## 4 Maintaining Maximum Coverage and Connectivity Scheme
### Design of the MCC-ACO Algorithm
If variable \(\vartheta\) is used to indicate the number of unfamiliar monitoring points, then we can use the improved heuristic value \(\eta_{ij}\) from the following equation:  
\[\eta_{ij}=1+\vartheta\left(s_{j}\right)+0.1\times Dist_{ij} \tag{12}\]  
where the first term is a constant whose main function is to prevent the denominator in Equation (10) from having a value of zero. The second term represents the potential number of events that can be monitored in the sensing radius of the next step, which aids in driving the node \(\eta_{ij}\) to move to the maximum candidate points and accelerate the convergence of the algorithm. In addition, we added the third term for the heuristic value (\(Dist_{ij}\) represents the Euler distance between the current node and the candidate node) and introduced the scaled distance variable. The function of this variable is to ensure that the choice of the next point is at a further grid point in order to obtain a sparse deployment network and reduce the total number of sensors. It should be noted that the valid candidate points refer to grid points that can monitor events. The movement relationship that comprises Equations (10) and (12) is termed a pseudorandom rule. This rule tends to select short grid points with high pheromone density as a movement direction. (3)  
Pheromone updating  
During the process of the event search, the node will release pheromones in the paths taken by it, and the local updating of pheromones is carried out using Equation (13):  
\[\tau_{ij}\left(t+1\right)=\left\{ \begin{array}{ll}\Psi&\tau_{ij}\left(t+1 \right)>\Psi\\ \left(1-\rho\right)\tau_{ij}\left(t\right)+\rho\sum\limits_{k=1}^{c}\Delta \tau_{ij}^{k}&else\end{array} \right. \tag{13}\]  
where \(\Psi\) is the threshold value and \(c\) is the number of mobile search nodes. \(\rho\) represents the evaporation coefficient of the existing pheromones and reflects the evolution status of the entire ant colony system. The magnitude of this variable directly affects the global search capabilities and convergence speed of the MCC-ACO, and its value is determined mainly based on practical experience and repeated debugging. \(1-\rho\) is used to simulate the natural evaporation process of the pheromones that are left on the movement path. The aim of introducing the natural evaporation process is to ensure that poor grid point selections will be removed slowly, along with the evaporation of the existing pheromones as the iteration progresses, in order to reduce the \"attractiveness\" to subsequent nodes. Therefore, the evaporation coefficient has a \"survival of the fittest\" function. \(\Delta\tau_{ij}^{k}\) represents the added pheromone on a grid point after the node has passed that grid point. This value is determined by the solution provided by that node. Nodes that have completed the task with the lowest number of hops will leave the highest level of pheromones on the grid point. This will attract the following ants and cause them to follow these grid points to continue to search for better solutions. Therefore, \(\Delta\tau_{ij}^{k}\) has a role as an empirical value, and it is defined as:  
\[\Delta\tau_{ij}^{k}=\
icefrac{{Q}}{{\mathcal{L}}} \tag{14}\]  
where \(Q\) is a constant that represents the number of tracks left by the ants, and its value is not affected by changes in \(\Delta\tau_{ij}^{k}\). \(L\) represents the distance of the movement of the node in the current iteration. It should be noted that the variable \(L\) shows the degree of global optimization. When the node has completed its task, the length of its movement distance is calculated before updating the pheromones in the grid points through which the node has passed. Thus, a good movement path for a node will attract subsequent node iterations, and the nodes will use this mutual exchange and attraction to complete the optimization behavior for a colony.