Abstract
We study isolated vertices and connectivity in the random intersection graph . A Poisson convergence for the number of isolated vertices is determined at the threshold for absence of isolated vertices, which is equivalent to the threshold for connectivity. When and , we give the asymptotic probability of connectivity at the threshold for connectivity. Analogous results are well known in ErdΕs-RΓ©nyi random graphs.
1. Introduction
The classical random graph , introduced by ErdΕs and RΓ©nyi in the late 1950s, consists of a fixed set of vertices and edges that exist with a certain probability , independently from each other. Since then many other random graph models with dependent edges have been developed. Among them, random intersection graph [1, 2] is defined as follows. Consider a set with vertices and another universal set with elements. Define a bipartite graph with independent vertex sets and . Edges between and exist independently with probability . The random intersection graph derived from is defined on the vertex set with vertices adjacent if and only if there exists some such that both and are adjacent to in .
Appropriately scaling the parameter as with some , Singer-Cohen [1] establishes connectivity thresholds for : the threshold lies at and for and , respectively. The result also reveals an asymptotic equivalence of graph connectivity and absence of isolated vertices in , that is, the zero-one law for the absence of isolated vertices is equal to that for connectivity. This is familiar in ErdΕs-RΓ©nyi model; see [3, 4] for more details. The study in the present paper is in continuation of Chapter 3 in [1]. Taking our cue from existing results for ErdΕs-RΓ©nyi graphs (e.g., [4, Corollary 3.31] and [3, Theorem 7.3]), we aim to explore similar results for the properties of isolated vertices and connectivity in .
The connectivity thresholds of another class of random intersection graphs , called random key graphs or uniform random intersection graphs, have been investigated recently [5, 6]. Both and can be viewed as subclasses of a general model [7]. In [8], the authors determine a zero-one law for the absence of isolated vertices in , which again turns out to be equivalent to that for graph connectivity [6]. Moreover, they show a Poisson convergence for the number of isolated vertices, which refines the corresponding zero-one law and leads to a βdouble exponentialβ result.
In this paper, we deal with the asymptotic distribution of the number of isolated vertices and address the connectivity probability in with , . A Poisson approximation result (Theorem 2.1) for the number of isolated vertices is obtained by utilizing the Stein-Chen method, which yields convergence to a Poisson random variable. The isolated vertices threshold [1, Proposition 3.2] now readily follows from our Theorem 2.1 by an easy monotonicity argument. In addition, based on a strong equivalence theorem [9] relating the and models we derive an approximation of the probability of connectivity at the threshold when (see Theorem 2.3), which is analogous to the well-known βdouble exponentialβ result of ErdΕs and RΓ©nyi [10].
Other related works regarding model have been reported. For example, [11, 12] examines the limiting distribution of the degree of a typical vertex, [13] treats the evolution of the order of the largest component, and random weights are assigned to the vertices in [14] to get general degree distributions.
The rest of the paper is organized as follows. Our main results are presented in Section 2. Sections 3 and 4 contain technical proofs of Theorems 2.1 and 2.3, respectively. Throughout the paper we set for some .
2. Main Results
In this section we provide our main results. Let denote the number of isolated vertices in and let be a Poisson random variable with parameter . Denote by and the mean and variance of random variable , respectively. Recall that the Poisson random variable has the unusual property that the mean and variance are both equal to the parameter .
Theorem 2.1. In the model , let where . If , then one has as , where represents convergence in distribution.
The upcoming corollary is immediate from Theorem 2.1.
Corollary 2.2. In the model with determined through (2.1), suppose . Then one gets
For a parallel βdouble exponentialβ result for connectivity when is large, we have the following.
Theorem 2.3. In the model with and determined through (2.1) (i.e., ), assume that . Then one has
These results complement those presented in [1] and get further insight into the evolutionary similarities and differences between and models. A natural question would be to ask what happens for connectivity probability when is small. This is currently under investigation.
3. Proof of Theorem 2.1
For , let and . Therefore, counts the number of isolated vertices in as defined in Section 2. We will demonstrate the asymptotic Poisson distribution of by employing the Stein-Chen method [15].
Before proceeding, we first introduce some definitions and notations. Let and denote the cardinality of a set . For two integer-valued random variables and , the total variation distance between them (more correctly, between their distributions and ) is given by Let be a finite set of indices and let be a family of random indicator variables. We say are positively related (c.f. [15]) if, for each , there exist random indicator variables with the distributions such that for every . It is notable that βpositively relatedβ is much stronger than βpositively correlatedβ. Suppose and are sequences of positive real numbers. We write if .
A useful result obtained by the Stein-Chen method is the following.
Lemma 3.1 (see [4, 15]). Suppose that , where the are positively related random indicator variables. Then one has
The next lemma collects some well-known approximations that are used in this paper.
Lemma 3.2. If , then ; and if , then .
In the sequel, we estimate the expectation of random variable .
Lemma 3.3. Suppose . Under the assumptions of Theorem 2.1, one gets
Proof. The probability that a vertex is isolated can be computed as
where the index represents the number of vertices in which are adjacent to in . Hence
For , we have
as . Thus by Lemma 3.2, we obtain
as .
For , note that
as . By using Lemma 3.2, we have
as . The proof is then complete.
Proof of Theorem 2.1. The triangular inequality for the total variation distance implies
By a coupling argument ([16, page 58]) and Lemma 3.3, we have
as . Combining this with (3.11), we now only need to prove
First, we claim that are positively related. To see this, define
for every , where represents the elements in which are adjacent to in ( is possibly empty). The random graphs and are coupled in a natural way. Conditional on the isolation of vertex in , any vertex is not adjacent to vertices of in . Hence, we have
For every , if then . Consequently, we get .
By Lemma 3.1, the binary nature and exchangeability of the random variables involved, we find that
The cross term in (3.16) is shown to be given by
where counts the number of vertices in adjacent to neither 1 or 2 in , leaving vertices in adjacent to exactly one of 1, 2.
Combining (3.5), (3.16), and (3.17) readily gives
For , we have similarly as in the proof of Lemma 3.3,
as . Thereby, it follows from Lemma 3.2 that
Applying this to (3.18), we obtain
as .
For , we get as in the proof of Lemma 3.3,
as . Hence, from Lemma 3.2 we have
Applying this to (3.18), we have
as , which concludes the proof.
4. Proof of Theorem 2.3
Let The following lemma drawn from [9] states an equivalence of and models.
Lemma 4.1 (see [9]). Let and be such that for some . For any and any graph property , as if it follows that
We recall the following classical result for connectivity threshold of .
Lemma 4.2 (see [10]). Let be fixed and . Then as .
Proof of Theorem 2.3. In view of Lemmas 4.1 and 4.2, it suffices to prove that
as .
By the assumptions, we find that
where denotes . Since
as , by Lemma 3.2, the right-hand side of (4.6)
which concludes the proof.
Acknowledgment
The author would like to thank Karen Singer-Cohen for her warm encouragement and supplying a copy of [1].