## 2 The Model
### Memory-Enhanced Arts and Knowledge Acquisition  
In our model, ants employ a navigation strategy that combines the use of pheromones and the calculated desirability of edges to make informed decisions about their movement within the environment. Pheromones act as a communication device among ants, directing them towards paths with higher concentrations of pheromones. Conversely, desirability is internally determined based on acquired information \(I_{ij}(t)\) about edge weights, achieved through two mechanisms. The first memory mechanism employed by ants involves traversing edges and storing the weights in their memory as _prior knowledge_\(\bar{w}_{p}^{k}\). As ants explore their environment, they accumulate information about the weights of the edges they traverse. The _prior knowledge_ represents the average weight of the visited edges, computed by summing the weights along a generic path (\(\pi(t)\)) and dividing by the number of edges in that path (\(n\)). This process, outlined in Equation (4), enables ants to establish a historical understanding of the environment. The _prior knowledge_ mechanism allows ants to form preferences for certain paths based on their past experiences. In essence, this memory mechanism reflects the ants' ability to internalize and use information about previously encountered paths, shaping their decision-making in navigating the environment. \[\bar{w}_{p}^{k}=\frac{1}{n}\sum_{i=1}^{n}w(\pi_{i},\pi_{i+1}); \tag{4}\]  
The second memory mechanism involves the acquisition of information about edge weights at the nearest endpoint, leading to the formation of their _local knowledge_\(\bar{w}_{l}^{k}\). Ants discern information (\(I_{ij}(t)\)) about the weights of edges in their immediate vicinity, considering the number of neighboring edges (\(m\)) at the position where this information is present (Figure 1). The _local knowledge_ is then determined as the average weight of these neighboring edges, as expressed in Equation (5). This mechanism allows ants to adapt their decision-making based on real-time information acquired from their immediate surroundings. By considering the specific context at the nearest endpoint, ants enhance their awareness of the current environment and adjust their desirability calculations accordingly. The combination of _prior knowledge_ and _local knowledge_ mechanisms enables ants to strike a balance between past experience and current information, facilitating efficient navigation and adaptation to their dynamic surroundings,  
\[\bar{w}_{l}^{k}=\frac{1}{m}\sum_{i=1}^{m}w(\pi_{i},\pi_{i+1}). \tag{5}\]  
The overall desirability value \(\eta_{ij}(t)\) is then determined using a decision tree (Equation (6)), considering the availability of information \(I_{ij}(t)\) and the type of knowledge:  
\[\eta_{ij}(t)= \begin{cases}\frac{1}{\bar{w}_{ij}}&\text{if $I_{ij}\
eq 0$ and $T\
eq 0$}\\ \frac{1}{\bar{w}}&\text{if $I_{ij}=0$ and $T\
eq 0$}\\ 1&\text{if $I_{ij}=0$ and $T=0$}\end{cases}  \tag{6}\]  
The _global knowledge_ (\(\bar{w}\)) is the mean of the prior and local knowledge:  
\[\bar{w}=\frac{\bar{w}_{p}^{k}+\bar{w}_{l}^{k}}{2}. \tag{7}\]  
When an ant is on a node \(i\), it calculates \(\eta_{ij}(t)\) considering the edge weight, with higher desirability for lower weights. If information \(I_{ij}(t)\) is available at either nearest endpoint, the ant inversely relates desirability to the edge weight. In scenarios where information \(I_{ij}(t)\) about neighboring edges is absent, the ant estimates desirability as the inverse of its global knowledge. This estimation is derived by averaging prior and local knowledge. Figure 2b illustrates this. Figure 1: Ant \(k\) positioned on node \(i\) perceives information \(I_{ij}(t)\) (green dot) regarding the weight of edge \((i,s)\) only when it is located at the nearest endpoint (**a**); otherwise, it remains unaware of it (**b**).