Minimum Node Degree and k-Connectivity in Wireless Networks with Unreliable Links

In this paper, we consider a wireless network with unreliable links and investigate its minimum node degree and k-connectivity. In such network, n nodes are uniformly distributed in a region, and edges are established for any two nodes within a certain distance and with a probabilistically active link in between. On a torus topology, we present a zero-one law for the property of minimum degree being at least k, leading to a zero-law for k-connectivity and an intermediate result towards a one-law. On a square with boundary effect, we establish a one-law for minimum degree. Our results are derived from rigorous analysis and also confirmed by the simulation, and they provide guidelines for the design of wireless networks.