## 3 Multi-hop Cluster Based Stable Backbone Trees (MCBT)
### Validation of the Proposed Method and the Effect of Weight Factor (\(\omega\))  
If the flooding value selected in Floodmin-\(2d\) is the value computed in the \(0^{th}\) round, the node will declare itself a cluster head during cluster head selection. This node has higher residual energy or degree, since its flooding value is selected in Floodmax-\(d\). This is clear, if we examine the process of cluster creation. Nodes selected as cluster heads can be classified into two types. The first class includes nodes with the maximum flooding value within \(d\)-hop. The second class includes nodes with the maximum flooding value among \(d\)-hop neighbors' neighbors. Consider node \(x\) in the first class; node \(x\) updates the flooding values of its \(d\)-hop neighbors during FloodMax. Therefore, all nodes within the \(d\)-hop coverage area of node \(x\) will record the flooding value of node \(x\) in their FloodingArray. At the conclusion of Floodmin, node \(x\) will declare itself a cluster head and some \(d\)-hop neighbors will select node \(x\) as their cluster head. Consider node \(y\) in the second class. Although flooding value of node \(y\) in its FloodingArray is changed by a larger flooding value, its flooding value in the \(0^{th}\) round continues to propagate and consumes all of the smaller node's flooding values within its \(d\)-hop neighbors. Therefore, at the completion of Floodmin, node \(y\) receives its flooding value in the \(0^{th}\) round again. Every node in the second class will elect themselves as cluster heads based on the first cluster head rule, and some \(d\)-hop neighbors will choose node \(y\) as their cluster head. Hence, any node that survives the Floodmax stage will elect itself as a cluster head. That is, Floodmax enables us to select favorable nodes with a higher flooding value. This allows a stable backbone to be constructed and balances the energy consumption around the cluster heads for WSNs. We simulate our algorithm to confirm that the backbone is composed of nodes with higher energy and degree. The parameters of the simulation are shown in Table 2 of Section 4. Figure 3 shows the results of the average and standard deviation of the backbone nodes' residual energy as the weight factor changes. As the weight factor approaches 1, the weight of the nodes' residual energy increases and the average residual energy of the backbone nodes also increases (solid line in Figure 3). As the weight factor approaches 1, nodes with higher residual energy are selected as cluster heads in advance and the distribution of residual energy among backbone nodes is even (dotted line in Figure 3). Figure 4 shows the average and standard deviation of degree of backbone nodes as the weight factor changes. As the weight factor approaches 0, the weight of the node's degree increases and the average degree of the backbone nodes also increases (solid line in Figure 4). As the weight factor approaches 0, the nodes with higher degree are selected as cluster heads in advance. The distribution among degree of backbone nodes could be even (dotted line in Figure 4). This weight policy can be implemented to optimize in various environments of WSNs by changing the weight factor, \(\omega\). As shown in Figure 5, we could create an appropriate backbone using the flooding value. For example, in some WSNs, a cluster head frequently handles heavy inter-cluster traffic and coordinates many nodes. In this case, a longer expected lifetime for sensor networks can be achieved by giving a weight factor value approaching 1. Although cluster heads consume more energy than ordinary nodes, the network lifetime could be extended due to their higher residual energy. Alternatively, assuming that each node periodically transmits the sensing information to the cluster head in a multi-hop manner, the nodes near the cluster head tend to exhaust their energy sooner due to the energy consumption in forwarding packets. In this scenario, a value approaching 0 is assigned to the weightfactor.