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.
Another notable work that is related to ours is by @cite_13 , who studied the @math -connectivity and @math -robustness in the inhomogeneous random key graph. A graph is said to be @math -connected if it remains connected after removal (i.e., ) of any @math nodes. Thus, the results obtained in @cite_13 ensure the reliability of the network against the failure of any @math nodes, for some integer constant @math . Since @math -vertex-connectivity implies @math -edge-connectivity, the network is ensured to be reliable against the failure of at least @math edges, for some integer constant @math . Our work complements these results by considering the case when edge fails with probability @math , so that the total number of failed links is possibly infinite; e.g., as many as @math links may fail.
@cite_13
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We consider secure and reliable connectivity in wireless sensor networks that utilize the heterogeneous random key predistribution scheme. We model the unreliability of wireless links by an on off channel model that induces an Erdős-Renyi graph, while the heterogeneous scheme induces an inhomogeneous random key graph. The overall network can thus be modeled by the intersection of both graphs. We present conditions (in the form of zero-one laws) on how to scale the parameters of the intersection model, so that with high probability: i) all of its nodes are connected to at least @math other nodes, i.e., the minimum node degree of the graph is no less than @math , and ii) the graph is @math -connected, i.e., the graph remains connected even if any @math nodes leave the network. These results are shown to complement and generalize several previous results in the literature. We also present numerical results to support our findings in the finite-node regime. Finally, we demonstrate via simulations that our results are also useful when the on off channel model is replaced with the more realistic disk communication model .