## 3 SWIPT-Based Multi-Hop Uneven Clustering Routing Protocol Design for RF EH-CRSNs  
### Network Model  
The sink is located at the center of a circular network with a radius of \(R\), where \(K\) identical CRSN nodes and \(m\) PUs are evenly and randomly scattered. The network is segmented into \(z\) concentric rings, each with a width of \(R_{t}\), and the rings are sequentially arranged from the innermost to the outermost as ring 1, ring 2, \(\ldots\), ring \(z\). A semi-Markov ON/OFF model is employed for mimicking the dynamic spectrum utilization patterns of PUs [34], while the energy consumption quantification for CRSN nodes is based on the established energy usage model in wireless communications [35]. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**References**} & \multirow{2}{*}{**Target Network**} & \multirow{2}{*}{\begin{tabular}{c} **Protocol** \\ **Type** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Considering** \\ **Control** \\ **Overhead** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Applying** \\ **SWIPT** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **Applying** \\ **Downlink EH** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Single-** \\ **Hop/Multi-Hop** \\ \end{tabular} } \\ \hline
[11; 13; 14] & CRSNs &
\begin{tabular}{c} Uniform \\ (Centralized) \\ \end{tabular} & \(\times\) & \(\times\) & \(\times\) & Single-hop \\
[12] & CRSNs &
\begin{tabular}{c} Uniform \\ (Centralized) \\ \end{tabular} & \(\surd\) & \(\times\) & \(\times\) & Single-hop \\
[15] & CRSNs &
\begin{tabular}{c} Uniform \\ (Distributed) \\ \end{tabular} & \(\times\) & \(\times\) & \(\times\) & Single-hop \\
[16; 17; 19] & CRSNs &
\begin{tabular}{c} Uniform \\ (Distributed) \\ \end{tabular} & \(\times\) & \(\times\) & \(\times\) & Multi-hop \\
[18] & CRSNs &
\begin{tabular}{c} Uniform \\ (Distributed) \\ \end{tabular} & \(\times\) & \(\times\) & \(\times\) & Single-hop \\
[20] & CRSNs &
\begin{tabular}{c} Uniform \\ (Hybrid) \\ \end{tabular} & \(\times\) & \(\times\) & \(\times\) & Single-hop \\
[21] & CRSNs &
\begin{tabular}{c} Uniform \\ (Hybrid) \\ \end{tabular} & \(\times\) & \(\times\) & \(\times\) & Multi-hop \\
[22; 23] & CRSNs & Uneven & \(\times\) & \(\times\) & \(\times\) & Single-hop \\
[24; 25] & CRSNs & Uneven & \(\times\) & \(\times\) & \(\times\) & Multi-hop \\
[26] & WSNs & Uneven & \(\times\) & \(\times\) & \(\times\) & Multi-hop \\
[27] & CRSNs & Uneven & \(\surd\) & \(\surd\) & \(\surd\) & Multi-hop \\
[28] & \(-\) & \(-\) & \(-\) & \(\surd\) & \(\surd\) & — \\
[29] & WSNs &
\begin{tabular}{c} Uniform \\ (Centralized) \\ \end{tabular} & \(\times\) & \(\times\) & \(\surd\) & Single-hop \\
[30] & WSNs & Uneven & \(\times\) & \(\times\) & \(\surd\) & Multi-hop \\
[31] & WSNs &
\begin{tabular}{c} Uniform \\ (Distributed) \\ \end{tabular} & \(\times\) & \(\times\) & \(\surd\) & Single-hop \\
[32] & CRSNs & \(-\) & \(\times\) & \(\times\) & \(\surd\) & — \\
[33] & CRSNs &
\begin{tabular}{c} Uniform \\ (Centralized) \\ \end{tabular} & \(\times\) & \(\times\) & \(\surd\) & Single-hop \\ Ours & CRSNs & Uneven & \(\surd\) & \(\surd\) & \(\surd\) & Multi-hop \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of the characteristics of existing clustering routing protocols. data from neighbors within the same ring and cluster radius, the energy \(E_{forward}(j)\) used in forwarding data for outer rings, and the energy \(E_{control}(j)\) expended in control information exchange, as shown in Equations (3)-(5). These calculations, alongside the node's remaining energy \(E_{res}(j)\) and the energy harvested \(E_{EH}(j)\), are used to derive the EH-based energy state function \(EH_{\_}ESF(j)\), as indicated in Equation (6).