Analyzing Connectivity of Heterogeneous Secure Sensor Networks

We analyze the connectivity of a heterogeneous secure sensor network that uses key predistribution to protect communications between sensors. For this network on a set nu(n) of n sensors, suppose there is a pool P-n consisting of P-n distinct keys. The n sensors in Vn are divided into m groups A(1), A(2),..., A(m). Each sensor v is independently assigned to exactly a group according to the probability distribution with P[v is an element of A(i)] = a(i) for i = 1, 2,..., m, where Sigma(m)(i=1) a(i) = 1. Afterwards, each sensor in group A(i) independently chooses K-i,K-n keys uniformly at random from the key pool P-n, where K-1,K-n <= K-2,K-n <=... <= K-m,K-n. Finally, any two sensors in nu(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. gan on connectivity in the following aspects. First, our result is more broadly applicable; specifically, we consider Km, n/ K1, n = o(root n), while [1] requires K-m,K-n/K-1,K-n = o(ln n). Put differently, K-m,K-n/K-1,K-n in our paper examines the case of circle dot(n(x)) for any x < 1/2 and circle dot (ln n)(y)) for any y > 0, while that of [1] does not cover any T(nx), and covers any circle dot (ln n)(y)) for only 0 < y < 1. This improvement is possible due to a delicate coupling argument. Second, although both studies show that a critical scaling for connectivity is that the term b(n) denoting Sigma(m)(j=1) {a(j) [1- (K-j,n(Pn-K1,n))/((Pn)(Kj,n))]} equals ln n/n our paper considers any of b(n) = o(ln n/n), b(n) = circle dot (ln n/n) and b(n) = omega(ln n/n), while [1] evaluates only b(n) = circle dot(ln n/n). Third, in terms of characterizing the transitional behavior of connectivity, our scaling b(n) = ln n+ beta n/n for a sequence beta n is more fine-grained than the scaling b(n) similar to cln n/n for a constant c not equal 1 of [1]. In a nutshell, we add the case of c = 1 in b(n) similar to c ln n/n, where the graph can be connected or disconnected asymptotically, depending on the limit of beta(n). Finally, although a recent study by Eletreby and Ya. gan [2] uses the fine-grained scaling discussed before for a more complex graph model, their result (just like [1]) also demandsK(m,n)/K-1,K-n = o(ln n), which is less general than K-m,K-n/K-1,K-n = o(root n) addressed in this paper.