  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.
