A Tutorial-Cum-Survey on Percolation Theory With Applications in Large-Scale Wireless Networks

Connectivity is an important key performance indicator and a focal point of research in large-scale wireless networks. Due to path-loss attenuation of electromagnetic waves, direct wireless connectivity is limited to proximate devices. Nevertheless, connectivity among distant devices can still be attained through a sequence of consecutive multi-hop communication links, which enables routing and disseminating legitimate information across wireless ad hoc networks. Multi-hop connectivity is also foundational for data aggregation in the Internet of things (IoT) and cyberphysical systems (CPS). On the downside, multi-hop wireless transmissions increase susceptibility to eavesdropping and enable malicious network attacks. Hence, security-aware network connectivity is required to maintain communication privacy, detect and isolate malicious devices, and thwart the spreading of illegitimate traffic (e.g., viruses, worms, falsified data, illegitimate control, etc.). In 5G and beyond networks, an intricate balance between connectivity, privacy, and security is a necessity due to the proliferating IoT and CPS, which are featured with massive number of wireless devices that can directly communicate together (e.g., device-to-device, machine-to-machine, and vehicle-to-vehicle communication). In this regards, graph theory represents a foundational mathematical tool to model the network physical topology. In particular, random geometric graphs (RGGs) capture the inherently random locations and wireless interconnections among the spatially distributed devices. Percolation theory is then utilized to characterize and control distant multi-hop connectivity on network graphs. Recently, percolation theory over RGGs has been widely utilized to study connectivity, privacy, and security of several types of wireless networks. The impact and utilization of percolation theory are expected to further increase in the IoT/CPS era, which motivates this tutorial. Towards this end, we first introduce the preliminaries of graph and percolation theories in the context of wireless networks. Next, we overview and explain their application to various types of wireless networks.