An Improved Sparrow Algorithm Based on Small Habitats in Cooperative Communication Power Allocation

Abstract

To solve the power allocation problem of multiple relays in cooperative communication, a sparrow improvement algorithm based on small habitats is proposed. First, the small-habitat rule increases the diversity of the energy classification of communication nodes. The problem of the premature convergence of the algorithm is solved. Subsequently, the joint jump rule is designed to replace the local search rule, and the local search rule of the sparrow algorithm is modified to avoid the local wandering phenomenon. The improved search method solves the local solution problem of the most available function. Lastly, a validation experiment of the performance of the algorithm is carried out according to the IEEE CEC 2017 benchmark document set. The simulation verifies the practicality of various algorithms. The maximum evaluation number (max FE) of the objective function is calculated to compare the performance of various algorithms. The experimental results show that the improved algorithm can increase the diversity of species. The multi-point search capability and global merit search capability are improved. Additionally, the convergence speed and computational accuracy of the algorithm are improved. The results show that the improved method is effective in controlling power in collaborative communication. The energy control algorithm has some practicality. The Friedman and Wilcoxon test criteria are applied and the convergence speed and computational accuracy of the improved algorithm are shown to be higher than those of other algorithms. This indicates that the improved power control method has some practicality in collaborative communication.

1. Introduction

Low-power communication is an important research direction in the field of cooperative communication. Power allocation methods have become an important technology for reducing energy consumption [1,2]. Clustered networks are a class of networks with a high number of communication nodes, such as communication networks composed of drones. Due to the number of nodes, the energy that the network uses for communication is limited. Therefore, the rational allocation of energy consumed by nodes for communication in cluster networks is a hot topic of research for scholars [3]. The existing research results cannot be applied to the energy allocation of large-scale cluster networks, and improving the utilization of energy in cooperative communication is an urgent problem.

Research on power control is still ongoing and scholars have obtained a variety of results. In the early days, the number of communication nodes in the network was limited so the early research methods were mainly power-averaging strategies [4]. As the number of communication nodes increased, the channel state information changes became uneven and the probability of network errors increased so scholars began to focus on non-average power allocation algorithms [5,6,7]. Recent studies have shown that the gradient descent method is no longer able to solve the multicarrier power allocation function [8]. Artificial intelligence-based algorithms were introduced into communication engineering for solving multi-objective optimization problems [9] and have since been used to calculate the optimal functions of parameters such as the system BER, outage probability, and channel gain [10,11]. However, the above methods are prone to poor algorithm convergence, local optima, and high complexity and have been discarded.

In recent years, scholars have developed a keen interest in the study of meta-inspired search methods for energy allocation in multi-objective optimization problems and the sparrow algorithm and seagull algorithm have become the mainstream algorithms used to solve these types of problems [12]. In 2020, Xue, J. and Shen, B. first proposed the sparrow algorithm to solve multiple local optimization problems [13]. Because of its fast convergence speed and high accuracy in finding the best solution, this algorithm became a classical algorithm for solving multi-objective optimization. Since then, many scholars have continuously optimized its computational complexity and the sparrow algorithm has become a meta-inspired mainstream algorithm. Zhao, J. and Tang, D. et al. proposed the Chaos Theory Improved Sparrow Search Algorithm [14]. The algorithm overcomes the drawback of the traditional SSA method, which is prone to falling into local optima. The study improved the accuracy of the search results and convergence speed but the application environment required by the algorithm is more demanding. Ji, Y., Tu, J., and Zhou et al. improved the sparrow algorithm by combining Corsi variance and backward learning [15]. The algorithm has a faster convergence speed and more detailed search capability. Hamid, M. and Tavakkoli-Moghaddam, R. et al. proposed an improved sparrow algorithm based on a hybrid sine cosine approach [16]. The search range of the algorithm was improved by at least 49 orders of magnitude. However, the improved algorithm requires a high number of hardware devices. Duan, Y. and Liu, C. proposed a sparrow search algorithm based on the Sobol sequence and the vertical and horizontal interleaving strategy [17]. The convergence speed of this algorithm was nearly doubled. However, the algorithm cannot be applied to complex large-scale communication. Zhang, H. and Tang, M. et al. designed a sparrow search algorithm incorporating the Corsi variance [18]. The algorithm has high convergence speed and accuracy but the algorithm cannot solve engineering problems. Li, W. and Shi, R. designed an improved sparrow search algorithm with multi-strategy fusion [19]. The algorithm increases the diversity of populations and speeds up convergence. Karimullah, S. and Vishnuvardhan, D. proposed an adaptive variational sparrow search algorithm [20]. The algorithm can complete the multivariate function-solving problem with more input parameters. Although the above research completes the fast search problem, the sparrow algorithm still needs to be improved in terms of convergence speed and the optimization search process.

At present, it can be seen from

Table 1. Comparison of classical literature on improved sparrow algorithm.

Innovation Points	Research Content	Literature
Original algorithm	The algorithm searches and feeds in two parts; the nodes with high energy are defined as the searchers and the other nodes are defined as the feeders.	[13]
Searcher rules	The searcher algorithm was modified and the complexity of the algorithm was reduced.	[14,15,16]
Expanded search	The algorithm expanded the scope of the search by designing a joint search method.	[17,19]
Improved speed	A pre-processing process was added to improve search speed.	[18]
Improved search	Jump search was proposed to prevent the occurrence of local optima.	[20]
Cross-application	Integration with other algorithms to simplify the process of the algorithm.	[14,15]

that the improved sparrow algorithm for solving optimization functions still faces challenges such as premature convergence and local wandering. In addition, a clustered network has a large number of communication nodes and the sparrow algorithm is prone to local optima when solving. Therefore, the sparrow algorithm needs to be improved in order to be used for solving these types of problems. A sparrow algorithm based on small-habitat genetic factors has been proposed to solve the problem of premature convergence. The small-habitat rule can increase population diversity, speed up algorithm convergence, and prevent the occurrence of local optima during the optimization process. Since the original algorithm tends to fall into local optima, it has been improved as follows: (1) moving the convergence operation to a distant point, and (2) reducing the smooth movement of the nearest point and replacing it with a comparison of the nearest point jumps in multiple clusters. Simulation experiments were conducted to compare the effectiveness of the improved algorithm, and the experimental results showed that the improved algorithm outperformed other algorithms in terms of convergence speed and complexity. This study shows that the improved sparrow algorithm is practical and advanced for solving large-scale cluster network energy allocation applications.

2. Mathematical Model of Cooperative Communication

The cooperative communication process shown in Figure 1 consists of two main phases. The first phase of communication consists of the source node broadcasting a signal to the relay node, which is usually the base station or other signal source. The second phase of communication involves the relay node forwarding signals to the destination node, which is often a UAV or other type of mobile equipment [21,22]. In this study, the source node is defined as S. The relay node is defined as R. The set R consists of {R1,R2,R3,..., Rn}. The destination node is defined as D.

Optimal Relay Cooperative Communication

Energy distribution in cooperative communication occurs mainly in the second stage of communication, as shown in Figure 2. During the first stage of communication, the signals broadcast by the source node S are received by both the relay node R and the destination node D, resulting in signal gain for both nodes. During the second stage of communication, the signal from the best relay node is accepted by the destination node [5,6,23]. The signals received by the nodes in both phases can be expressed as a mathematical equation in Figure 2.

In the first stage of collaborative communication, the signal received by the destination node is represented as discussed in [5].

$$y_{\mathrm{s}d}=h_{sd}\sqrt{P_1}x+z_{sd}$$

(1)

where y denotes the signal received by the node, h denotes the link channel gain, P₁ denotes the energy consumed by the transmission, and Z denotes the channel noise.

The signal received by the relay node is represented as [6]

$$y_{s\mathrm{r}}=h_{s\mathrm{r}}\sqrt{P_1}x+z_{sr}$$

(2)

In the second phase of cooperative communication, the signal received by the destination node is represented as discussed in [5,6]

$$y_{\mathrm{r}d}=h_{rd}·\sqrt{P_2}·G·y_{s\mathrm{r}}+z_{rd}$$

(3)

In the formula, the amplification and forwarding coefficient of the relay node is

$$G={({(p_1+{\left|h_{sr}\right|}^2+z_{sr})}^{-1})}^{-\frac{1}{2}}$$

(4)

The destination section re-decodes the different signals according to the maximum integration ratio principle [4,5,6]. The expression of the final signal is

$$y_{\mathrm{s}d}=\frac{1}{N_{sd}}(h_{sd}\sqrt{P_1}{y}_{sd}+h_{sr}{h}_{rd}\sqrt{P_1}G{y}_{rd})$$

(5)

In collaborative communication, the energy of the system is mainly used by relay nodes to forward signals. To simplify the study process, it is assumed that the energy consumed by multiple relay nodes during cooperative communication can be mathematically expressed equivalently as

$${\mathrm{y}}_{\mathrm{c},\mathrm{k}}={\mathrm{w}}_{\mathrm{k}}{\mathrm{H}}_{\mathrm{c},\mathrm{k}}{\mathrm{t}}_{\mathrm{k}}{\mathrm{s}}_{\mathrm{c},\mathrm{k}}+{\mathrm{a}}_{\mathrm{r}\mathrm{c},\mathrm{k}}{\mathrm{w}}_{\mathrm{k}}\mathrm{b}\left({\mathrm{j}}_{\mathrm{c}}\right){\mathrm{a}}^{\mathrm{T}}\left({\mathrm{q}}_{\mathrm{r}}\right){\mathrm{u}}_{\mathrm{k}}{\mathrm{s}}_{\mathrm{r},\mathrm{k}}+{\mathrm{w}}_{\mathrm{k}}{\mathrm{n}}_{\mathrm{c},\mathrm{k}}$$

(6)

where W is the weight factor and the value is related to the number of nodes. k is the number of relay nodes and c is the best relay node number. H is the channel matrix involved in signal forwarding, and b and a are the energy consumption coefficients of the relay nodes for signal forwarding and link distribution, respectively. H, a, and b are solved as follows:

$${\mathrm{H}}_{\mathrm{c},\mathrm{k}}=\left[\begin{array}{ccc}{H}_{c,1,1,k}& \cdots & H_{c,1,N_T,k}\\ ⋮& \ddots & ⋮\\ H_{c,N_R,1,k}& \cdots & H_{c,N_R,N_T,k}\end{array}\right]$$

$$\begin{array}{l}\mathrm{b}\left({\phi }_c\right)={\left[1,e^{-j\pi \sin {\phi }_c},\cdots ,e^{-j\pi \left(N_R-1\right)\sin {\phi }_c}\right]}^T\hfill \\ \mathrm{a}\left({\theta }_r\right)={\left[1,e^{-j\pi \sin {\theta }_r},\cdots ,e^{-j\pi \left(N_T-1\right)\sin {\theta }_r}\right]}^T\hfill \end{array}$$

The energy of the communication system is limited. The energy is mainly supplied to the relay nodes for signal forwarding so the cooperative communication power allocation is mainly in the forwarding phase. Assume that the transmit power of the source node is P₁, the transmit power of the relay node is P₂, and the total power of the system is P. The constraints are P₁ + P₂ = P. The method of determining P₂ is the problem to be solved in this study.

The communication links are all independent and the links contain relay nodes that can be duplicated [24]. The energy consumed by the relay nodes is link-dependent. The minimum signal-to-noise ratio is the criterion for link communication, and the relationship between the energy consumed by each relay node and the link signal-to-noise ratio can be abstracted into a mathematical expression. The formula for the minimum SNR in the link where the K-th relay is located is as follows:

$$\begin{array}{ll}{\mathrm{SIN}\mathrm{R}}_{\mathrm{c},\mathrm{k}}& =\frac{\mathrm{E}\left\{{\left|{\mathrm{w}}_k^{\prime }{\mathrm{H}}_{\mathrm{c},\mathrm{k}}{\mathrm{t}}_{\mathrm{k}}{\mathrm{s}}_{\mathrm{c},\mathrm{k}}\right|}^2\right\}}{\mathrm{E}\left\{{\left|{\mathrm{a}}_{\mathrm{rc},\mathrm{k}}{\mathrm{w}}_k^{\prime }\mathrm{b}\left({\mathsf{\phi }}_{\mathrm{c}}\right){\mathrm{a}}^{\mathrm{T}}\left({\mathsf{\theta }}_{\mathrm{r}}\right){\mathsf{\pi }}_{\mathrm{k}}{\mathrm{s}}_{\mathrm{r},\mathrm{k}}{+\mathrm{w}}_k^{\prime }{\mathrm{n}}_{\mathrm{c},\mathrm{k}}\right|}^2\right\}}\\ & =\frac{{\mathrm{P}}_{\mathrm{c},\mathrm{k}}{\left|{\mathrm{w}}_k^{\prime }{\mathrm{H}}_{\mathrm{c},\mathrm{k}}{\mathrm{t}}_{\mathrm{k}}\right|}^2}{{\mathrm{w}}_k^{\prime }{\mathrm{R}}_{\mathrm{u},\mathrm{k}}{\mathrm{w}}_{\mathrm{k}}{+\mathsf{\sigma }}_{\mathrm{n},\mathrm{c},\mathrm{k}}^2{\mathrm{w}}_k^{\prime }{\mathrm{w}}_{\mathrm{k}}}\end{array}$$

(7)

where E denotes the energy, R is the energy utilization matrix, and σ is the energy distribution function.

$$\begin{array}{l}{\mathrm{R}}_{\mathrm{u},\mathrm{k}}=\mathrm{E}\left\{\overline{\mathrm{C}}{\mathrm{u}}_{\mathrm{k}}{\mathrm{u}}_{\mathrm{k}}^{\prime }\overline{\mathrm{C}}\prime \right\}\\ {\sigma }^2={\mathrm{a}}_{\mathrm{r}\mathrm{c},\mathrm{k}}\mathrm{b}\left({\mathrm{j}}_{\mathrm{c}}\right){\mathrm{a}}^{\mathrm{T}}\left({\mathrm{q}}_{\mathrm{r}}\right){\mathrm{s}}_{\mathrm{r},\mathrm{k}}\end{array}$$

The energy distribution function for multi-relay cooperative communication is constructed by calculating the signal-to-noise ratio of multiple links. The mathematical expression of the energy function for relay forwarding under the total power constraint is as follows:

$$\arg \min \left[{\mathrm{f}(\mathrm{P}}_2)\right]={\displaystyle {\sum }_{i=1}^K\left[K_i·\Delta f_i{\log }_2(1+{\mathrm{SNR}}_i)+{\delta }^k\right]}$$

(8)

$$s.t. P=P_1+P_2$$

After analysis, the power distribution function is a multivariate function. There are many optimal solutions for the multivariate function so the optimal solution cannot be solved using the gradient descent method. It is necessary to choose the best energy distribution coefficient among many nodes.

3. Optimal Power Allocation Solution

The smart algorithm is a mainstream algorithm for computing optimal solutions for multivariate functions [25,26,27]. Because the sparrow algorithm can calculate the optimal solution using only the location information, the improved sparrow algorithm is often used to calculate the energy distribution function. However, the sparrow algorithm has two main limitations. One is that it can converge prematurely and the other is a wandering phenomenon that can slow down the solution speed.

3.1. Small-Habitat Delineation Groups

The sparrow algorithm can suffer from premature convergence because of its reliance on a single population. Increasing population diversity is a common method for avoiding this issue. Cluster networks require a finite number of populations and the small-habitat rule is the best way to increase the population diversity of cluster networks.

The microhabitat law is a method of dividing populations in biology [28]. The rules for dividing populations are designed to mimic the activities of similar organisms. High-quality individuals are identified and filtered using a suitable individual distance threshold. Then, more populations are generated by mutation and hybridization between these good individuals. The penalty function in the rule is modified to divide the nodes for cluster communication. After dividing the network into several populations according to the rule that each node is less than the average node distance, the excellent nodes are selected by calculating the sharing functions G(X,Y) for each population.

$$G(X,Y)=\{\begin{array}{c}\begin{array}{cc}1& ,\mathrm{d}(\mathrm{X},\mathrm{Y})=0\end{array}\\ \begin{array}{cc}1-(\frac{\mathrm{d}(\mathrm{X},\mathrm{Y})}{\mathrm{d}}{)}^{\mathsf{\alpha }}& ,\mathrm{d}(\mathrm{X},\mathrm{Y})⩾\mathrm{d}\end{array}\\ \begin{array}{cc}0& ,\mathrm{d}(\mathrm{X},\mathrm{Y})⩾\mathrm{d}\end{array}\end{array}$$

(9)

where d(X,Y) is the Euclidean distance between any two different individuals X and Y; d is the set distance parameter; and α is the group division parameter, which is usually a positive real number. After the function is applied, each individual shares the function with groups using shared functions.

A diversity of species rule is established using the following formula. The cluster network consists of S populations and the number of node populations is increased by an improved minor-habitat rule.

$$\tilde{\delta }(X)=s\widehat{\delta }(X)/{\displaystyle \sum _{i=1}^{s}G}\left(X,Y_i\right)$$

(10)

A diverse population of sparrows was created to solve the energy function using the above equation. The energy function of the population satisfies the matrix conditional mathematical expression [29].

$$X=\left[\begin{array}{cccc}{x}_1^1& x_1^2& \cdots & x_1^d\\ x_2^1& x_2^2& \cdots & x_2^d\\ \cdots & \cdots & \cdots & \cdots \\ x_n^1& x_n^2& \cdots & x_n^d\end{array}\right]$$

(11)

The adaptive function energy function was established based on the positions of the original species, with the expansion of the population within the range of similar species. The mathematical formulation of the adaptive function is expressed as follows, where Fx represents the fitness function value [30]:

$$Fx=\left[\begin{array}{cccc}f[x_1^1& x_1^2& \cdots & x_1^d]\\ f[x_2^1& x_2^2& \cdots & x_2^d]\\ \cdots & \cdots & \cdots & \cdots \\ f[x_n^1& x_n^2& \cdots & x_n^d]\end{array}\right]$$

(12)

3.2. Sparrow Algorithm for Solving Energy Distribution

The Sparrow Search Algorithm (SSA) is an intelligent algorithm used to solve multivariate equations in parallel that was proposed in 2020. Its advantage is that it relies on location information to solve multiple optimal individual values. Its limitations are that the computational process is time-consuming and local optima can occur [13].

There are two types of members in the SSA, searchers and feeders. The searcher’s job is to find food for the group and the feeder’s job is to feed. The searcher has a high energy level and the group size of the feeder is not fixed. The identities of the two are interchangeable. The feeder relies on distance information to follow the searcher. The individual with the highest energy is defined as the searcher and is solved using an iterative formula. The iterative formulas are organized by different distances into the Formula (13).

$$X_{i,j}^{t+1}=\{\begin{array}{c}{X}_{i,j}\cdot \exp \left(-\frac{i}{\alpha \cdot \mathrm{itermax} }\right), \mathrm{if} R2<ST\\ X_{i,j}+Q\cdot L\hspace{1em}, \mathrm{if} \hspace{1em}R2\ge ST\end{array}$$

(13)

where t represents the current number of iterations, X^i,j represents the position information of the i-th sparrow population in the j-th dimension, alpha represents a random number from 0 to 1, itermax represents the maximum number of iterations, and Q represents a random number that obeys a normal distribution. L is a 1*d matrix with all elements of 1; R2 is between 0 and 1, indicating the early warning value of the sparrow population position; and is between 0.5 and 1, indicating the safety value of the sparrow population position.

When R2 < ST, the early warning value is less than the safe value. At this time, there are no predators in the foraging environment and the finder can carry out extensive search operations; when R2 > ST, some sparrows in the population have discovered predators and sent them to the population. The other sparrows in the group issue an alert and all sparrows need to fly to a safe area for foraging.

During the foraging process, some joiners will always monitor the finder. When the finder finds better food, the joiner will compete with it. If successful, the finder’s food will be obtained immediately. Otherwise, the joiner will follow Formula (14) for location updates:

$$X_{i,j}^{t+1}=\{\begin{array}{cc}Q.\exp \left(\frac{X_{\mathrm{worst} }-X_{i,j}^t}{i^2}\right),if& i>n/2\\ X_P^{t+1}+\left|X_{i,j}-X_P^{t+1}\right|A^{+}+.L& ,\mathrm{otherwise} \end{array}$$

(14)

where XP represents the optimal position found by the current discoverer, Xworst represents the current global worst position, and A represents a 1*d matrix whose elements are randomly assigned to 1 or −1 and satisfy the relationship outlined below.

L is a 1*d matrix with all 1s. When i > n/2, the i-th joiner has not obtained food and is in a hungry state and needs to fly to other places for foraging to obtain more energy.

In the sparrow population, the number of sparrows aware of danger accounts for 10% to 20% of the total number of sparrows. The positions of these sparrows are randomly generated, and the position of the sparrows that are aware of danger is continuously updated according to Formula (15).

The location of the vigilante is updated as follows:

$$X_{i,j}^{t+1}=\{\begin{array}{c}{X}_{best }^t+\beta \left|X_{i,j}^t-X_{best}^t\right|, if \hspace{1em} fi >fg\\ X_{i,j}^t+K\cdot \left(\frac{\left|X_{i,j}^t-X_{worst}^t\right|}{(fi-fiv)+\epsilon }\right), if \hspace{1em}fi=fg\end{array}$$

(15)

where X_best represents the current global optimal position, which is a random number that obeys the standard normal distribution and is used as a step-size control parameter; beta is a random number between −1 and 1, fi is the fitness value of the current sparrow individual, fg is the global best fitness value, and f is the global worst fitness value. Unwashed is a constant that avoids having a denominator of 0. When fi > fg, the sparrow is at the edge of the population at this time and is extremely vulnerable to predators. When fi = fg, the sparrow in the middle of the population is also in danger. At this time, it needs to be close to other sparrows to reduce the risk of predation.

This study found that the slow convergence of the sparrow algorithm is due to its tendency to hover between the optimal solution position and the original position, requiring necessary selection to approach the optimal point. Because the sparrow algorithm directly jumps to the current optimal solution when solving the optimal solution rather than moving smoothly, it tends to suffer from the local optima phenomenon. Therefore, the algorithm has been improved to address this issue.

3.3. Sparrow Search Algorithm Improvements

Since the original algorithm tends to converge to local optima, the algorithm needs to be improved in order to solve the multivariate energy function. The first improvement is to remove the iterations that converge toward the distant point. Another improvement is to reduce the jumping process at the optimal value position, and the smooth moving search becomes the main solution action.

$$x_{i,d}^{t+1}=\{\begin{array}{l}{x}_{i,d}^t\cdot (1+Q),R_2<ST\hfill \\ x_{i,d}^t+Q,R_2\ge ST\hfill \end{array}$$

(16)

At the same time, the predator’s status is constantly updated through the following formula.

$$x_{i,d}^{t+1}=x{b}_{i,d}^t+\frac{1}{D}{\displaystyle \sum _{d=1}^D(\mathrm{rand}\{-1,1\}\cdot \left(\left|x{b}_{i,d}^t-x_{i,d}^t\right|\right)}$$

(17)

In the cooperative sparrow search algorithm (CSSA), the systematic search behavior includes all nodes. The search behavior starts from the first position and spreads to the local best points around it. The best nodes within the population are searched and the nodes of other populations are compared with the best values. The smoothing search operation was proposed to solve the intra-population wandering phenomenon.

$$x_{i,d}^{t+1}=\{\begin{array}{l}{x}_{i,d}^t+\beta \cdot \left(x_{i,d}^t-x{b}_{i,d}^t\right),f_i\ne f_g\hfill \\ x_{i,d}^t+\beta \cdot \left(x{w}_{i,d}^t-x{b}_{i,d}^t\right),f_i=f_g\hfill \end{array}$$

(18)

It was found that the sparrow would flee to a random position between the optimal position and the worst position if it was in the optimal position, and to a random position between it and the optimal position if it was not in the optimal position. The improved pseudocode is designed in Algorithm 1.

Algorithm 1: the framework of the CSSA

Input:

G: the maximum iterations

PD: the number of producers

SD: the number of sparrows who perceive the danger

R2: the alarm value

n: the number of sparrows

Initialize a population of n sparrows and define its relevant parameters.

Output:        Xbest: the location of the optimal solution.        Fg: the value of the best adaptation

1: while (t < G)

2:  Rank the fitness values

3:   R2 = rand(1)

4:    for i = 1: PD

5:      Using Equation (3), update the sparrow’s location;

6:    end for

7:    for i = (PD + 1): n

8:      Using Equation (4), update the sparrow’s location;

9:    end for

10:      for l = 1: SD

11:       Using Equation (5), update the sparrow’s location;

12:   end for

13:   Get the current new location;

14:   If the new location is better than before, update it;

15:      t = t + 1

16:   end while

17: Return Xbest, fg.

4. Algorithm Performance Simulation Verification

A comprehensive experiment was carried out to verify the effectiveness of the designed collaborative sparrow algorithm. The experimental design consists of three parts: the benchmark design, simulation, and discussion of results. A benchmark principle is determined by referring to the IEEE CEC standard, and three different types of benchmark functions are introduced that contain high-dimensional single-peaked functions, as shown in

Table 2. Classical test problems.

Name	Function	Type	Range	Min
Ackley	${\mathrm{f}}_1\left(\mathrm{x}\right)=20-20\exp -0.2({\displaystyle {\sum }_{\mathrm{i}=1}^{\mathrm{D}}{\mathrm{x}}_{\mathrm{i}}^2·\frac{1}{D}}{)}^{\frac{1}{2} }-\exp \frac{1}{D}\left({\displaystyle {\sum }_{\mathrm{i}=1}^{\mathrm{D}}\cos }\left({2\mathrm{px}}_{\mathrm{i}}\right)\right)+\mathrm{e}$	M	[−100, 100]	0
Cigar	${\mathrm{f}}_2\left(\mathrm{x}\right)={\mathrm{x}}_1^2{+10}^6{\displaystyle {\sum }_{\mathrm{i}=1}^{\mathrm{D}}{\mathrm{x}}_{\mathrm{i}}^2}$	U	[−100, 100]	0
Levy	${\mathrm{f}}_3{\left(\mathrm{x}\right)=\sin }^2\left({3\mathsf{\pi }\mathrm{x}}_1\right)+\left|{\mathrm{x}}_{\mathrm{D}}-1\right|\ast \left({1+\sin }^2\left({3\mathsf{\pi }\mathrm{x}}_{\mathrm{D}}\right)\right)+{\displaystyle {\sum }_{\mathrm{i}=1}^{\mathrm{D}-1}\left[{\left({\mathrm{x}}_{\mathrm{i}}-1\right)}^2\left({1+\sin }^2\left({3\mathsf{\pi }\mathrm{x}}_{\mathrm{i}+1}\right)\right)\right]}$	M	[−100, 100]	0

; high-dimensional multi-peaked functions; and low-dimensional multi-peaked functions [31,32,33,34]. Five meta-inspired mainstream algorithms are compared to verify the effectiveness of the proposed algorithm. The simulation results are presented by comparing the max FE values in order to verify the feasibility and superiority of the improved algorithm.

4.1. Parameter Settings

A collection of benchmark functions was created to test the effectiveness of the improvement method. The benchmark test functions contained multimodal and unimodal types. Considering that the sparrow algorithm is a variant of the metaheuristic algorithm, with reference to the requirements of the metaheuristic test and the CEC 2017 test suite [31,35,36], the set of benchmark functions used only contained three typical unconstrained optimization problems. These benchmark problems contained one multimodal multi-peak type, one unimodal multi-peak type, and one multimodal single-peak type.

Table 3. Parameter settings for all algorithms.

Algorithm	Parameters	Literature
SSA	Comparison Algorithm	[31]
CSSA	-	-
BSA	Mixrate = 1.00, F = 3 ∗ randn	[32]
GWO	A linearly decreases from 2 to 0	[33]
LSA	-	[34]
CSA	Flight length = 2, awareness probability = 0.1	[35]

presents information on the three types of test functions, including the name, expression, type, search space (range), and global optimum of each function.

4.2. Comparison Algorithm Selection

All the comparison functions are shown in Table 3. The complexity values of the mainstream algorithms, i.e., PSO (particle swarm optimization), GWO (gray wolf optimization), and WOA (whale optimization algorithm), were calculated to compare the effectiveness of the CSSA. In the simulation calculations, the dimensionalities of the algorithms were set to 30 and 50. In addition, the nonparametric Wilcoxon rank-sum test was used to calculate the significance of the differences for all the algorithms [31,33,34,35,37]. The search performance of the improved algorithms in the different dimensions was shown to be improved. Lastly, the convergence curves of the various algorithms were compared under the actions of the different composite functions.

In the test study, five algorithms were compared by computing a hybrid function. The hybrid function is a composite function that eliminated the iterative effects of the algorithms and was used to compare important measures of improvements in the effectiveness of the CSSA. The multi-peak classical benchmark functions cited in Table 1 were used to evaluate the exploration capability of the five leading search algorithms, whereas users of the single-peak problem verified the scalability of the five search algorithms.

4.3. Chapter Name Remains Unchanged

A summary of the results for the 30-dimensional problem is presented in

Table 4. Comparison of the mean error and standard deviation of the test function values. (30 dimensions).

Fn	SSA (Original Value)	CSSA	BSA	GWO	CSA	LSA
f1	5.09 × 10³ (5.68 × 10³)	5.27 × 10³ (5.89 × 10³)	1.37 × 10³ (9.89 × 10³)	1.85 × 10³ (1.27 × 10³)	1.01 × 10³ (5.04 × 10³)	4.51 × 10³ (4.86 × 10³)
f2	3.59 × 10³ (1.13 × 10³)	3.81 × 10³ (1.89 × 10³)	8.69 × 10³ (1.37 × 10³)	5.68 × 10³ (1.03 × 10³)	2.50 × 10³ (7.50 × 10³)	7.35 × 10³ (2.24 × 10³)
f3	4.02 × 10³ (1.19 × 10³)	4.54 × 10³ (1.89 × 10³)	9.43 × 10³ (5.66 × 10³)	5.62 × 10³ (1.74 × 10³)	2.11 × 10³ (7.68 × 10³)	2.18 × 10³ (1.28 × 10³)

. For functions f1, f2, and f3, all algorithms found the global optimum. All algorithms showed similar performance for the above functions. A summary of the results for the 50-dimensional problem is presented in

Table 5. Comparison of the mean error and standard deviation of the test function values. (50 dimensions).

Fn	SSA (Original Value)	CSSA	BSA	GWO	CSA	LSA
f1	5.87 × 10³ (8.08 × 10³)	6.12 × 10³ (6.84 × 10³)	5.14 × 10³ (2.18 × 10³)	8.38 × 10³ (3.58 × 10³)	3.10 × 10³ (1.20 × 10³)	3.30 × 10³ (6.48 × 10³)
f2	1.17 × 10³ (3.23 × 10³)	4.66 × 10³ (2.53 × 10³)	1.85 × 10³ (2.89 × 10³)	1.09 × 10³ (2.10 × 10³)	6.99 × 10³ (1.32 × 10³)	1.68 × 10³ (3.31 × 10³)
f3	1.30 × 10³ (2.60 × 10³)	5.31 × 10³ (2.16 × 10³)	3.66 × 10³ (1.61 × 10³)	7.36 × 10³ (4.33 × 10³)	9.34 × 10³ (2.27 × 10³)	8.02 × 10³ (3.76 × 10³)

. All algorithms showed better performance for functions f1 and f2. For function f3, the CSSA algorithm outperformed the other algorithms. In both tables, the mean and standard deviation of the absolute error values are shown for the two different dimensional test functions.

According to the results, all algorithms easily handled the optimization problem for the unimodal single-peaked functions. For the solution process of the multimodal low-peaked functions, the search ability and search for the optimal function were similar, but for the solution problem of the multimodal multi-peaked functions, the search ability of the improved CSSA was more successful than that of the mainstream algorithms. This shows that the improved CSSA could explore the global optimum for searching multidimensional multi-peaked functions.

Figure 3 shows the convergence profiles of the six competing algorithms on the 30-dimensional f1, f2, and f3 functions. The different colored curves in the figure indicate the convergence effects of the six algorithms, the BSA, GWO, LSA, CSA, SSA, and CSSA. It can be seen in the figure that the above algorithms achieved high-speed convergence in the 30-dimensional f1-f3 functions, and the improved CSSA avoided premature convergence due to the introduction of the small-habitat algorithm.

In the figure, it can be seen that the SSA had better search ability compared with the BSA, GWO, LSA, and CSA algorithms, but the convergence ability was still better with the improved CSSA. The main reason for this phenomenon is that the CSSA algorithm added a small-habitat algorithm to keep up with the diversity of the population and the algorithm did not terminate by converging prematurely.

The complexities of multiple algorithms were simulated according to IEEE CEC 2017. The complexities of the compared algorithms under the 50-dimensional conditions are shown in Figure 4. The ascending order of algorithm complexity was GWO, BSA, LSA, CSA, SSA, and CSSA. The curves in the figure indicate a gradual decrease in the complexity of the comparison algorithm, which was due to the stabilization of the algorithm’s ability in performing subsequent searches. The improved algorithm had a clear advantage because it balanced the comparison of each local optimal solution using smooth jumps and did not fall prey to a wandering solution.

In summary, the CSSA significantly improved the performance of the three types of benchmark functions. The method had good stability and robustness, especially for functions f2 and f3. The performance of the CSSA was 10 orders of magnitude higher than the other four algorithms, which gives it obvious advantages. This research verified the performance of the proposed algorithm under the same conditions. The results showed the high accuracy and search capability of the comparison algorithm (CSSA), which proves its feasibility and superiority.

5. Conclusions

An algorithm based on sparrow foraging was proposed to solve the problem of uneven power distribution in cooperative communication. The study introduced the division method of small-habitat groups, increased population diversity, and effectively solved the premature convergence problem of the original algorithm. By improving the sparrow algorithm, the wandering process between multiple optimal points in the global optimization was effectively eliminated and the convergence speed and accuracy of the algorithm in the late stage were accelerated. Experimental results showed that the improved algorithm had certain advantages over the mainstream algorithms in convergence speed and accuracy.

Due to the limitations of simulation software and hardware, meta-inspired search algorithms have not yet been compared at the same time in the literature. The study of improved algorithms and other search algorithms is the subject of future work. However, the performance of this algorithm was proven by comparing it with five mainstream algorithms. In conclusion, the improved CSSA algorithm is advanced and practical in cooperative communication in cluster networks.

6. Patents

The patent entitled “The heterogeneous cognitive wireless sensor network cluster routing method” is disclosed under CN110708735B.