Connectivity results for sensor networks under a random pairwise key predistribution scheme

We investigate the connectivity of wireless sensor networks under the random pairwise key predistribution scheme of Chan et al. Under the assumption of full visibility, this reduces to studying connectivity in the so-called random K-out graph H(n;K); here n is the number of nodes and K < n is an integer parameter affecting the number of keys stored at each node. We show that if K ≥ 2 (resp. K = 1), the probability that H(n; K) is a connected graph approaches 1 (resp. 0) as n goes to infinity. This is done by establishing an explicitly computable lower bound on the probability of connectivity. From this bound we conclude that with K ≥ 2, the connectivity of the network can already be guaranteed by a relatively small number of sensors with very high probability. This corrects an earlier analysis based on a heuristic transfer of classical connectivity results for Erdős-Rényi graphs.