On Connectivity in a General Random Intersection Graph

Abstract

There has been growing interest in studies of general random intersection graphs. In this paper, we consider a general random intersection graph $\mathbb{G}(n,\overrightarrow{a}, \overrightarrow{K_n},P_n)$ defined on a set $\mathcal{V}_n$ comprising $n$ vertices, where $\overrightarrow{a}$ is a probability vector $(a_1,a_2,\ldots,a_m)$ and $\overrightarrow{K_n}$ is $(K_{1,n},K_{2,n},\ldots,K_{m,n})$. This graph has been studied in the literature including a most recent work by Ya\u{g}an [arXiv:1508.02407]. Suppose there is a pool $\mathcal{P}_n$ consisting of $P_n$ distinct objects. The $n$ vertices in $\mathcal{V}_n$ are divided into $m$ groups $\mathcal{A}_1, \mathcal{A}_2, \ldots, \mathcal{A}_m$. Each vertex $v$ is independently assigned to exactly a group according to the probability distribution with $\mathbb{P}[v \in \mathcal{A}_i]= a_i$, where $i=1,2,\ldots,m$. Afterwards, each vertex in group $\mathcal{A}_i$ independently chooses $K_{i,n}$ objects uniformly at random from the object pool $\mathcal{P}_n$. Finally, an undirected edge is drawn between two vertices in $\mathcal{V}_n$ that share at least one object. This graph model $\mathbb{G}(n,\overrightarrow{a}, \overrightarrow{K_n},P_n)$ has applications in secure sensor networks and social networks. We investigate connectivity in this general random intersection graph $\mathbb{G}(n,\overrightarrow{a}, \overrightarrow{K_n},P_n)$ and present a sharp zero-one law. Our result is also compared with the zero-one law established by Ya\u{g}an.