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In this paper, we investigate the secure connectivity of wireless sensornetworks utilizing the heterogeneous random key predistribution scheme, whereeach 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 isgiven $K_i$ cryptographic keys selected uniformly at random from a key pool ofsize $P$. After deployment, two nodes can communicate securely if they share atleast one cryptographic key. We consider the wireless connectivity of thenetwork using a heterogeneous on-off channel model, where the channel between aclass-$i$ 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 cryptographickey and ii) have a wireless channel in between that is on. We model the overallnetwork using a composite random graph obtained by the intersection ofinhomogeneous random key graphs (IRKG) $\mathbb{K}(n;\pmb{\mu},\pmb{K},P)$ withinhomogeneous Erd\H{o}s-R\'enyi graphs (IERG) $\mathbb{G}(n;\pmb{\mu},\pmb{\alpha})$. The former graph is naturally induced by the heterogeneousrandom key predistribution scheme, while the latter is induced by theheterogeneous on-off channel model. More specifically, two nodes are adjacentin the composite graph if they are i) adjacent in the IRKG i.e., share acryptographic key and ii) adjacent in the IERG, i.e., have an availablewireless channel. We investigate the connectivity of the composite random graphand present conditions (in the form of zero-one laws) on how to scale itsparameters so that it i) has no secure node which is isolated and ii) issecurely connected, both with high probability when the number of nodes getslarge. We also present numerical results to support these zero-one laws in thefinite-node regime.

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