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 . We present a randomized algorithm for dynamic graph connectivity. With failure probability less than @math (for any constant @math we choose), our solution has worst case running time @math per edge insertion, @math per edge deletion, and @math per query, where @math is the number of vertices. The previous best algorithm has worst case running time @math per edge insertion and @math per edge deletion. The improvement is made by reducing the randomness used in the previous result, so that we save a @math factor in update time. Specifically, kapron2013dynamic uses @math copies of a data structure in order to boost a success probability from @math to @math . We show that, in fact though, because of the special structure of their algorithm, this boosting via repetition is unnecessary. Rather, we can still obtain the same correctness guarantee with high probability by arguing via a new invariant, without repetition. We investigate the connectivity of a wireless sensor network (WSN) secured by the heterogeneous key predistribution scheme under an independent on off channel model. The heterogeneous scheme induces an inhomogeneous random key graph, denoted by @math and the on off channel model induces an Erdős-Renyi graph, denoted by @math . Hence, the overall random graph modeling the WSN is obtained by the intersection of @math and @math . We present conditions on how to scale the parameters of the intersecting graph with respect to the network size @math such that the graph i) has no isolated nodes and ii) is connected, both with high probability (whp) as the number of nodes gets large. Our results are supported by a simulation study demonstrating that i) despite their asymptotic nature, our results can in fact be useful in designing finite -node WSNs so that they achieve secure connectivity whp; and ii) despite the simplicity of the on off communication model, the probability of connectivity in the resulting WSN approximates very well the case where the disk model is used. The classic all-terminal network reliability problem posits a graph, each of whose edges fails independently with some given probability. The goal is to determine the probability that the network becomes disconnected due to edge failures. This problem has obvious applications in the design of communication networks. Since the problem is @math -complete and thus believed hard to solve exactly, a great deal of research has been devoted to estimating the failure probability. In this paper, we give a fully polynomial randomized approximation scheme that, given any n-vertex graph with specified failure probabilities, computes in time polynomial in n and @math an estimate for the failure probability that is accurate to within a relative error of @math with high probability. We also give a deterministic polynomial approximation scheme for the case of small failure probabilities. Some extensions to evaluating probabilities of @math -connectivity, strong connectivity in directed Eulerian graphs and @math -way disconnection, and to evaluating the Tutte polynomial are also described. This version of the paper corrects several errata that appeared in the previous journal publication [D. R. Karger, SIAM J. Comput., 29 (1999), pp. 492--514].