  We analyze connectivity of a heterogeneous secure sensor network that uses
key predistribution to protect communications between sensors. For this network
on a set $\mathcal{V}_n$ of $n$ sensors, suppose there is a pool
$\mathcal{P}_n$ consisting of $P_n$ distinct keys. The $n$ sensors in
$\mathcal{V}_n$ are divided into $m$ groups $\mathcal{A}_1, \mathcal{A}_2,
\ldots, \mathcal{A}_m$. Each sensor $v$ is independently assigned to exactly a
group according to the probability distribution with $\mathbb{P}[v \in
\mathcal{A}_i]= a_i$ for $i=1,2,\ldots,m$, where $\sum_{i=1}^{m}a_i = 1$.
Afterwards, each sensor in group $\mathcal{A}_i$ independently chooses
$K_{i,n}$ keys uniformly at random from the key pool $\mathcal{P}_n$, where
$K_{1,n} \leq K_{2,n}\leq \ldots \leq K_{m,n}$. Finally, any two sensors in
$\mathcal{V}_n$ establish a secure link in between if and only if they have at
least one key in common. We present critical conditions for connectivity of
this heterogeneous secure sensor network. The result provides useful guidelines
for the design of secure sensor networks.
  This paper improves the seminal work [1] (IEEE Transactions on Information
Theory 2016) of Ya{\u{g}}an on connectivity in several aspects (omitted due to
arxiv abstract's character limitation; see the paper for details).
