## 3 SWIPT-Based Multi-Hop Uneven Clustering Routing Protocol Design for RF EH-CRSNs
### Complexity Analysis of ES-MUCRP
This process necessitates extensive control information exchanges among adjacent nodes, leading to significant energy usage. As a result, the first node death occurs earlier in WCM, DSAC, and NSAC protocols than in ES-MUCRP protocol, with a sharp decline in the number of active nodes in subsequent rounds. Fuzzy C-means and IMOCRP represent centralized, single-hop clustering routing protocols for CRSNs. These protocols mandate that each  
Figure 5: Comparison of the execution time across various protocols. surviving CRSN node transmits information such as remaining energy to the sink. The sink is responsible for choosing CHs and communicating the clustering results to all CRSN nodes. Consequently, their total control overhead per round is equivalent to the count of surviving nodes. The first node death in CogLEACH happens later than that in ES-MUCRP. This is due to the fact that the control overhead for CHs selection and cluster formation in CogLEACH is roughly twice the number of active nodes, which is relatively modest. In CogLEACH, each CH broadcasts temporary and final CHs announcements, and nodes not designated as CHs send temporary join requests and final confirmation to their CH. Despite lower energy usage under CogLEACH, Fuzzy C-means, and IMOCRP, restriction to single-hop communication with the sink substantially restricts the network scalability and monitoring capability. 3. According to the aforementioned analysis, the total control overhead per round of ES-MUCRP is close to that of WCM. However, as shown in Figure 4, its total energy consumption per round is much lower than that of WCM, NSAC, and DSAC. This is because nodes under ES-MUCRP exchange control information within the cluster radius during CHs selection and cluster construction stage, while nodes in other competing protocols exchange information within \(R_{t}\). Since the cluster radius is smaller than \(R_{t}\), nodes consume less energy in CHs selection and cluster construction. Moreover, in order to make full use of the direct communication between CHs in ring 1 and the sink, in ES-MUCRP, CHs in ring 1 send their state messages directly to the sink in route selection stage. The sink receives, aggregates, and broadcasts the message, which can reduce the number of control messages received by CHs in ring 2 and the energy consumption of competing for accessing CCC in ring 1. Thus, node energy is saved, and the network lifetime is prolonged. 4. From Figure 5, we can observe that the execution time of DSAC is the longest among all competing protocols. This results from its excessive control information exchange for cluster merging and multi-hop route selection. More competing nodes within \(R_{t}\) increases the time required for successful channel access. ES-MUCRP can reduce the time required for control information exchange by controlling the cluster radius, but more effective data gathering nodes will inevitably increase the execution time of data transmission. This is the price to pay for guaranteeing powerful network surveillance capability. Nonetheless, data packets in ES-MUCRP can still reach the sink within the round time. 5. Additionally, from Figure 2 and Table 3, it is observable that the first node failure in S-MUCRP occurs at round 893. Beyond round 2588, ES-MUCRP protocol maintains a notably higher number of surviving nodes compared to S-MUCRP protocol. Upon calculation, it is found that compared to S-MUCRP protocol, ES-MUCRP protocol exhibits a 37.02% increase in the number of surviving nodes. A detailed analysis of the specific reasons is as follows. By integrating a linear EH mechanism, ES-MUCRP protocol enables CRSN nodes to identify the optimal RF energy source from either the sink or a PU occupying a channel based on the node's location information before cluster formation and route establishment in each round. This RF EH supplements the node's residual energy, delaying node depletion. The energy-intensive activities reduce the residual energy of some nodes to below the dormancy threshold in the network employing ES-MUCRP protocol.