k-Connectivity in Random Graphs induced by Pairwise Key Predistribution Schemes

Abstract

Random key predistribution schemes serve as a viable solution for facilitating secure communication in Wireless Sensor Networks (WSNs). We analyze reliable connectivity of a heterogeneous WSN under the random pairwise key predistribution scheme of Chan et al. According to this scheme, each of the n sensor nodes is classified as type-1 (respectively, type-2) with probability μ (respectively, 1 − μ) where 0 < μ < 1. Each type-1 (respectively, type-2) node is paired with 1 (respectively, Kn) other node selected uniformly at random; each pair is then assigned a unique pairwise key so that they can securely communicate with each other. A main question in the design of secure and heterogeneous WSNs is how should the parameters n, μ, and Kn be selected such that resulting network exhibits certain desirable properties with high probability. Of particular interest is the strength of connectivity often studied in terms of k-connectivity; i.e., with k = 1, 2, …, the property that the network remains connected despite the removal of any k − 1 nodes or links. In this paper, we answer this question by analyzing the inhomogeneous random K-out graph model naturally induced under the heterogeneous pairwise scheme. It was recently established that this graph is 1-connected asymptotically almost surely (a.a.s.) if and only if Kn = ω(1). Here, we show that for k = 2, 3, …, we need to set ${K_n} = \frac{1}{{1 - \mu }}\left( {\log n + (k - 2)\log \log n + \omega (1)} \right)$ for the network to be k-connected a.a.s. The result is given in the form of a zero-one law indicating that the network is a.a.s. not k-connected when ${K_n} = \frac{1}{{1 - \mu }}\left( {\log n + (k - 2)\log \log n - \omega (1)} \right)$. We present simulation results to demonstrate the usefulness of the results in the finite node regime.