are launched, but their rapid exit is not reflected in the curve. In the subsequent time interval (\(5000<T<10\),000), there is an increase in the number of ants, marked by fluctuations in the curves.