## 3 SWIPT-Based Multi-Hop Uneven Clustering Routing Protocol Design for RF EH-CRSNs
### Complexity Analysis of ES-MUCRP
The activation of the energy status control mechanism transitions these nodes from an active state to a sleep state, wherein they only engage in EH, thus preventing early node death due to substantial energy depletion from data transfers. Due to the introduction of the inter-cluster SWIPT mechanism, ES-MUCRP protocol allows nodes requiring relay to assess the necessity of energy replenishment through inter-cluster SWIPT, taking into account their own energy situation and that of the relay nodes. After round 2588, some relay nodes experience a sharp decline in energy due to extensive data relaying and forwarding. The activation of the inter-cluster SWIPT mechanism prevents the early death of relay nodes in critical positions and ensures a more uniform distribution of residual energy among the inter-cluster routing nodes. To demonstrate the effectiveness of the intra-cluster and inter-cluster SWIPT mechanisms, as well as the energy status management mechanism of ES-MUCRP, this paper introduces the metric of average network energy consumption variance to measure the balance degree of energy consumption among nodes. Simulation results are displayed in Figure 6. Specifically, first, the average network energy consumption \(aveE(r)\) for all nodes in the current round \(r\) is calculated according to Equation (34). \[aveE(r)=\frac{\sum\limits_{s=1}^{r}TotalE(s)}{r}, \tag{34}\]  
where \(TotalE(s)\) is the sum of the differences between the initial energy and the residual energy at the end of round \(s\) for all nodes, which is the cumulative network energy consumption of CRSNs as formulated in Equation (35). \[TotalE(s)=\sum\limits_{j=1}^{K}(E_{initial}(j)-E_{res}(j)) \tag{35}\]  
Subsequently, the average network energy consumption variance \(varE(r)\) for all nodes in round \(r\) is computed, as specified in Equation (36). \[\text{var}E(r)=\frac{\sum\limits_{s=1}^{r}\sum\limits_{t=1}^{s}\left(TotalE(t)- aveE(s)\right)^{2}}{r} \tag{36}\]  
Figure 6 indicates that beyond round 2588, the average network energy consumption variance of ES-MUCRP protocol is notably lower than S-MUCRP protocol, suggesting that ES-MUCRP protocol achieves a more uniform distribution of network energy consumption across nodes. Calculations show that compared to S-MUCRP protocol, the balance degree of energy consumption in ES-MUCRP protocol has improved by 17.69%. Besides network lifetime, the number of CRSN nodes that effectively gather data serves as a crucial indicator of clustering protocol performance, as it reflects the network  
Figure 6: Comparison of average network energy consumption variance for ES-MUCRP and S-MUCRP protocols. monitoring capabilities of each protocol. This paper records the number of nodes effectively collecting data under each protocol, as illustrated in Figure 7.  
Figure 7 reveals that the number of nodes effectively collecting data in ES-MUCRP protocol is significantly higher than that of CogLEACH, DSAC, WCM, NSAC, Fuzzy C-means, and IMOCRP protocols, and the reasons are given below. As a multi-hop clustering routing protocol, DSAC initially enables a larger count of nodes to route data to the sink through multiple hops, resulting in a higher number of nodes effectively collecting data. However, substantial control overhead leads to rapid energy depletion of nodes, with a marked decline in the number of surviving nodes and nodes effectively collecting data after round 618. CogLEACH, NSAC, Fuzzy C-means, and IMOCRP protocols function as single-hop CRSNs clustering routing protocols, limiting effective data collection to nodes that can reach the sink in a single hop. Furthermore, the number of nodes effectively collecting data in WCM and NSAC protocols drastically decreases along with the swift reduction in the number of surviving nodes. While building clusters, Fuzzy C-means fails to consider channel availability. Random channel selection may lead to the absence of a common channel between CMs and their CH, severely impacting the successful delivery of data packets.