We investigate the secure connectivity of wireless sensor networks under a heterogeneous random key predistribution scheme and a heterogeneous channel model. In particular, we study a random graph formed by the intersection of an inhomogeneous random key graph with an inhomogeneous Erd o s-R 'enyi graph. The former graph is naturally induced by the heterogeneous random key predistribution scheme while the latter graph constitutes a heterogeneous on off channel model; wherein, the wireless channel between a class- @math node and a class- @math node is on with probability @math independently. We present conditions (in the form of zero-one laws) on how to scale the parameters of the intersection model so that it has no isolated node with high probability as the number of nodes gets large. We also present numerical results to support these zero-one laws in the finite-node regime.
We consider the network reliability problem in secure wireless sensor networks that employ a heterogeneous random key predistribution scheme. This scheme is introduced recently, as a generalization of the Eschenauer-Gligor scheme, to account for the cases when the network comprises sensors with varying level of resources or connectivity requirements; e.g., regular nodes vs. cluster heads. The scheme induces the inhomogeneous random key graph, denoted G(n;μ,K, P ), where each of the n nodes are independently assigned to one of r classes according to a probability distribution μ = (μ1, . . . , μr) and then a class-i node is assigned Ki keys uniformly at random from a pool of size P ; two nodes that share a key are then connected by an edge. We analyze the reliability of G(n;μ,K, P ) against random link failures. Namely, we consider G(n;μ,K, P, α) formed by deleting each edge of G(n;μ,K, P ) independently with probability 1 − α, and study the probability that the resulting graph i) has no isolated node; and ii) is connected. We present scaling conditions on K , P , and α such that both events take place with probability zero or one, respectively, as the number of nodes gets large. We also present numerical results to support these zero-one laws in the finite-node regime.