# A Multi-Hop Clustering Mechanism for Scalable IoT Networks
## 3 Multi-Hop Clustering for IoT Networks
### The Proposed Mechanism
In Figure 2f, it is shown that when the number of entire nodes is 600, the number of selected coordinators is smaller for larger \(TR\) with a given \(H\), and vice versa. The ratio of coordinators to the entire nodes becomes smaller as the node density increases for all \(TR\) and \(H\) values in Figure 3. Moreover, Figures 2a-e and 3a-e show that the performance of the proposed mechanism in a large scale network correspond to the one in a small scale network seen in Figure 1a,b respectively. For example, the number of the selected coordinators increases as the number of the IoT nodes, i.e., \(\left|N\right|\) grows regardless of TR as seen in Figures 1a and 2a-e for \(H=1\). Furthermore, it is smaller for larger \(TR\) and \(H\). The reason is shown in Figure 4, that is, the average number of member nodes connected to a coordinator increases as the node density, \(TR\), or \(H\) increases. To check the effectiveness of \"the coordinator reassignment step\" of the proposed mechanism, the hop count from a node to its coordinator is measured before and after the coordinator reassignment step and the average values are compared in Figure 5. If the \(TR\) value is small and/or the node density is low, the connectivity among nodes becomes poor and it is more likely that a node connected to a coordinator to which the hop count is less than the maximum hop count constraint. Therefore, the difference between the maximum hop count constraint and the average hop count between nodes

Figure 1: Number (**a**) and ratio (**b**) of selected coordinators in terms of \(\left|N\right|\). and their coordinators is larger for a smaller TR value and for a lower node density at a given \(TR\) value in Figure 5.

Figure 3: Performance comparison for various \(H\) values = {1, 2, 3, 4, 5} measured as ratio of coordinators to the entire nodes under varying \(|N|\); for (**a–e**) \(TR\) = 20, 40,..., 100, respectively, and for (**f**) the ratio of coordinators to the entire nodes in terms of various \(TR\)s with \(|N|\) = 600. Figure 2: Performance comparison for various \(H\) values = {1, 2, 3, 4, 5} measured as the number of coordinators under varying \(|N|\); for (**a–e**) \(TR\) = 20, 40,..., 100, respectively, and for (**f**) the number of coordinators in terms of various \(TR\)s with \(|N|\) = 600. Figure 4: Performance comparison for various \(H\) values = [1, 2, 3, 4, 5] measured as average number of member nodes for a coordinator under varying \(\left\lvert N\right\rvert\) ; for **(a-e)**\(TR\) = 20, 40,..., 100, respectively, and for **(f)** average number of member nodes for a coordinator in terms of various \(TR\)s with \(\left\lvert N\right\rvert\) =600. Figure 5: Average hop counts before and after the coordinator reassignment step (\(x\) axis: number of IoT nodes (\(\left\lvert N\right\rvert\)), \(y\) axis: number of hops). In Figure 5, the difference between the average hop count values before and after the coordinator reassignment step becomes larger for larger \(H\) values and as the node density increases and/or the \(TR\) becomes larger. This is because more candidate coordinators are available within the maximum hop count constraint as the maximum hop count constraint or the degree of connectivity among the network nodes increases. That is, the effect of the coordinator reassignment step becomes greater as \(H\), \(TR\), or the node density becomes larger. ## 5 Conclusions

We proposed a multi-hop clustering mechanism for IoT networks to minimize the number of required Internet connections. Specifically, the proposed mechanism tries to select the minimum number of coordinators, which take the role of a representative node for the cluster, i.e., having the Internet connection on behalf of the rest of the nodes in the cluster and maps a partition of the IoT nodes onto the selected set of coordinators to minimize the total distance between the nodes and their respective coordinator under a given constraint in terms of the maximum hop count between the IoT nodes and their respective coordinator.