In this paper, we investigate the secure connectivity of wireless sensor networks utilizing the heterogeneous random key predistribution scheme, where each sensor node is classified as class- @math with probability @math for @math with @math and @math . A class- @math sensor is given @math cryptographic keys selected uniformly at random from a key pool of size @math . After deployment, two nodes can communicate securely if they share at least one cryptographic key. We consider the wireless connectivity of the network using a heterogeneous on-off channel model, where the channel between a class- @math node and a class- @math node is on (respectively, off) with probability @math (respectively, @math ) for @math . Collectively, two sensor nodes are adjacent if they i) share a cryptographic key and ii) have a wireless channel in between that is on. We model the overall network using a composite random graph obtained by the intersection of inhomogeneous random key graphs (IRKG) @math with inhomogeneous Erd o s-R 'enyi graphs (IERG) @math . The former graph is naturally induced by the heterogeneous random key predistribution scheme, while the latter is induced by the heterogeneous on-off channel model. More specifically, two nodes are adjacent in the composite graph if they are i) adjacent in the IRKG i.e., share a cryptographic key and ii) adjacent in the IERG, i.e., have an available wireless channel. We investigate the connectivity of the composite random graph and present conditions (in the form of zero-one laws) on how to scale its parameters so that it i) has no secure node which is isolated and ii) is securely connected, both with high probability when the number of nodes gets large. We also present numerical results to support these zero-one laws in the finite-node regime.
In comparison with the existing literature on similar models, our result can be seen to extend the work by Eletreby and Ya g an in @cite_8 (respectively, @cite_18 ). Therein, the authors established a zero-one law for the @math -connectivity (respectively, @math -connectivity) of @math , i.e., for a wireless sensor network under the heterogeneous key predistribution scheme and a uniform on-off channel model. Although these results form a crucial starting point towards the analysis of the heterogeneous key predistribution scheme under a wireless connectivity model, they are limited to uniform on-off channel model where all channels are on (respectively, off) with the same probability @math (respectively, @math ). The heterogeneous on-off channel model accounts for the fact that different nodes could have different radio capabilities, or could be deployed in locations with different channel characteristics. In addition, it offers the flexibility of modeling several interesting scenarios, such as when nodes of the same type are more (or less) likely to be adjacent with one another than with nodes belonging to other classes. Indeed, by setting @math for @math and each @math , our results reduce to those given in @cite_8 .
@cite_18 @cite_8
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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 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.