## 5 Experiment and Result  
Average Overlapping Degree (\(AOD\)) is defined as the average overlap degree between any two overlapping clusters in the network. Let us consider that \(A\) and \(B\) are the cluster head nodes in two clusters. We assume that \(OD\) is the maximum overlapping degree between the two clusters, where \(OD\
eq 0\). Two symmetrical circles with radius \(R\) = maximum \(\beta_{C}\) = maximum \(\beta_{X_{i}}\), where \(\beta_{C}\) is the range of communication between sensors, which are present in cluster \(A\) or \(B\). Notice that \(OD\) is specifically defined for clusters that overlap (\(OD\
eq 0\)). We calculate the \(AOD\) as the mean value of this random variable \(OD\). \(d_{(A,B)}\) is the Euclidean distance between the two nodes \(A\) and \(B\). In other words, \(d_{(A,B)}\) is a continuous random variable that can contain values ranging from 0 to \(2R\). In the case of \(d_{(A,B)}\) = 0, there should be complete overlaps between two clusters; on the other hand, when \(d_{(A,B)}\) is greater than or equal to \(2R\), there should be no overlap. In the simulation study, we investigated the degree of overlap with increasing \(\Delta\) value and without increasing \(\Delta\) value. Our method might not achieve complete overlap (100%). Moreover, there could be instances where a direct path between the source and destination is unavailable at certain times, but with \(\Delta\), achieving satisfactory \(AOD\) is feasible. In Figure 3A,B, it is shown that as the number of clusters increases, the average overlap between clusters increases. However, Figure 3B demonstrates a significant improvement in the overlapping degree between clusters when comparing clustering with and without increasing \(\Delta\). Therefore, this method enhances routing and reduces delay. During several practical experiments, utilizing 60 clusters, we observed a 34% improvement with increasing \(\Delta\) compared to without increasing \(\Delta\), and we also found an almost 98% overlap between clusters. Figure 4 shows the efficient performance ACRP method in the DR compared to other clustering methods such as QoS-OLSR [17], CBVRP [21], kRop [13], and AMRBC [20]. ACRP routing protocols illustrate a very high and approximately stable DR during the simulation. This is because ACRP considers the encounter rate, giving more weight to nodes in high-density areas to be considered as cluster heads. This strategy increases the likelihood of a high delivery ratio, contributing to the consistent performance observed across different node densities. Figure 4 illustrates that the ACRP, when deployed with 250 nodes, achieved a DR improvement exceeding 28% in comparison to QoS-OLSR. Furthermore, the AMRBC exhibited suboptimal performance, lagging behind other protocols. In contrast, ACRP outperformed AMRBC by a substantial margin, achieving an impressive 51% enhancement. Additionally, when considering the highest number of nodes (250),  
Figure 3: Overlapping percentages of ACRP ((**A**), without increasing \(\Delta\) value and (**B**), with increasing \(\Delta\) value). ACRP exhibited superiority over both kRop and CBVRP, with enhancements exceeding 5% and 13%, respectively. Our approach selects the cluster head based on the highest encounter rate and the highest remaining energy. This prioritizes both network connectivity and energy efficiency, ensuring that the cluster head effectively manages message forwarding while conserving energy resources. In contrast, the KRop method, which relies on the k-Means algorithm, may suffer from suboptimal cluster formations due to sensitivity to initial cluster centroid selection. Similarly, in the CBVRP approach, clusters formed may not consistently optimize communication efficiency or network coverage. However, with ACRP, by enforcing constraints on cluster overlap and member count, we aim to create optimal clusters that maximize both communication efficiency and coverage. The end-to-end average delay refers to the time taken for a packet to traverse a network from its source to its destination.