## 1 Introduction
[PERSON] [3,13] studied persistent monitoring problems in discrete and continuous environments, and established two optimization models, aiming to enhance the monitoring frequency. [PERSON] [14] studied the persistent monitoring problem of multiple UAVs and established a mathematic model based on the optimization of the maximum environmental recognition accuracy, which was then solved by a heuristic algorithm. [PERSON] [15] studied a similar problem, i.e., UAV data collection, allowing UAVs to visit some targeted location with high priority more than once in a single cycle. An optimization by maximizing the average period reward was formulated, and the precise solution combining dynamic programming and mixed integer linear programming was achieved. Subsequently, considering the scalability of the algorithm and improving its efficiency, an approximate solution was proposed for the nodes with specific visiting times [16]. [PERSON] [17] also discussed the algorithm scalability where a genetic algorithm was used to obtain the approximate solution that showed better scalability than a precise method through experiments. [PERSON] [18] studied a multi-UAV cooperative path planning problem with monitoring data transport for the purpose of minimizing the time delay between data being captured by UAVs and the arrival of the data at the base station. [PERSON] [19] considered the monitoring frequency constraint and set the fixed horizon to a given number, \(k\), which assumes that the UAV can only access \(k\) nodes in each cycle. However, once there exists an intelligent intrusion in the monitoring environment, the monitoring privacy will have already been destroyed. The above persistent monitoring studies considered monitoring frequency constraints, but only focused on the monitoring performance or coverage rate of the given area [20] and did not consider monitoring security issues in an adversarial environment. With regard to the concern for monitoring security, one also needs to consider how easy the monitoring strategy can be acquired by intelligent intruders. The privacy of the persistent monitoring process is of great concern, especially in some applications where intelligent adversaries or intruders might occur. At present, there are at least two ideas in the field of monitoring security. One is to improve the existing deterministic strategy for the path planning problem and use random algorithms instead, such as Markov chains, or random walk theory. The other is to establish a game model and a balance scheme between the competing players. [PERSON] [21] proposed a Markov strategy, which is a polynomial-time algorithm, and their research is motivated by reducing the probability of being invaded at a weak task position as much as possible. Entropy has also been introduced in path planning [22]. For example, [PERSON] [23] and [PERSON] [24] studied the entropy rate maximization problem based on Markov chains. Stackelberg game theory was used by [PERSON] [25] to formulate an optimal solution to the path planning problem for a single robot on a security patrol, while assuming only one intruder. Security game theory has been proposed for the study of the persistent monitoring path planning problem in ecological protection [26]. The main motivation for their study on patrol and monitoring strategies is to obtain an unpredictable trajectory, which was finally obtained through maximum entropy. With the aforementioned observations, some studies on persistent monitoring path planning only concern the complete coverage rate, and some studies consider the monitoring frequency, but the final paths often fall in a fixed monitoring period which makes the monitoring regularity completely exposed to intrusions. The other study considers monitoring security, but they still do not consider monitoring frequency constraints. To bridge the gap between the monitoring frequency and monitoring security, this study will comprehensively consider both sides simultaneously, that is, improving monitoring path privacy while increasing monitoring frequency.