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 n ) 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 K n 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 K n = 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.