## 3 Experiments and Results
### Pheromone Distribution Analysis  
We now study the correlation between the temporal distribution of pheromones on the edges and the number of ants navigating the network during the simulation. This involves the systematic computation of the mean and standard deviation of pheromones on the edges, as well as counting how many ants are still in the network. \begin{table}
\begin{tabular}{c c c c c} \hline  & \multicolumn{2}{c}{\(\rho=0.01\)} & \multicolumn{2}{c}{\(\rho=0.001\)} \\ \hline
**Group** & **Success Rate** & **Cost** & **Success Rate** & **Cost** \\ \hline
1 & **1.000** & 1162.1375 & 0.950 & 1135.9887 \\
2 & 0.925 & 1158.7023 & **1.000** & 1111.2629 \\
3 & 0.975 &  
The mean and standard deviation of pheromones over time provide insights into how ants distribute the pheromone along the edges. The average quantity of pheromone deposited by ants is directly proportional to the number of ants in the network. As the number of ants increases, the quantity of pheromone deposited along the edges also increases. Conversely, when some ants exit the network, the pheromone is deposited in a lower quantity. The standard deviation of the pheromone, on the other hand, measures the degree of heterogeneity of the pheromone on the edges. A low value indicates a higher uniformity of the pheromone, leading to greater difficulty for the ants to exit the network since the edges have more or less the same level of pheromone. Conversely, high values denote a higher degree of heterogeneity of the pheromone and this information can be used by ants to exit the network. Finally, the number of ants that are present in the network at each instant of time allows us to better visualize the groups of ants that enter the network at regular intervals and their persistence in the network. This process was carried out for each group configuration and both values of \(\rho\). The resulting data are presented through three plots discussed before, which are obtained by selecting the best experiment that maximizes the number of ants that exit the network, considering a total of 10 experiments performed for each configuration. In Figure 3, the outcomes for \(g=1\) and \(n_{g}=200\) are illustrated. In the initial phase of the simulation, there is an increase in the mean of pheromones, coinciding with active exploration by a substantial number of ants in the environment. Subsequently, a decline occurs, more pronounced, with an evaporation rate of 0.01 and a slower stabilization at 0.001. This pattern underscores the deep connection between mean pheromone levels and the presence of ants in the network. As ants exit the network, there is a corresponding decrease in the pheromone quantity on edges, particularly clear at a higher evaporation rate. This decrease is attributed to the reduced ability of the remaining ants to counteract the ongoing pheromone evaporation. The second plot mirrors these dynamics, illustrating changes in the standard deviation. Also, in this scenario there is an initial rise in the value during the simulation's early phase, signifying an increase in the heterogeneity of pheromones as ants release them at the edges. As the ants gradually exit the network, the standard deviation decreases when \(\rho=0.01\) and stabilizes when \(\rho=0.001\). This is consistent with the idea that a smaller ant population in the network results in a diminished ability to counteract the evaporation rate, ultimately leading to a more uniform distribution of pheromone levels across the edges. The third plot depicts the evolving number of ants in the network over time, starting at 200, corresponding to the initial exploration group in the \(g=1\) configuration. As time progresses, ants exit with varied trends influenced by the \(\rho\) value. Notably, these trends undergo an inversion around \(14,000\) ticks. In Figure 4, the plots for \(g=2\) and \(n_{g}=100\) exhibit a trend similar to the previous configuration. Both mean pheromones and standard deviation show an initial increase during the simulation.