  We study the secure and reliable connectivity of wireless sensor networks
under the heterogeneous pairwise key predistribution scheme. This scheme was
recently introduced as an extension of the random pairwise key predistribution
scheme of Chan et al. to accommodate networks where the constituent sensors
have different capabilities or requirements for security and connectivity. For
simplicity, we consider a heterogeneous network where each of the $n$ sensors
is classified as type-1 (respectively, type-2) with probability $\mu$
(respectively, $1-\mu)$ where $0<\mu<1$. Each type-1 (respectively, type-2)
node selects 1 (respectively, $K_n$) other nodes uniformly at random to be
paired with; according to the pairwise scheme each pair is then assigned a
unique pairwise key so that they can securely communicate with each other. We
establish critical conditions on $n, \mu$, and $K_n$ such that the resulting
network has minimum node degree of at least $k$ with high probability in the
limit of large network size. Our result constitutes a zero-one law for the
minimum node degree of the recently introduced inhomogeneous random K-out graph
model. This constitutes a crucial step towards establishing a similar zero-one
law for the $k$-connectivity of the graph; i.e., for the property that the
network remains connected despite the failure of any $k-1$ nodes or links. We
present numerical results that indicate the usefulness of our results in
selecting the parameters of the scheme in practical settings with finite number
of sensors.
