## 2 System Model  
In this section, a multi-hop CRN is considered, which consists of \(M\) PUs and \(N\) SUs. The licensed spectrum bands are assigned to PUs, and each Primary User (PU) occupies the specific sub-band with regularity. SUs have no authority to utilize any frequency bands and opportunistically access channels only when PUs are inactive. The communication infrastructure is abstracted as a directed graph \(G(N,L)\), where each vertex \(i\in N\) denotes a Secondary User (SU) node in the network. The directed link \((i,j)\in L\) on the graph represents a communication link between the SU nodes \(n_{i}\) and \(n_{j}\) that exists when the nodes are within transmission range of each other and hold at least one common channel.  
The architecture of the CRN scenario is shown in Figure 1. In the networking scenario, the multi-hop CRN coexists with two centralized PU networks. PUs communicate with the PU base station through direct transmission. SUs transmit data flows from the source to the destination node in a multi-hop manner via CR routers. As Figure 1 shows, there exists an expert source node which has learned stable routing policy. On this foundation, a new source node generates data streams, and the location of a small number of intermediate nodes changes simultaneously.  
The available spectrum bands comprise the Data Transmission Channel (DTC) and the Common Control Channel (CCC). SU utilizes DTC for data transmission and accesses it in the absence of the PU. CCC is used to exchange command information between SU nodes, and it can be used at any time. Furthermore, the PU's occupation is modeled as an independent and identical alternation between two stages, i.e., ON status, when the DTC is occupied, and OFF status, when the DTC is idle [22]. The probability density function of the OFF periods is demonstrated as follows:  
\[g(\tau)=\left\{\begin{array}{cc}\lambda e^{-\lambda\tau}&\tau\geq 0\\ 0&\tau<0\end{array}\right. \tag{1}\]  
where \(\lambda\) is the mean OFF period of the PU. Hence, the probability that the PU arrives during the data transmission of SU \(n_{i}\) is calculated as  
\[\begin{array}{ll}P_{\textit{collision}}&=1-P(t\geq\tau_{i})\\ &=1-\int_{\tau_{i}}^{\infty}g(t)dt\\ &=1-e^{-\lambda\tau_{i}}\end{array} \tag{2}\]  
Here \(\tau_{i}\) indicates the single-hop transmission latency of SU \(n_{i}\) and is calculated as:  
\[\tau_{i}=\frac{S_{packet}}{B\log_{2}\!\left(1+\frac{S_{i}p_{i}}{\xi+\sigma_{i \text{j}}^{\text{PU}}}\right)} \tag{3}\]  
Figure 1: Multi-hop cognitive radio networking scenario.  
where \(S_{packet}\) represents data packet size, \(B\) denotes the bandwidth of DTC, \(g_{ijk}\) is the power gain between the node \(n_{i}\) and \(n_{j}\), \(\phi_{ijc}^{PUI}\) represents the adjacent channel interference of PUs at node \(n_{i}\), and \(\xi\) is the AWGN power.  
To capture the energy efficiency of SUs, the Power Consumption Ratio (PCR) of SU \(n_{i}\) is introduced as follows:  
\[\eta_{i}=\frac{p_{i}}{B\log_{2}\!\left(1+\frac{S_{ijk}p_{i}}{\xi+\phi_{ijc}^{ PUI}}\right)} \tag{4}\]  
PCR indicates power consumption as unit throughput is obtained. The parameters described in (4) are the same as in (3), which will not be repeated again.