## 2 Related Works
### Cognitive Radio Model  
The demand for spectrum resources has significantly increased as a result of the quick expansion of wireless communication services brought on by the introduction of 5G communications. However, low spectrum utilization and the wastage of important resources have been caused by the fixed spectrum allocation mechanism and the limited supply of spectrum resources. In order to address these issues, [PERSON] et al. proposed Cognitive Radio (CR) [45] as a means to improve spectrum utilization. Cognitive radio allows wireless communication systems to be aware of their surroundings, to secondarily utilize the unused spectrum holes of authorized users, to adapt to environmental changes and to automatically adjust system parameters, and to utilize the spectrum in a more flexible and efficient way. Existing spectrum allocation models include graph-theoretic coloring models [46; 47], bargaining mechanism models [48], game theory models [49], etc. Models for spectrum distribution can optimize system advantages, but it is challenging to precisely manage user fairness; therefore, they cannot guarantee the absence of flaws like those related to user fairness and unethical user competition. Scholars have used different swarm intelligence algorithms to optimize spectrum allocation, such as genetic algorithms (GA) [50], particle swarm algorithms (PSO) [4], butterfly optimization algorithms (BOA) [13], cat swarm algorithms (CSO) [51] and other intelligent optimization algorithms, for spectrum allocation problems in cognitive radio. Most of the above algorithms suffer from premature convergence, stagnation and iterative solution drawbacks when solving the problem. To overcome these problems and maintain population diversity, an improved golden eagle algorithm based on a graph-theoretic coloring model is applied to the optimization of the spectrum allocation model in this paper. A conversion function is used to convert the search space from a continuous search space to a discrete search space. A suitable conversion function is required to increase the performance of the binary algorithm. Firstly, the conversion function is algorithm independent and does not affect the search capability of the algorithm; secondly, the computational complexity of the algorithm does not change. For continuous-to-discrete conversions, this work employs the Sigmoid function, which can be written as follows:  
\[S\Big{(}x_{j}^{i}(t)\Big{)}=\frac{1}{1+e^{-x_{j}^{i}(t)}}, \tag{10}\]  
where \(x_{j}^{i}(t)\) is the position of the \(j\)th dimension in the \(i\)th individual in the golden eagle population at the \(t\)th generation. The function of spectrum allocation is to build a reasonable allocation of available spectrum (channels) for cognitive users, which can satisfy the resource demand of cognitive users while effectively avoiding interference with primary users and maximizing the use of resources. Assuming that there are N secondary users (SUs) and M available channels in a certain region, the relevant definition of the graph-theoretic coloring model is as follows:  
**Theorem 1**. : _Idle matrix \(\boldsymbol{L}\ \boldsymbol{L}=\left\{l_{n,m}|l_{n,m}\in\left\{0,1\right\} \right\}_{N\times M}\), where \(l_{n,m}=1\) indicates spectrum \(m\) is available to SU \(n\) and \(l_{n,m}=0\) indicates spectrum \(m\) is not available to SU \(n\)._  
**Theorem 2**. : _Spectrum efficiency matrix \(\boldsymbol{B}\ \boldsymbol{B}=\left\{b_{n,m}|b_{n,m}\in R^{+}\right\}_{N\times M}\), where \(b_{n,m}\) is a positive real number representing the network benefit that SU \(n\) can receive after obtaining spectrum \(m\)._  
**Theorem 3**. : _Interference constraint matrix \(\boldsymbol{C}\ \boldsymbol{C}=\left\{c_{n,k,m}|c_{n,k,m}\in\left\{0,1\right\} \right\}_{N\times N\times M^{\prime}}\) where \(\boldsymbol{C}\) represents individual user conflicts on the \(m\)th spectrum band, \(c_{n,k,m}=1\) indicates that the simultaneous use of channel \(m\) by SU \(n\) and SU \(k\) causes interference, while \(c_{n,k,m}=0\) means no interference.