Connectivity of Wireless Sensor Networks Secured by the Heterogeneous Random Pairwise Key Predistribution Scheme

We investigate the connectivity of wireless sensor networks secured by the heterogeneous random pairwise key predistribution scheme. In contrast to the homogeneous scheme proposed by Chan et al., where each node is paired (offline) with K other nodes chosen uniformly at random; herein, each node is classified as class-1 with probability µ or class-2 with probability 1 - µ, for 0 < µ < 1, independently. Then, each class-1 (respectively, class-2) node is paired (offline) with K1 (respectively, K2) other nodes selected uniformly at random. We consider the particular case when K1 = 1 and K 2 = K. The heterogeneous random pairwise scheme induces an inhomogeneous random K-out graph H(n; µ, Kn), where n denotes the number of nodes and Kn denotes a scaling of K with respect to the network size n. Hence, establishing the connectivity of wireless sensor networks secured by the heterogeneous random pairwise scheme maps to deriving conditions on how to scale Kn with respect to the network size n such that the graph is connected with high probability as n tends to infinity. With K1 = 1, we show that i) when Kn = K for all n = 2,3, … for a positive, finite integer K with K ≥ 2, the resulting graph is not connected with a positive probability. In this case, we derive a tight upper bound on the probability of connectivity and verify the results via simulations. Moreover, ii) we prove that when Kn is chosen such that $lim_{n\rightarrow \infty}K_{n}=\infty$, the graph is connected with high probability as n tends to infinity.