Critical Intensity for Unbounded Sequential Localizability

Locations of mobile agents are often requisite information for wireless applications such as sensor networks and Internet of Things (IoT). As the network size increases, verifying the localizability of all nodes in a network quickly becomes intractable. In this article, we turn to analyzing the unbounded localizability of infinite stochastic networks under sequential localization methods. Specifically, we prove the existence of the phase transition on the probability of localizing an unbounded subnetwork from a bounded initial anchor set in Poisson point process networks. The phase transition occurs when the node intensity of the network reaches a critical intensity, which is determined by the adopted sequential localization method. Furthermore, we develop a simulation method to obtain tight upper and lower bounds of the critical intensity for two-dimensional (2-D) networks with high confidence, and provide the numerical bounds under several typical sequential localization methods. We also show by simulation that the percentage of localizable nodes increases rapidly near the critical intensity, which provides guidelines for network design and deployment.