Lévy flights is a random walk where the step-lengths have a probability distribution that is heavy-tailed. It has been shown that Lévy flights can maximize the efficiency of resource searching in uncertain environments, and also movements of many foragers and wandering animals have been shown to follow a Lévy distribution. The reason mainly comes from that the Lévy distribution, has an infinite second moment, and hence is more likely to generate an offspring that is farther away from its parent. However, the investigation into the efficiency of other different heavy-tailed probability distributions in swarm-based searches is still insufficient up to now. For swarm-based search algorithms, randomness plays a significant role in both exploration and exploitation, or diversification and intensification. Therefore, it's necessary to discuss the optimal randomness in swarm-based search algorithms. In this study, CS is taken as a representative method of swarm-based optimization algorithms, and the results can be generalized to other swarm-based search algorithms. In this paper, four different types of commonly used heavy-tailed distributions, including Mittag-Leffler distribution, Pareto distribution, Cauchy distribution, and Weibull distribution, are considered to enhance the searching ability of CS. Then four novel CS algorithms are proposed and experiments are carried out on 20 benchmark functions to compare their searching performances. Finally, the proposed methods are used to system identification to demonstrate the effectiveness. Swarm-based 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 real-life 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 swarm-based 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 swarm-based 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 step-lengths have a probability distribution that is heavytailed. At each iteration process, CS firstly searches for new solutions in Lévy flights random walk. Secondly, CSproceeds to obtain new solutions in local exploitative random walk. After each random walk, a greedy strategy is used to select...
