Secure Connectivity of Heterogeneous Wireless Sensor Networks Under a Heterogeneous On-Off Channel Model

Abstract: 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-$i$ with probability $\mu_i$ for $i=1,\ldots,r$ with $\mu_i>0$ and $\sum_{i=1}^r \mu_i=1$. A class-$i$ sensor is given $K_i$ cryptographic keys selected uniformly at random from a key pool of size $P$. 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-$i$ node and a class-$j$ node is on (respectively, off) with probability $\alpha_{ij}$ (respectively, $1-\alpha_{ij}$) for $i,j=1,\ldots,r$. 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) $\mathbb{K}(n;\pmb{\mu},\pmb{K},P)$ with inhomogeneous Erd\H{o}s-R\'enyi graphs (IERG) $\mathbb{G}(n;\pmb{\mu}, \pmb{\alpha})$. 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.