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 V
_n
of n sensors, suppose there is a pool P
_n
consisting of P
_n
distinct keys. The n sensors in V
_n
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 ∈ A
_i
] = a
_i
for i = 1, 2, . . . , m, where Σ
_i=1
^m
a
_i
= 1. Afterwards, each sensor in group A
_i
independently chooses K
_i,n
keys uniformly at random from the key pool P
_n
, where K
_1,n
≤ K
_2,n
≤ . . . ≤ K
_m,n
. Finally, any two sensors in 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 Yagan on connectivity in the following aspects. First, our result is more broadly applicable; specifically, we consider K
_m,n
/K
_1,n
= o(√n), while [1] requires K
_m,n
/K
_1,n
= o(ln n). Put differently, K
_m,n
/K
_1,n
in our paper examines the case of Θ(n
^x
) for any x <; 1/2 and Θ((ln n)
^y
) for any y > 0, while that of [1] does not cover any Θ(n
^x
), and covers Θ((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 Σ
_j=1
^m
{a
_j
[1 - (
_Kj,n
^P
n-K1,n
)/(
_K
j,n
^P
_n
)]} equals ln n/n, our paper considers any of b
_n
= o(ln n/n), b
_n
= Θ(ln n/n), and b
_n
= ω(ln n/n), while [1] evaluates only b
_n
= Θ(ln n/n). Third, in terms of characterizing the transitional behavior of connectivity, our scaling b
_n
= ln n+ β
_n
/n for a sequence β
_n
is more fine-grained than the scaling b
_n
~ c ln n/ n for a constant c ≠ 1 of [1]. In a nutshell, we add the case of c = 1 in b
_n
~ c ln n/n, where the graph can be connected or disconnected asymptotically, depending on the limit of βn. Finally, although a recent study by Eletreby and Yagan [2] uses the fine-grained scaling discussed before for a more complex graph model, their result (just like [1]) also demands K
_m,n
/K
_1,n
= o(ln n), which is less general than K
_m,n
/K
_1,n
= o(√n) addressed in this paper.