Analyzing connectivity of heterogeneous secure sensor networks

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).