1 Introduction
Swarmbased search algorithms have attracted great interest of researchers in fields of computational intelligence, artificial intelligence, optimization, data mining, and machine learning during the last two decades
[1]. Moreover, the swarm intelligence algorithms and artificial intelligence have been successfully used in complex reallife applications, such as wind farm decision system, social aware cognitive radio handovers, feature selection, truck scheduling and so on
[2, 3, 4, 5]. Up to now, a lot of swarmbased search algorithms have been presented, including artificial bee colony (ABC) [6], cuckoo search (CS)[7], firefly algorithm (FA) [8], particle swarm optimization (PSO)
[9] and so on.Among the existing swarmbased search algorithms, CS is presented in terms of the obligate brood parasitic behavior of some cuckoo species and the Lévy flight behavior of some birds and fruit flies. CS searches for new solutions by performing a global explorative random walk together with a local exploitative random walk. CS is famous for utilizing Lévy flights in its global explorative random walk. Lévy flights play a critical role in enhancing randomness, as Lévy flights is a random walk where the steplengths have a probability distribution that is heavytailed. At each iteration process, CS firstly searches for new solutions in Lévy flights random walk. Secondly, CS proceeds to obtain new solutions in local exploitative random walk. After each random walk, a greedy strategy is used to select a better solution from the current and newly generated solutions according to their fitness values. Due to the salient features such as simple concept, limited parameters, and implementation simplicity, CS has aroused extensive attention and has been accepted as a simple but efficient optimization technique for solving optimization problems. Accordingly, many new CS variants have been continuously presented recently [10, 11, 12, 13, 14]. However, there’s still a lot of space in designing newly improved or enhanced techniques to help to increase the accuracy and convergence speed and enhance the searching stability for the original CS algorithm.
In nature, the movements of many foragers and wandering animals have been shown to follow a Lévy distribution [15]
rather than Gaussian distribution. It is found that foragers frequently take a large step to enhance the searching efficiency since it is the natural evolution for millions of years. Inspired by the mentioned natural phenomena, CS is proposed by combination with Lévy, where the steplength is drawn from a heavytailed probability distribution and large steps frequently take place flights. In fact, before CS, the idea of Lévy flights has been applied in
[16] to solve a problem of nonconvex stochastic optimization, due to big jumps of the Lévy flights process. In this way, it can enhance the searching ability compared with Gaussian distribution where large steps seldom happen. More exactly, we have to say the foragers should move following a heavytailed distribution since Lévy distribution is a simple heavytailed distribution which is easy to analyze. There are many other heavytailed distributions such as MittagLeffler distribution, Pareto distribution, Cauchy distribution, and Weibull distribution, and large steps still frequently happen when using them to generate the steps. For swarmbased optimization algorithms, randomness plays a significant role in both exploration and exploitation, or diversification and intensification [17]. Therefore, it’s necessary to discuss the optimal randomness in swarmbased search algorithms.In this paper, we mainly focus on the discussion on the impact of different heavytailed distributions on the performance of swarmbased search algorithms. In the study, CS is taken as a representative method of swarmbased optimization algorithms, and the results can be generalized to other swarmbased search algorithms. At first, some basic definitions of the heavytailed distributions and how to generate the random numbers according to the given distribution are provided. Then by replacing the Lévy flight with steps generated from other heavytailed distributions, four different randomnessenhanced CS algorithms (namely CSML, CSP, CSC, and CSW) are presented by applying MittagLeffler distribution, Pareto distribution, Cauchy distribution and Weibull distribution. Finally, dedicated experimental studies are carried out on a test suite of 20 benchmark problems with unimodal, multimodal, rotated and shifted properties to compare the performance of different variant algorithms. The experimental results demonstrate that the four proposed randomnessenhanced CS algorithms show a significant improvement over the original CS algorithm. This suggests that the performance of CS can be improved by means of integrating different heavytailed probability distributions rather than Lévy flights into it. Moreover, comparisons of CSML, CSP, CSC, and CSW with other optimization algorithms are also performed. At last, an application problem of parameter identification of unknown fractionalorder chaotic systems is further considered. Based on the observations and results analysis, the randomnessenhanced CS algorithms are able to exactly identify the unknown specific parameters of the fractionalorder system with better effectiveness and robustness. The randomnessenhanced CS algorithms can be regarded as an efficient and promising tool for solving the realworld complex optimization problems besides the benchmark problems.
The remainder of this paper is organized as follows. The principle of the original CS algorithm is described in Section 2. Section 3 gives details of four randomnessenhanced CS algorithms after a brief review of several commonly used heavytailed distributions. Experimental results and discussions of randomnessenhanced CS algorithms are presented in Section 4. Finally, Section 5 summarizes the conclusions and future work.
2 Cuckoo Search Algorithm
Cuckoo search (CS), developed by Yang and Deb, is considered to be a simple but promising stochastic natureinspired swarmbased search algorithm [7, 18]. CS is inspired by the intriguing brood parasitism behaviors of some species of cuckoos, and is enhanced by Lévy flights instead of simple isotropic random walks. Cuckoos are considered to be fascinating birds not only for their beautiful sounds but also for their aggressive reproduction strategy. Some cuckoo species lay their eggs in host nests, and at the same time, they may remove host birds’ eggs in order to increase the hatching probability of their own eggs. For simplicity in describing the standard CS, there are three idealized rules as follows [7]: (1) Only one egg is laid by each cuckoo bird at a time, and dumped in a randomly chosen nest; (2) The next generations of cuckoos search for new solutions using the best nests with highquality; (3) The number of available host nests is fixed, and the egg laid by a cuckoo is discovered by the host bird with a probability . In this condition, the host bird can either remove the egg or simply abandon the nest and build a completely new nest.
The purpose of CS is to substitute a notsogood solution in the nests with the new and potentially better solutions (cuckoos). At each iteration process, CS employs a balanced combination of a local random walk and the global explorative random walk under control of a switching parameter . A greedy strategy is used after each random walk to select better solutions from the current and newly generated solutions based on their fitness values.
2.1 Lévy Flights Random Walk
At generation , a global explorative random walk carried out by using Lévy flights can be defined as follows:
(1) 
where denotes a new solution generated in Lévy flights random walk, and is the best solution obtained so far. is the step size related to the scales of the problem of interest, is the best solution obtained so far, the product represents entrywise multiplications, and is defined according to a simple powerlaw formula as follows:
(2) 
where
is a random variable,
is a stability index. Moreover, it is worth mentioning that the wellknown Gaussian and Cauchy distribution are its special cases when its stability index is respectively set to 2 and 1.In practice, can be updated as follows:
(3) 
where is suggested as 1.5 [18], and
are random numbers drawn from a normal distribution with mean of 0 and standard deviation of 1,
denotes the gamma function, and is expressed as:(4) 
2.2 Local Random Walk
The local random walk can be defined as:
(5) 
where and are two different selected random solutions, and
are two independent random numbers with uniform distribution, and
is a Heaviside function. The local random walk utilizes a far field randomization to generate a substantial fraction of new solutions which are sufficiently far from the current best solution. The pseudocode of the standard CS algorithm is given in Algorithm 1.3 RandomnessEnhanced CS Algorithms
The standard CS algorithm uses Lévy flights in global random walk to explore the search space. The Lévy step is taken from the Lévy distribution which is a heavytailed probability distribution. In this case, a fraction of large steps is generated, which plays an important role in enhancing the search capability of CS. Although many foragers and wandering animals have been shown to follow a Lévy distribution [15], the investigation into the impact of other different heavytailed probability distributions on CS is still insufficient up to now. This motivates us to make an attempt to apply the wellknown MittagLeffler distribution, Pareto distribution, Cauchy distribution and Weibull distribution to the standard CS algorithm, by using which, more efficient searches are supposed to take place in the search space thanks to the long jumps. In this section, a brief review of several commonly used heavytailed distributions is given, and then the scheme of the randomnessenhanced CS algorithms is introduced.
3.1 Commonly Used HeavyTailed Distributions
This subsection provides the definition of heavytailed distribution and several examples of commonly used heavytailed distributions.
Definition 1 (HeavyTailed Distribution).
The distribution of a realvalued random variable is said to have a heavy right tail if the tail probabilities
decay more slowly than those of any exponential distribution, i.e., if
(6) 
for every . Heavy left tails are defined in a similar way [19].
Example 1 (MittagLeffler Distribution).
A random variable is said to subject to MittagLeffler distribution if its distribution function has the following form
(7) 
where , , and for . For , the MittagLeffler distribution is a heavytailed generalization of the exponential, and reduces to the exponential distribution when .
A MittagLeffler random number can be generated using the most convenient expression proposed by Kozubowski and Rachev [20]:
(8) 
where is the scale parameter, are independent uniform random numbers, and is a MittagLeffler random number.
Example 2 (Pareto Distribution).
A random variable is said to subject to Pareto distribution if its cumulative distribution function has the following expression:
(9) 
where is the scale parameter, is the shape parameter (Pareto’s index of inequality).
Example 3 (Cauchy Distribution).
A random variable is said to subject to Cauchy distribution if its cumulative distribution function has the following expression:
(10) 
where is the location parameter, is the scale parameter.
Example 4 (Weibull Distribution).
A random variable is said to subject to Weibull distribution if it has a tail function as follows:
(11) 
where is the scale parameter, is the shape parameter. If and only if , the Weibull distribution is a heavytailed distribution.
3.2 Improving CS with Different HeavyTailed Probability Distributions
For swarmbased search algorithms, randomness plays a significant role in both exploration and exploitation, or diversification and intensification [17]. It’s very necessary to discuss the optimal randomness in swarmbased search algorithms. Randomness is normally realized by employing pseudorandom numbers, based on some common stochastic processes. Generally, randomization is achieved by simple random numbers that are drawn from a uniform distribution or a normal distribution. But in other cases, more sophisticated randomization approaches are considered, for example, random walks and Lévy flights. Here, we have to say the foragers should move following a heavytailed distribution since Lévy distribution is a simple heavytailed distribution which is easy to analyze. There are many other heavytailed distributions such as MittagLeffler distribution, Pareto distribution, Cauchy distribution, and Weibull distribution, and large steps still frequently happen when using them to generate the steps. In this paper, we mainly focus on the discussion on the impact of different heavytailed distributions on the performance of swarmbased search algorithms. In the study, CS is taken as a representative method of swarmbased optimization algorithms, and the results can be generalized to other swarmbased search algorithms.
In this section, four randomnessenhanced cuckoo search algorithms are proposed in this paper. Specifically, the following modified CS methods are considered: (1) CS with the MittagLeffler distribution, denoted as CSML; (2) CS with the Pareto distribution, denoted as CSP; (3) CS with the Cauchy distribution, denoted as CSC; (4) CS with the Weibull distribution, referred to CSW. In the modified CS methods, the aforementioned four different heavytailed probability distributions are respectively used to be integrated into CS instead of the original Lévy flights in the global explorative random walk. By using these heavytailed probability distributions, the updating equation (1) can be reformulated as follows
(12) 
(13) 
(14) 
(15) 
where in Equation (12) denotes a random number drawn from MittagLeffler distribution; in Equation (13) represents a random number drawn from Cauchy distribution; in Equation (14) denotes a random number drawn from Cauchy distribution; in Equation (15) means a random number drawn from Weibull distribution. Compared with the standard CS algorithm, the differences of randomnessenhanced cuckoo search methods lie in line 9 from Algorithm 1.
Remark 1.
In this paper, our emphasis is to study the effects of different heavytailed distributions on the swarmbased search algorithms.
Remark 2.
Since CS is a popular swarmbased search algorithm, we only use it as an representative. Similar analyses for optimal randomness can be applied to other swarmbased algorithms.
Remark 3.
The source code of randomnessenhanced cuckoo search algorithms (namely CSML, CSP, CSC, CSW), written in Matlab, is available at
https://www.mathworks.com/matlabcentral/fileexchange/71758optimalrandomnessinswarmbasedsearch.
4 Experimental Results
This study focuses on discussing the effectiveness and efficiency of the proposed randomnessenhanced CS algorithms. To fulfill this purpose, extensive experiments are carried out on a test suite of 20 benchmark functions, which are chosen from the literature [21, 22]. The superiority of randomnessenhanced CS algorithms over the standard CS is tested, then a scalability study and comparison with other optimization algorithms are performed. Finally, an application to parameter identification of fractionalorder chaotic systems is also investigated.
4.1 Experimental Setup
For parameter settings of CS, CSML, CSP, CSC and CSW, the probability is set to 0.25 [7], the scaling factor is set to 0.01. The proposed randomnessenhanced CS algorithms introduce new parameters to CS: the scale parameter and the MittagLeffler index in CSML; the scale parameter and the shape parameter in CSP; the location parameter and the scale parameter in CSC; the scale parameter and the shape parameter in CSW. As for these newly introduced parameters, their values are given in Table 1 after analysis in Section 4.2. Moreover, the population size satisfies where denotes the dimension of the problem unless a change is mentioned. In the experimental studies, the maximum number of function evaluations (namely MaxFEs) is taken as the termination criterion and set to . All the algorithms are evaluated for 50 times and the averaged experimental results are recorded for each benchmark function respectively. Besides, two nonparametric statistical tests for independent samples are taken to detect the differences between the proposed algorithm and the compared algorithms. The tests contain the Wilcoxon signedrank test at the 5% significance level and the Friedman test. The symbol ”‡”, ”†” and ”=” respectively denote the average performance gained by the chosen approach is weaker than, better than, and similar to the compared algorithm. Meanwhile, the best experimental results for each benchmark problem are marked in boldface, for clarity.
Distribution  Algorithm  Parameters 

MittagLeffler distribution  CSML  , 
Pareto distribution  CSP  , 
Cauchy distribution  CSC  , 
Weibull distribution  CSW  , 
4.2 Parameter Tuning
From Section 3.2, it’s obvious that each of the four randomnessenhanced CS algorithms brings two new userdefined parameters, for example, the scale parameter and the MittagLeffler index in CSML. To illustrate the impact of these two parameters on the optimization results and to offer reference values to users of our algorithm, parameter analyses are conducted in advance and corresponding experiments are performed on unimodal function and multimodal function with dimension set to 30. The optimal value of selected benchmark functions is 0. is the default value for MaxFEs. 15 independent runs are carried out for each parameter setting to reduce statistical sampling effects. The experimental results are plotted in Figure 1. For simplicity of description, only the result of parameter tuning for CSML is shown here, and the same operation is conducted on CSP, CSC, and CSW. In Figure 1(a), varies within interval in steps of 0.5, varies from 0.1 to 0.9 in steps of 0.1, and ‘Error’ represents the average error to the optimal value.
From Figure 1(a), we can see that the MittagLeffler index , in general, has a slight effect on the performance of CSML, whereas the value of scale parameter shows a more significant impact on the experimental results. According to the right part of each subfigure in Figure 1(a), the larger the value of scale parameter is, the better the performance of CSML will be. In view of the above considerations, we set the values of and to 0.8 and 4.5 for all the experiments being conducted in the next subsections. For Pareto distribution, Cauchy distribution and Weibull distribution, the same parameter analysis is performed according to Figures 1(b) 1(c) 1(d). The userdefined parameter values for all the randomnessenhanced CS algorithms are listed in Table 1.
4.3 Performance Evaluation of RandomnessEnhanced CS Algorithms
In this section, lots of experiments are performed in order to probe into the effectiveness and efficiency of different heavytailed distributions on the performance of CS, and meanwhile, to decide the optimal randomness in improving CS. In our experiments, the standard CS and four proposed randomnessenhanced CS algorithms (namely, CSML, CSP, CSC, and CSW) are tested on 20 test functions where is set to 30. The experimental results are presented in Table 2.
According to Table 2, it can be clearly found that CS with different heavytailed probability distributions provides significantly better results when compared with the original CS. Specifically speaking, in terms of the total number of ””, CS is inferior to CSML, CSP, CSC, and CSW on 17, 17, 16 and 16 test functions, similar to CSML, CSP, CSC and CSW on 1, 1, 2 and 1 test functions, and superior to CSML, CSP, CSC, and CSW on 2, 2, 2 and 3 test functions, respectively. It is worth noting that CSML, CSP, CSC and CSW are capable of achieving the global optimum on test problem and , while CS doesn’t. Moreover, all the pvalues are less than 0.05. These results suggest that CSML, CSP, CSC, and CSW are able to significantly improve the performance of CS for the test functions at . The comprehensive ranking orders are CSW, CSC, CSML, CSP, and CS in a descending manner. This indicates that the integration of different heavytailed probability distributions into CS not only retains the merit of CS, but also performs even better. Besides, the Weibull distribution performs the best in enhancing the search ability of CS, that is, CSW is supposed to be the optimal randomness in improving CS among all the comparison methods for solving benchmark problems at .
Fun  CS  CSML  CSP  CSC  CSW 

9.58E31  4.90E54  4.74E59  1.17E57  4.40E51  
1.20E+01  5.22E+00  3.10E+00  2.74E+00  8.62E+00  
7.70E13  1.06E14  1.07E14  9.56E15  8.28E15  
7.11E17  0.00E+00  0.00E+00  0.00E+00  0.00E+00  
2.32E+01  1.38E+01  1.88E+01  1.49E+01  8.34E+00  
1.57E+03  5.37E+02  1.32E+03  4.80E+02  3.56E+01  
3.76E01  2.96E01  3.00E01  2.84E01  2.20E01  
3.73E+02  2.00E+02  2.49E+02  2.27E+02  1.93E+02  
2.07E03  1.57E32  1.57E32  2.07E03  1.57E32  
4.82E28  1.35E32  1.35E32  1.35E32  1.35E32  
6.48E30  0.00E+00  0.00E+00  0.00E+00  0.00E+00  
1.05E02  1.10E03  2.77E04  1.40E03  1.23E02  
2.17E+06  3.04E+06  2.99E+06  3.25E+06  3.61E+06  
1.79E+03  4.98E+02  3.58E+02  4.02E+02  5.51E+02  
3.17E+03  2.44E+03  1.98E+03  2.11E+03  1.94E+03  
2.78E+01  1.57E+01  9.91E+00  1.23E+01  1.59E+01  
1.34E03  2.22E03  5.79E03  3.73E03  2.49E03  
2.09E+01  2.09E+01  2.09E+01  2.09E+01  2.09E+01  
2.84E+01  1.30E+01  2.74E+01  1.28E+01  6.81E+00  
1.69E+02  1.21E+02  1.31E+02  1.18E+02  1.03E+02  
  17/1/2  17/1/2  16/2/2  16/1/3  
pvalue    8.97E03  1.00E02  1.00E02  1.87E02 
Avg. rank  4.35  2.78  2.88  2.58  2.43 
To further discuss the convergence speed of the four randomnessenhanced CS algorithms, several test problems (namely , , and ) at are selected to plot the convergence curves of the averages of the function error values within MaxFEs over 50 independent runs, which are presented in Figure 2. From Figure 2, it can be observed that CSML, CSP, CSC, and CSW converge outstandingly faster than CS according to the convergence curves. In summary, it can be concluded that the standard CS algorithm can be improved by integrating different heavytailed probability distributions rather than Lévy distribution into it.
Besides, to analyze the reasons for different performances among the four proposed randomnessenhanced CS algorithms, the jump lengths of CS, CSML, CSP, CSC, and CSW (namely, , , , , and ) are depicted in Figure 3, where the parameters are given in Table 1 and the scaling factor is set to . From Figure 3, it can be observed that (1) Lévy distribution and Cauchy distribution are onesided distribution where all the random numbers are positive, and the other three distributions are twosided; (2) large steps frequently take place for all distributions; (3) since the tail of Weibull distribution is the lightest, the extreme large steps (compared with its mean) are less likely to happen.
4.4 Scalability Study
In this section, a scalability study comparing with the standard CS algorithm is conducted in order to study the effect of problem size on the performance of the four proposed randomnessenhanced CS algorithms. We carry out experiments on the 20 benchmark functions with dimension set to 10 and 50. When , the population size is chosen as ; meanwhile, when , the population size is selected as . All the other control parameters are kept unchanged. The experimental results achieved by CS and four proposed randomnessenhanced CS algorithms at and are listed in Tables 3 and 4, respectively, and the results of the Wilcoxon signedrank test are also given in the tables.
Fun  CS  CSML  CSP  CSC  CSW 

4.87E26  3.39E31  4.21E48  2.04E46  2.48E46  
9.63E01  3.02E+01  1.12E01  1.75E01  2.97E01  
4.16E11  2.50E14  4.37E15  4.44E15  4.44E15  
3.44E02  0.00E+00  2.22E02  2.07E02  1.44E02  
3.00E+00  6.93E+01  2.25E+00  2.95E01  2.26E09  
6.72E+01  3.80E+03  1.38E+01  6.91E03  1.27E04  
1.04E01  4.78E01  9.99E02  9.99E02  9.99E02  
2.40E+01  9.89E+02  1.54E+01  1.01E+01  5.72E+00  
1.96E16  2.01E28  4.71E32  4.71E32  4.71E32  
4.86E23  9.17E30  1.35E32  1.35E32  1.35E32  
4.13E26  0.00E+00  0.00E+00  0.00E+00  0.00E+00  
8.16E14  3.73E+02  1.33E21  4.54E19  1.51E16  
2.08E+02  1.70E+07  7.20E+02  6.78E+02  8.31E+02  
1.01E05  1.96E+04  1.46E09  1.20E08  4.82E08  
9.30E05  6.82E+03  6.13E10  5.11E09  9.27E09  
9.78E01  4.11E+01  6.38E01  3.48E01  2.69E01  
5.33E02  1.07E03  5.91E02  4.72E02  4.39E02  
2.04E+01  2.11E+01  2.04E+01  2.04E+01  2.03E+01  
2.75E+00  7.37E+01  1.80E+00  1.79E01  2.35E10  
1.99E+01  2.89E+02  1.63E+01  1.59E+01  1.43E+01  
  7/0/13  17/1/2  18/1/1  19/1/0  
Avg. rank  4.10  4.28  2.28  2.25  2.10 
According to Table 3, CSML, CSP, CSC, and CSW are significantly better than CS on 7, 17, 18 and 19 test functions, similar to CS on 0, 1, 1 and 1 test functions, and worse than CS on 13, 2, 1 and 0 test functions, respectively. The comprehensive ranking orders in the case of are CSW, CSC, CSP, CS, and CSML in descending manner. The results show that the performance improvement of using different heavytailed probability distributions persists expect CSML when the problem dimension reduces to 10. In the case of , it can be observed from Table 4 that CSML, CSP, CSC and CSW perform better than CS on 16, 14, 16 and 16 test functions, to CS on 1, 1, 1 and 1 test functions, and worse than CS on 3, 5, 3 and 3 test functions, respectively. Meanwhile, the corresponding comprehensive ranking orders when are CSC, CSML, CSP, CSW and CS. In general, we can draw conclusions that the advantages of four randomnessenhanced CS algorithms over the standard CS are overall stable when the problem dimension increases, except CSML which deteriorates to a certain extent when set to 10. Furthermore, regarding to the different comprehensive ranking orders obtained at every dimension, it is pointed out that CS with Lévy flights seems not the optimal randomness when compared with those using different heavytailed probability distributions in CS.
Fun  CS  CSML  CSP  CSC  CSW 

3.79E17  3.47E31  7.41E36  5.75E32  2.73E24  
4.22E+01  3.07E+01  2.82E+01  2.99E+01  3.41E+01  
2.85E02  2.43E14  2.05E14  2.05E14  7.40E13  
1.93E10  0.00E+00  0.00E+00  0.00E+00  0.00E+00  
8.44E+01  6.80E+01  8.69E+01  7.54E+01  7.37E+01  
4.87E+03  4.14E+03  6.05E+03  4.38E+03  2.38E+03  
6.69E01  4.68E01  4.87E01  4.22E01  3.68E01  
1.36E+03  9.58E+02  1.21E+03  1.09E+03  1.13E+03  
8.13E03  6.74E28  1.04E27  7.21E30  1.47E23  
3.25E14  1.02E29  1.57E32  1.44E30  2.01E23  
1.40E16  0.00E+00  0.00E+00  0.00E+00  3.57E24  
2.34E+02  3.57E+02  1.86E+02  4.49E+02  8.59E+02  
8.53E+06  1.66E+07  1.47E+07  1.83E+07  1.85E+07  
2.72E+04  1.99E+04  1.72E+04  1.91E+04  1.88E+04  
1.06E+04  6.95E+03  6.49E+03  6.65E+03  6.30E+03  
6.38E+01  3.90E+01  4.15E+01  3.63E+01  4.43E+01  
1.30E03  1.81E03  3.56E03  2.43E03  3.64E03  
2.11E+01  2.11E+01  2.11E+01  2.11E+01  2.11E+01  
1.24E+02  7.04E+01  1.27E+02  7.47E+01  6.50E+01  
3.87E+02  2.87E+02  3.13E+02  2.85E+02  2.69E+02  
  16/1/3  14/1/5  16/1/3  16/1/3  
Avg. rank  4.10  2.58  2.70  2.55  3.08 
4.5 Comparison with Other Optimization Algorithms
In order to demonstrate the superiority of the four proposed randomnessenhanced CS algorithms, we compare the performance of CSML, CSP, CSC and CSW with several classical stateoftheart optimization algorithms, namely ABC[6], DE[23], FA[8], FPA[24] and PSO[9], by conducting numerical experiments on the 20 benchmark functions at dimension . In our experimental study, MaxFEs set to is taken as the termination criterion, and population size is set to 30. For ABC, the number of food sources , maximum number of trial for abandoning a source ; for PSO, inertia weight defined as and denotes iteration number, acceleration constants , . The MATLAB source codes of FA and FPA are obtained from [17]. And the results of DE are taken from the literature [21] which has the same termination criterion. The comparative simulation results of all the optimization algorithms are listed in Table 5. Additionally, Table 5 records the statistical results obtained by both of the Wilcoxon signedrank test and the Friedman test for 20 benchmark functions at dimension .
The experimental results in Table 5 clearly demonstrate that all of the four proposed randomnessenhanced CS algorithms perform better on the majority of benchmark functions. More specifically, CSW is overall the best, CSC is the second best, CSML is the third best, and CSP is the fourth best followed by DE, PSO, ABC, FPA, and FA. This suggests that the proposed randomnessenhanced CS algorithms are also highly competitive when compared with other optimization algorithms.
Fun  CSML  CSP  CSC  CSW  ABC  DE  FA  FPA  PSO 

4.90E54  4.74E59  1.17E57  4.40E51  6.28E157  5.73E17  2.90E+03  8.65E04  2.05E06  
5.22E+00  3.10E+00  2.74E+00  8.62E+00  3.00E+02  5.20E+01  7.91E+07  1.16E+02  3.99E+01  
1.06E14  1.07E14  9.56E15  8.28E15  2.51E01  1.37E09  1.07E+01  1.57E+00  1.90E+01  
0.00E+00  0.00E+00  0.00E+00  0.00E+00  5.33E+00  2.66E03  2.71E+01  2.39E02  9.76E03  
1.38E+01  1.88E+01  1.49E+01  8.34E+00  1.99E01  2.55E+01  2.20E+02  4.05E+01  1.21E+02  
5.37E+02  1.32E+03  4.80E+02  3.56E+01  1.65E+02  4.90E+02  1.00E+04  3.49E+03  1.13E+04  
2.96E01  3.00E01  2.84E01  2.20E01  2.98E+00  2.52E01  5.60E+00  1.74E+00  8.94E01  
2.00E+02  2.49E+02  2.27E+02  1.93E+02  6.79E+03  3.10E+02  1.30E+13  5.14E+04  7.04E+02  
1.57E32  1.57E32  2.07E03  1.57E32  8.18E+06  4.56E02  4.58E+02  4.41E01  4.09E01  
1.35E32  1.35E32  1.35E32  1.35E32  1.38E15  1.44E01  3.94E+05  2.47E+00  7.69E01  
0.00E+00  0.00E+00  0.00E+00  0.00E+00  3.81E+03  3.87E14  6.39E+04  9.32E03  1.08E05  
1.10E03  2.77E04  1.40E03  1.23E02  4.19E+03  8.50E02  9.45E+04  8.61E+00  3.79E14  
3.04E+06  2.99E+06  3.25E+06  3.61E+06  2.90E+07  3.63E+06  1.71E+09  1.00E+05  1.53E+06  
4.98E+02  3.58E+02  4.02E+02  5.51E+02  1.26E+05  5.54E+01  1.23E+05  5.16E+03  1.11E+03  
2.44E+03  1.98E+03  2.11E+03  1.94E+03  1.45E+04  1.08E+03  4.28E+04  1.74E+03  5.66E+03  
1.57E+01  9.91E+00  1.23E+01  1.59E+01  1.62E+02  6.67E+01  2.85E+10  2.69E+03  4.17E+01  
2.22E03  5.79E03  3.73E03  2.49E03  4.70E+03  7.59E03  1.22E+04  4.70E+03  2.71E+03  
2.09E+01  2.09E+01  2.09E+01  2.09E+01  2.12E+01  2.09E+01  2.12E+01  2.10E+01  2.09E+01  
1.30E+01  2.74E+01  1.28E+01  6.81E+00  1.65E01  2.43E+01  3.89E+02  9.23E+01  1.21E+02  
1.21E+02  1.31E+02  1.18E+02  1.03E+02  4.58E+02  7.33E+01  6.42E+02  1.91E+02  2.23E+02  
Avg. rank  3.85  3.95  3.60  3.35  7.20  5.50  9.78  7.58  6.95 
4.6 Application to Parameter Identification of FractionalOrder Chaotic Systems
In this section, the four proposed randomnessenhanced CS algorithms (namely, CSML, CSP, CSC, and CSW) are applied to identify unknown parameters of fractionalorder chaotic systems, which is a critical issue in chaos control and synchronization. Our main task of this section is to further demonstrate that improving CS with different heavytailed probability distributions can also effectively tackle the realworld complex optimization problems besides the benchmark problems. In fact, by using a nonLyapunov way according to problem formulation suggested in [25], the nonlinear function optimization can be converted to from parameter identification of uncertain fractionalorder chaotic systems.
In the numerical simulation, the fractionalorder financial system [26] under the Caputo definition is taken for example, which can be described as
(16) 
where and are fractional orders and systematic parameters. When , , and initial point , the system above is chaotic. Figure 4 depicts the distribution figure of system (16) for the objective function values.
Method  CS  CSML  CSP  CSC  CSW 

0.999999825481796  0.999999979386471  1.000000001165006  0.999999930875086  0.999999994619958  
1.75E07  2.28E08  1.17E09  6.91E08  5.38E09  
0.100000078306700  0.100000006492360  0.099999999732393  0.100000038684769  0.100000001325757  
7.83E07  1.12E07  2.68E09  3.87E07  2.06E08  
1.000000126069434  0.999999979588057  0.999999995606294  0.999999876500337  0.999999979353103  
1.26E07  4.61E08  4.39E09  1.23E07  1.33E08  
1.07E055.46E06  4.75E072.74E07  7.46E083.29E08  1.89E069.38E07  1.03E076.12E08 
The validation of the proposed methods in this paper is further proved by comparing CSML, CSP, CSC, and CSW with the standard CS algorithm for parameter identification. In the simulations, the maximum iteration number is set to 200 and the population size is set to 40. For the system to be identified, the step size is set to 0.005, and the number of samples set to 200. In addition, it is worth mentioning that the same computation effort is used in implementation for all the compared algorithms to make a fair comparison. Table 6 lists the statistical results of the average identified values, the corresponding relative error values, and the objective function values for system (16). From Table 6
, it can be clearly observed that all the four proposed randomnessenhanced CS algorithms outperform CS according to the average objective function values, and they are able to generate estimated values with much higher accuracy than CS. Besides, it can be seen that CSP surpasses CS, CSML, CSW, and CSC in obtaining the best average identified values, the corresponding relative error values, and the objective function values.
Moreover, Figure 5 shows the convergence curves of the relative error values of the estimated parameters and objective function values for the corresponding system via CSML, CSP, CSC, CSW, and CS. From Figures 5 5 5, the relative error values of the estimated values generated by the randomnessenhanced CS algorithms converge to zero more quickly than the original CS. This indicates that CS algorithms with the four different heavytailed probability distributions are able to obtain more accurate values of the estimated parameters. In terms of Figure 5, the objective function values of CSML, CSP, CSC, CSW also decline faster than CS, and among which CSP performs the best. It is noteworthy that CSW has a similar convergence curve of objective function values with CSP, and can converge to the nearby area of CSP. Therefore, CSW can still be considered as an efficient tool for solving optimization problems.
According to the foregoing discussion, it can be summarized that the randomnessenhanced CS algorithms are able to exactly identify the unknown specific parameters of the fractionalorder system (16) with better effectiveness and robustness, and CSP together with CSW may be treated as a useful tool for handling the problem of parameter identification.
5 Conclusions
The purpose of this paper is to discuss the optimal randomness in swarmbased search algorithms. In the study, CS is taken as a representative method of swarmbased optimization algorithms, and the results can be generalized to other swarmbased search algorithms. The impact of different heavytailed distributions on the performance of CS is investigated. By replacing Lévy flights with steps generated from other heavytailed distributions in CS, four different randomnessenhanced CS algorithms (namely CSML, CSP, CSC, and CSW) are presented by applying MittagLeffler distribution, Pareto distribution, Cauchy distribution and Weibull distribution, in order to improve the optimization performance of CS. The improvement in effectiveness and efficiency is validated through dedicated experiments. The experimental results indicate that all four proposed randomnessenhanced CS algorithms show a significant improvement in effectiveness and efficiency over the standard CS algorithm. Furthermore, the randomnessenhanced CS algorithms are successfully applied to system identification. In summary, CS with different heavytailed probability distributions can be regarded as an efficient and promising tool for solving the realworld complex optimization problems besides the benchmark problems.
Future promising topics can be directed to 1) theoretically analyze the global convergence of randomnessenhanced CS algorithms; 2) do a similar analyses to other swarmbased search algorithms for the optimal randomness; 3) since the search range is always finite for swarmbased search algorithms, it is necessary to study the optimal randomness in a finite range.
References
 [1] XinShe Yang. Natureinspired metaheuristic algorithms. Luniver press, 2010.
 [2] H Anandakumar and K Umamaheswari. A bioinspired swarm intelligence technique for social aware cognitive radio handovers. Computers & Electrical Engineering, 71: 925937, 2018.
 [3] Lucija Brezočnik, Iztok Fister, and Vili Podgorelec. Swarm intelligence algorithms for feature selection: a review. Applied Sciences, 8(9): 1521, 2018.
 [4] Xuejing Zhao, Chen Wang, Jinxia Su, and Jianzhou Wang. Research and application based on the swarm intelligence algorithm and artificial intelligence for wind farm decision system. Renewable Energy, 134: 681697, 2019.
 [5] Maxim A Dulebenets. A novel memetic algorithm with a deterministic parameter control for efficient berth scheduling at marine container terminals. Maritime Business Review, 2(4): 302330, 2017.
 [6] Dervis Karaboga and Bahriye Basturk. A powerful and efficient algorithm for numerical function optimization: artificial bee colony (abc) algorithm. Journal of global optimization, 39(3): 459471, 2007.
 [7] XinShe Yang and Suash Deb. Cuckoo search via Lévy flights. In Nature & Biologically Inspired Computing, pages 210214. IEEE, 2009.
 [8] XinShe Yang. Firefly algorithms for multimodal optimization. In International symposium on stochastic algorithms, pages 169178. Springer, 2009.

[9]
J Kennedy and R Eberhart. Particle swarm optimization (pso). In
Proc. IEEE International Conference on Neural Networks, Perth, Australia
, pages 19421948, 1995.  [10] Hongqing Zheng and Yongquan Zhou. A novel cuckoo search optimization algorithm based on gauss distribution. Journal of Computational Information Systems, 8(10): 41934200, 2012.
 [11] Lijin Wang, Yiwen Zhong, and Yilong Yin. Nearest neighbour cuckoo search algorithm with probabilistic mutation. Applied Soft Computing, 49: 498509, 2016.
 [12] Hojjat Rakhshani and Amin Rahati. Snapdrift cuckoo search: A novel cuckoo search optimization algorithm. Applied Soft Computing, 52: 771794, 2017.
 [13] Zhihua Cui, Bin Sun, Gaige Wang, Yu Xue, and Jinjun Chen. A novel oriented cuckoo search algorithm to improve dvhop performance for cyber physical systems. Journal of Parallel and Distributed Computing, 103: 4252, 2017.
 [14] Rohit Salgotra, Urvinder Singh, and Sriparna Saha. New cuckoo search algorithms with enhanced exploration and exploitation properties. Expert Systems with Applications, 95: 384420, 2018.

[15]
Toby J Richer and Tim M Blackwell. The Lévy particle swarm. In
2006 IEEE International Conference on Evolutionary Computation
, pages 808815. IEEE, 2006.  [16] Ilya Pavlyukevich. Lévy flights, nonlocal search and simulated annealing. Journal of Computational Physics, 226(2): 18301844, 2007.
 [17] XinShe Yang. Natureinspired optimization algorithms. Elsevier, 2014.
 [18] XinShe Yang and Suash Deb. Engineering optimisation by cuckoo search. International Journal of Mathematical Modelling and Numerical Optimisation, 1(4): 330343, 2010.
 [19] Sergey Foss, Dmitry Korshunov, Stan Zachary, et al. An introduction to heavytailed and subexponential distributions, volume 6. Springer, 2011.
 [20] Tomasz J Kozubowski and Svetlozar T Rachev. Univariate geometric stable laws. Journal of Computational Analysis and Applications, 1(2): 177217, 1999.
 [21] Nasimul Noman and Hitoshi Iba. Accelerating differential evolution using an adaptive local search. IEEE Transactions on evolutionary Computation, 12(1): 107125, 2008.
 [22] Ponnuthurai N Suganthan, Nikolaus Hansen, Jing J Liang, Kalyanmoy Deb, YingPing Chen, Anne Auger, and Santosh Tiwari. Problem definitions and evaluation criteria for the cec 2005 special session on realparameter optimization. KanGAL report, 2005005: 2005, 2005.

[23]
Rainer Storn and Kenneth Price. Differential evolutiona simple and efficient heuristic for global optimization over continuous spaces.
Journal of global optimization, 11(4): 341359, 1997.  [24] XinShe Yang. Flower pollination algorithm for global optimization. In International conference on unconventional computing and natural computation, pages 240249. Springer, 2012.
 [25] Fei Gao, Fengxia Fei, Xuejing Lee, Hengqing Tong, Yanfang Deng, and Hualing Zhao. Inversion mechanism with functional extrema model for identification incommensurate and hyper fractional chaos via differential evolution. Expert Systems with Applications, 41(4): 19151927, 2014.
 [26] WeiChing Chen. Nonlinear dynamics and chaos in a fractionalorder financial system. Chaos, Solitons & Fractals, 36(5): 13051314, 2008.
Comments
There are no comments yet.