The q-composite key predistribution scheme [1] is used prevalently for secure communications in large-scale wireless sensor networks (WSNs). Prior work [2]-[4] explores topological properties of WSNs employing the q-composite scheme for q = 1 with unreliable communication links modeled as independent on off channels. In this paper, we investigate topological properties related to the node degree in WSNs operating under the q-composite scheme and the on off channel model. Our results apply to general q and are stronger than those reported for the node degree in prior work even for the case of q being 1. Specifically, we show that the number of nodes with certain degree asymptotically converges in distribution to a Poisson random variable, present the asymptotic probability distribution for the minimum degree of the network, and establish the asymptotically exact probability for the property that the minimum degree is at least an arbitrary value. Numerical experiments confirm the validity of our analytical findings.
Krishnan @cite_16 and Krzywdzi ' n ski and Rybarczyk @cite_4 describe results for the probability of connectivity asymptotically converging to 1 in WSNs employing the @math -composite key predistribution scheme with @math (i.e., the basic Eschenauer--Gligor key predistribution scheme), not under the on off channel model but under the well-known disk model @cite_29 @cite_7 @cite_8 @cite_10 @cite_39 @cite_43 , where nodes are distributed over a bounded region of a Euclidean plane, and two nodes have to be within a certain distance for communication. Simulation results in our work @cite_33 indicate that for WSNs under the key predistribution scheme with @math , when the on-off channel model is replaced by the disk model, the performances for @math -connectivity and for the property that the minimum degree is at least @math do not change significantly.
@cite_4 @cite_33 @cite_7 @cite_8 @cite_29 @cite_39 @cite_43 @cite_16 @cite_10
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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 predistribution scheme of Eschenauer and Gligor (EG) is a typical solution for ensuring secure communications in a wireless sensor network (WSN). Connectivity of the WSNs under this scheme has received much interest over the last decade, and most of the existing work is based on the assumption of unconstrained sensor-to-sensor communications. In this paper, we study the k-connectivity of WSNs under the EG scheme with physical link constraints; k-connectivity is defined as the property that the network remains connected despite the failure of any (k - 1) sensors. We use a simple communication model, where unreliable wireless links are modeled as independent on off channels, and derive zero-one laws for the properties that i) the WSN is k-connected, and ii) each sensor is connected to at least k other sensors. These zero-one laws improve the previous results by Rybarczyk on the k-connectivity under a fully connected communication model. Moreover, under the on off channel model, we provide a stronger form of the zero-one law for the 1-connectivity as compared to that given by Yagan. The seminal q-composite key predistribution scheme [3] (IEEE S&P 2003) is used prevalently for secure communications in large-scale wireless sensor networks (WSNs). Yagan [12] (IEEE IT 2012) and we [15] (IEEE ISIT 2013) explore topological properties of WSNs employing the q-composite scheme in the case of q = 1 with unreliable communication links modeled as independent on off channels. However, it is challenging to derive results for general q under such on off channel model. In this paper, we resolve such challenge and investigate topological properties related to node degree in WSNs operating under the q-composite scheme and the on off channel model. Our results apply to general q, yet there has not been any work in the literature reporting the corresponding results even for q = 1, which are stronger than those about node degree in [12], [15]. Specifically, we show that the number of nodes with an arbitrary degree asymptotically converges to a Poisson distribution, present the asymptotic probability distribution for the minimum node degree of the network, and establish the asymptotically exact probability for the property that the minimum node degree is at least an arbitrary value. Numerical experiments confirm the validity our analytical findings. The q-composite key predistribution scheme [1] is used prevalently for secure communications in large-scale wireless sensor networks (WSNs). Prior work [2]-[4] explores topological properties of WSNs employing the q-composite scheme for q = 1 with unreliable communication links modeled as independent on off channels. In this paper, we investigate topological properties related to the node degree in WSNs operating under the q-composite scheme and the on off channel model. Our results apply to general q and are stronger than those reported for the node degree in prior work even for the case of q being 1. Specifically, we show that the number of nodes with certain degree asymptotically converges in distribution to a Poisson random variable, present the asymptotic probability distribution for the minimum degree of the network, and establish the asymptotically exact probability for the property that the minimum degree is at least an arbitrary value. Numerical experiments confirm the validity of our analytical findings.