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 <inline-formula><tex-math notation="LaTeX">$\mathcal {V}_n$ </tex-math></inline-formula> of <inline-formula><tex-math notation="LaTeX">$n$</tex-math></inline-formula> sensors, suppose there is a pool <inline-formula><tex-math notation="LaTeX">$\mathcal {P}_n$</tex-math></inline-formula> consisting of <inline-formula><tex-math notation="LaTeX">$P_n$</tex-math></inline-formula> distinct keys. The <inline-formula><tex-math notation="LaTeX">$n$</tex-math></inline-formula> sensors in <inline-formula> <tex-math notation="LaTeX">$\mathcal {V}_n$</tex-math></inline-formula> are divided into <inline-formula> <tex-math notation="LaTeX">$m$</tex-math></inline-formula> groups <inline-formula><tex-math notation="LaTeX">$\mathcal {A}_1, \mathcal {A}_2, \ldots, \mathcal {A}_m$</tex-math></inline-formula>. Each sensor <inline-formula> <tex-math notation="LaTeX">$v$</tex-math></inline-formula> is independently assigned to exactly a group according to the probability distribution with <inline-formula><tex-math notation="LaTeX">$\mathbb {P}[v \in \mathcal {A}_i]= a_i$ </tex-math></inline-formula> for <inline-formula><tex-math notation="LaTeX">$i=1,2,\ldots, m$</tex-math> </inline-formula>, where <inline-formula><tex-math notation="LaTeX">$\sum _{i=1}^{m}a_i = 1$</tex-math></inline-formula> . Afterwards, each sensor in group <inline-formula><tex-math notation="LaTeX">$\mathcal {A}_i$</tex-math> </inline-formula> independently chooses <inline-formula><tex-math notation="LaTeX">$K_{i,n}$</tex-math></inline-formula> keys uniformly at random from the key pool <inline-formula><tex-math notation="LaTeX">$\mathcal {P}_n$</tex-math> </inline-formula>, where <inline-formula><tex-math notation="LaTeX">$K_{1,n} \leq K_{2,n}\leq \ldots \leq K_{m,n}$ </tex-math></inline-formula>. Finally, any two sensors in <inline-formula><tex-math notation="LaTeX">$\mathcal {V}_n$ </tex-math></inline-formula> 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 <xref ref-type="bibr" rid="ref1">[1]</xref> (IEEE Transactions on Information Theory 2016) of Yağan on connectivity in the following aspects. First, our result is more broadly applicable; specifically, we consider <inline-formula><tex-math notation="LaTeX">$K_{m,n} / K_{1,n} = o(\sqrt{n})$</tex-math></inline-formula>, while <xref ref-type="bibr" rid="ref1">[1]</xref> requires <inline-formula><tex-math notation="LaTeX">$K_{m,n} / K_{1,n} = o(\ln n)$</tex-math></inline-formula>. Put differently, <inline-formula><tex-math notation="LaTeX">$K_{m,n} / K_{1,n}$ </tex-math></inline-formula> in our paper examines the case of <inline-formula><tex-math notation="LaTeX">$\Theta (n^{x})$</tex-math></inline-formula> for any <inline-formula><tex-math notation="LaTeX">$x <\frac{1}{2}$</tex-math> </inline-formula> and <inline-formula><tex-math notation="LaTeX">$\Theta \big ((\ln n)^y\big)$</tex-math> </inline-formula> for any <inline-formula><tex-math notation="LaTeX">$y > 0$</tex-math></inline-formula>, while that of <xref ref-type="bibr" rid="ref1">[1]</xref> does not cover any <inline-formula><tex-math notation="LaTeX">$\Theta (n^{x})$</tex-math></inline-formula>, and covers <inline-formula><tex-math notation="LaTeX">$\Theta \big ((\ln n)^y)$ </tex-math></inline-formula> for only <inline-formula><tex-math notation="LaTeX">$0<y<1$</tex-math> </inline-formula>. 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 <inline-formula><tex-math notation="LaTeX">$b_n$</tex-math> </inline-formula> denoting <inline-formula><tex-math notation="LaTeX">$\sum _{j=1}^{m} \left \lbrace a_j \left [1 -{{{P_n-K_{1,n}}\atopwithdelims (){K_{j,n}}}\big /{{P_n}\atopwithdelims (){K_{j,n}}}}\right] \right\rbrace$</tex-math> </inline-formula> equals <inline-formula><tex-math notation="LaTeX">$\frac{\ln n}{n}$</tex-math></inline-formula>, our paper considers any of <inline-formula><tex-math notation="LaTeX">$b_{n}=o\big (\frac{ \ln n }{n}\big)$</tex-math> </inline-formula>, <inline-formula><tex-math notation="LaTeX">$b_{n}=\Theta \big (\frac{ \ln n }{n}\big)$</tex-math> </inline-formula>, and <inline-formula><tex-math notation="LaTeX">$b_{n}=\omega \big (\frac{ \ln n }{n}\big)$</tex-math> </inline-formula>, while <xref ref-type="bibr" rid="ref1">[1]</xref> evaluates only <inline-formula> <tex-math notation="LaTeX">$b_{n}=\Theta \big (\frac{ \ln n }{n}\big)$</tex-math></inline-formula>. Third, in terms of characterizing the transitional behavior of connectivity, our scaling <inline-formula><tex-math notation="LaTeX"> $b_{n}=\frac{ \ln n + \beta _n}{n}$</tex-math></inline-formula> for a sequence <inline-formula> <tex-math notation="LaTeX">$\beta _n$</tex-math></inline-formula> is more fine-grained than the scaling <inline-formula> <tex-math notation="LaTeX">$b_{n}\sim \frac{c \ln n}{n}$</tex-math></inline-formula> for a constant <inline-formula> <tex-math notation="LaTeX">$c \ne 1$</tex-math></inline-formula> of <xref ref-type="bibr" rid="ref1">[1]</xref>. In a nutshell, we add the case of <inline-formula><tex-math notation="LaTeX">$c=1$</tex-math></inline-formula> in <inline-formula><tex-math notation="LaTeX">$b_{n}\sim \frac{c \ln n}{n}$</tex-math></inline-formula>, where the graph can be connected or disconnected asymptotically, depending on the limit of <inline-formula><tex-math notation="LaTeX"> $\beta _n$</tex-math></inline-formula>. Finally, although a recent study by Eletreby and Yağan <xref ref-type="bibr" rid="ref2">[2]</xref> uses the fine-grained scaling discussed before for a more complex graph model, their result (just like <xref ref-type="bibr" rid="ref1">[1]</xref>) also demands <inline-formula> <tex-math notation="LaTeX">$K_{m,n} / K_{1,n} = o(\ln n)$</tex-math></inline-formula>, which is less general than <inline-formula><tex-math notation="LaTeX">$K_{m,n} / K_{1,n} = o(\sqrt{n})$</tex-math></inline-formula> addressed in this paper.