The Number of Isolated Nodes in a Wireless Network with a Generic Probabilistic Channel Model

Abstract

A wireless node is called isolated if it has no links to other nodes. The number of isolated nodes in a wireless network is an important connectivity index. However, most previous works on analytically determining the number of isolated nodes were not based on practical channel models. In this work, we study this problem using a generic probabilistic channel model that can capture the behaviors of the most widely used channel models, including the disk graph model, the Bernoulli link model, the Gaussian white noise model, the Rayleigh fading model, and the Nakagami fading model. We derive the expected number of isolated nodes and further prove that their distribution asymptotically follows a Poisson distribution. We also conjecture that the nonexistence of isolated nodes asymptotically implies the connectivity of the network, and that the probability of connectivity follows the Gumbel function.