# On Connectivity and Energy Efficiency for Sleeping-Schedule-Based Wireless Sensor Networks
## 1 Introduction  
With the rapid development and maturation of wireless communication technology, micro-motor system low-power consumption and highly-integrated digital devices enable the realization of low-cost, low-power, and small-volume sensor nodes. Such sensor nodes, together with the wireless communication networks, form self-organizing wireless sensor networks (WSNs) and are widely used in battlefield surveillance, large-scale environmental monitoring, and target tracking in large areas. Sensors are responsible for monitoring the environment, transferring, and processing the data to the WSN base stations. Since the WSN nodes are generally powered by batteries, which are difficult to replace due to the large number and harsh working environment, one of the crucial issues in the construction of a large-scale wireless sensor network is the limited battery life cycle [1]. Low power consumption plays a critical role and the energy efficiency of nodes is one of the most critical design criteria for wireless sensor networks [2; 3]. For wireless sensor networks, connectivity is of great significance for wireless network planning, capacity design, topology control, and node power optimization. Therefore, it is of great theoretical and practical significance to ensure the connectivity of WSN and realize its low power consumption at the same time, to minimize the energy consumption of the network and improve the service life of the network. Percolation is a subject that studies various structures of graphs based on probability analysis. Its central problem is to study the threshold of connectivity of various graphs, which has large application value in social life and scientific research. In recent years, many researchers have conducted research work on ad hoc networks, most of them focusing on network coverage and network routing algorithms. [PERSON] et at. [PERSON] et al. (2018) analyzed the number of nodes in the large-scale network by using the theory of continuous flow and analyzed the connectivity of the large-scale component. In [PERSON] et al. (2018), for wireless sensor networks with randomly-placed nodes as the background, graph theory, and statistics are used as the primary research methods to study the relationship among the number of nodes, the communication radius of nodes, and the network connectivity. The paper established a corresponding mathematical model, carried out a large number of computer simulation experiments, and obtained the mathematical analytic formula between them. [PERSON] et al. [PERSON] et al. (2018) joined on the connectivity and coverage together, analyzed the characteristics of WSN with random deployment way, by considering the distribution of sensor nodes in the monitoring area as a Poisson point process. Aiming at the characteristics of random deployment, the paper gave the node density to satisfy the network connectivity requirement in the case of the node sensing range and communication range. According to the analysis results, it proposed a location-independent node self-scheduling connectivity covering algorithm (LISCCA). [PERSON] et al. [PERSON] et al. (2018) mainly studied the influence of the lognormal shadow wireless transmission model on the connectivity of the wireless network, introduced the connection function to describe the existence of wireless links between nodes, and then abstracted the partial connectivity issues as the seepage problem of special stochastic graph. Finally, the theoretical upper bound of the critical node density was deduced in a partially connected state to solve the problem of heavy maintenance in wireless sensor networks. [PERSON] et al. [PERSON] et al. (2018) proposed a networking algorithm with good connectivity performance, which contributed to the energy efficiency optimization of WSNs. [PERSON] et al. [PERSON] et al. (2018) also proposed an irregular cellular learning automaton (ICLA)-based algorithm to preserve sensors protection.