## 3 Proposed Method
However, due to estimation errors in contact probabilities and unpredictable sequencing of meetings among mobile nodes, it is possible for numerous small or large clusters to form, some of which may have significant overlap with other clusters. To address this problem, we propose a novel strategy where each cluster (\(C_{i}\)) maintains an Overlapping Degree (\(OD\)) with other clusters, indicating the count of shared members among them. Assume \(C_{i}\) represents the \(i\)-th cluster, \(\aleph_{q}\) represents the \(q\)-th common member, and \(\beta_{\chi_{i}}\) represents the transmission range of the \(i\)-th \(\chi\), whereas every cluster head maintains a consistent maximum transmission distance (\(\beta_{\chi_{i}}\leq R\)). In cases where there is no overlap (\(OD=0\)), the cluster head can adjust its transmission range (\(\beta_{\chi_{i}}\)) to establish communication with other clusters. Additionally, we introduce a limit of common members between clusters. After experimenting with different values, we found that a maximum of five common members between clusters provides ideal clustering. Put differently, we establish a boundary wherein clusters can share a maximum of five common members and a minimum of two. Having a cluster size larger than five common members results in less ideal conditions with significant overlap among clusters. Moreover, the cluster head should not be a common member (\(\chi_{i}\
eq\aleph_{q}\)) among the clusters. This ensures that the clusters remain ideal and do not overlap completely, thereby increasing their clustering effectiveness. Furthermore, clusters interact with each other through common members and also have members that are unique to each cluster. In other words, each cluster has common members and non-common members between other clusters. Additionally, if members within each cluster are unable to communicate with their cluster head, they have the capability to independently increase their transmission range. Whenever a node has difficulty communicating with its cluster head, it can extend its transmission range. Our experiment shows that even a slight increase in transmission range allows a node to communicate with the cluster head, given their Euclidean distance proximity. In addition, clusters must have minimal connectivity with other clusters to facilitate communication through shared members. Ultimately, the cluster must satisfy one of the constraints listed below. \[(2\leq~{}OD\leq 5)~{}OR~{}(M_{i}\geq 5) \tag{8}\]  
According to Equation (8), if the specific cluster \(C_{i}\) has less than two members in common, the \(\chi_{i}\) should increase the transmit range to be able to connect with other clusters. In this way, the communication between the clusters is permanent and it becomes possible to send the packet in the whole network through common members. Furthermore, if the cluster is unable to satisfy the specified constraint (\(2\leq~{}OD\leq 5\)), it must consist of a minimum of \(M_{i}=5\) members. This ensures that the cluster head can extend its transmission range in cases where the cluster comprises fewer than five members (\(M_{i}<5\)). For clarity, we highlight that these numbers are obtained based on our simulation studies and the value depends on the network specifications. This implies that the values could be different within a different network with different specifications to achieve better results. We assume that if \(\chi_{i}\) has fewer than two common members (\(OD<2\)), or if the cluster head has fewer than five members (\(Mi<5\)), the cluster head will increase its transmission range by the amount of \(\Delta\), where \(\Delta\) represents the minimum distance between a cluster head and its members. We expand the transmission range of the cluster head by integrating the value of \(\Delta\) until it fulfills one of the conditions specified in Equation (8).