We consider wireless sensor networks secured by the heterogeneous random key predistribution scheme under an on off channel model. The heterogeneous random key predistribution scheme considers the case when the network includes sensor nodes with varying levels of resources, features, or connectivity requirements; e.g., regular nodes vs. cluster heads, but does not incorporate the fact that wireless channel are unreliable. To capture the unreliability of the wireless medium, we use an on off channel model; wherein, each wireless channel is either on (with probability @math ) or off (with probability @math ) independently. We present conditions (in the form of zero-one laws) on how to scale the parameters of the network model so that with high probability the network is @math -connected, i.e., the network remains connected even if any @math nodes fail or leave the network. We also present numerical results to support these conditions in the finite-node regime.
Finally, our work improves upon the results by @cite_19 who established zero-one laws for the @math -connectivity of inhomogeneous random key graphs (therein, this model was referred to as the general random intersection graph). Indeed, by setting @math for each @math , our result recovers the results in @cite_19 . We also remark that the additional conditions required by main results of @cite_19 render them inapplicable in practical WSN implementations. This issue is explained in details in [Section 3.3] Yagan Inhomogeneous .
@cite_19
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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. Random key graphs form a class of random intersection graphs that are naturally induced by the random key predistribution scheme of Eschenauer and Gligor for securing wireless sensor network (WSN) communications. Random key graphs have received much attention recently, owing in part to their wide applicability in various domains, including recommender systems, social networks, secure sensor networks, clustering and classification analysis, and cryptanalysis to name a few. In this paper, we study connectivity properties of random key graphs in the presence of unreliable links. Unreliability of graph links is captured by independent Bernoulli random variables, rendering them to be on or off independently from each other. The resulting model is an intersection of a random key graph and an Erdős–Renyi graph, and is expected to be useful in capturing various real-world networks; e.g., with secure WSN applications in mind, link unreliability can be attributed to harsh environmental conditions severely impairing transmissions. We present conditions on how to scale this model’s parameters so that: 1) the minimum node degree in the graph is at least @math and 2) the graph is @math -connected, both with high probability as the number of nodes becomes large. The results are given in the form of zero-one laws with critical thresholds identified and shown to coincide for both graph properties. These findings improve the previous results by Rybarczyk on @math -connectivity of random key graphs (with reliable links), as well as the zero-one laws by Yagan on one-connectivity of random key graphs with unreliable links. The random key graph is a random graph naturally associated with the random key predistribution scheme introduced by Eschenauer and Gligor in the context of wireless sensor networks (WSNs). For this class of random graphs, we establish a new version of a conjectured zero-one law for graph connectivity as the number of nodes becomes unboundedly large. The results reported here complement and strengthen recent work on this conjecture by Blackburn and Gerke. In particular, the results are given under conditions which are more realistic for applications to WSNs. We consider the random graph induced by the random key predistribution scheme of Eschenauer and Gligor under the assumption of full visibility. We show the existence of a zero-one law for the absence of isolated nodes, and complement it by a Poisson convergence for the number of isolated nodes. Leveraging earlier results and analogies with Erdos-Renyi graphs, we explore similar results for the property of graph connectivity.