Problem of localisation in networks of randomly deployed nodes: asymptotic and finite analysis, and thresholds

Consider a two-dimensional domain S ? R² containing two sets of nodes from two statistically independent uniform Poisson point processes with constant densities ?_L and ?_NL. The first point process identifies the distribution of a set of nodes having information about their positions, hereafter denoted as L-nodes (localised-nodes), whereas the other is used to model the spatial distribution of nodes that need to localise themselves, hereafter denoted as NL-nodes (not localised-nodes). For simplicity, both kinds of nodes are equipped with the same kind of transceiver, and communicate over a channel affected by shadow fading. As a first goal, the authors derive the probability that a randomly chosen NL-node over S gets localised as a function of a variety of parameters. Then, the authors derive the probability that the whole network of NL-nodes over S gets localised. As with many other random graph properties, the localisation probability is a monotone graph property showing thresholds. In this work, the authors derive both finite (when the number of nodes in the bounded domain is finite and does not grow) and asymptotic thresholds for the localisation probability. In connection with the asymptotic thresholds, the authors show the presence of asymptotic thresholds on the network localisation probability in two different scenarios. The first refers to dense networks, which arise when the domain S is bounded and the densities of the two kinds of nodes tend to grow unboundedly. The second kind of thresholds manifest themselves when the considered domain increases but the number of nodes grow in such a way that the L-node density remains constant throughout the investigated domain. In this scenario, what matters is the minimum value of the maximum transmission range averaged over the fading process, denoted as d_max, above which the network of NL-nodes almost surely gets asymptotically localised.