  This article deals with localization probability in a network of randomly
distributed communication nodes contained in a bounded domain. A fraction of
the nodes denoted as L-nodes are assumed to have localization information while
the rest of the nodes denoted as NL nodes do not. The basic model assumes each
node has a certain radio coverage within which it can make relative distance
measurements. We model both the case radio coverage is fixed and the case radio
coverage is determined by signal strength measurements in a Log-Normal
Shadowing environment. We apply the probabilistic method to determine the
probability of NL-node localization as a function of the coverage area to
domain area ratio and the density of L-nodes. We establish analytical
expressions for this probability and the transition thresholds with respect to
key parameters whereby marked change in the probability behavior is observed.
The theoretical results presented in the article are supported by simulations.
