We consider the network reliability problem in wireless sensor networks secured by the heterogeneous random key predistribution scheme. This scheme generalizes Eschenauer-Gligor scheme by considering the cases when the network comprises sensor nodes with varying level of resources; e.g., regular nodes vs. cluster heads. The scheme induces the inhomogeneous random key graph, denoted @math . We analyze the reliability of @math against random link failures. Namely, we consider @math formed by deleting each edge of @math independently with probability @math , and study the probability that the resulting graph i) has no isolated node; and ii) is connected. We present scaling conditions on @math , @math , and @math such that both events take place with probability zero or one, respectively, as the number of nodes gets large. We present numerical results to support these in the finite-node regime.
@cite_5: We introduce a new random key predistribution scheme for securing heterogeneous wireless sensor networks. Each of the @math sensors in the network is classified into @math classes according to some probability distribution @math . Before deployment, a class- @math sensor is assigned @math cryptographic keys that are selected uniformly at random from a common pool of @math keys. Once deployed, a pair of sensors can communicate securely if and only if they have a key in common. We model the communication topology of this network by a newly defined inhomogeneous random key graph. We establish scaling conditions on the parameters @math and @math so that this graph: 1) has no isolated nodes and 2) is connected, both with high probability. The results are given in the form of zero-one laws with the number of sensors @math growing unboundedly large; critical scalings are identified and shown to coincide for both graph properties. Our results are shown to complement and improve those given by and for the same model, therein referred to as the general random intersection graph.
Our main results extend the work by Ya g an @cite_5 who established zero-one laws for the connectivity of inhomogeneous random key graph @math without employing a link-failure model. It is clear that, although a crucial first step in the study of heterogeneous key predistribution schemes, the assumption that all links are operational, i.e., reliable , is not likely to hold in most practical settings. In this regard, our work extends the results by Ya g an @cite_5 to more practical WSN scenarios where the unreliability of links are taken into account. In fact, by setting @math for each @math (i.e., by assuming that all links are reliable ), our results reduce to those given in @cite_5 .