POISSON APPROXIMATION FOR THE NUMBER OF ISOLATED COMPLETE GRAPHS IN A RANDOM INTERSECTION GRAPH

Let Wn,r be the number of isolated complete graphs of order r in a random intersection graph G(n,m, p). In this paper, we demonstrate that Wn,r can be approximated by Poisson distribution and give the bound of this approximation by using the Stein-Chen method.


Introduction
Given a set V with n vertices and another universal set U with m elements, define a bipartite graph B(n, m, p) with independent vertex sets V and U and edges between v ∈ V and u ∈ U existing independently with probability p.The random intersection graph G(n, m, p) , derived from B(n, m, p), is defined on the vertex set V with vertices v 1 , v 2 ∈ V adjacent if and only if there exist some u ∈ U such that both v 1 and v 2 are adjacent to u in B(n, m, p).Also define S i be a random subset of U such that each element of S i is adjacent to i ∈ V , in which case two vertices i, j ∈ V are adjacent if and only if S i ∩ S j = φ, and edge set E(G) is define as E(G) = {{i, j} : i, j ∈ V, S i ∩ S j = φ}.
The properties of G(n, m, p) were studied in [2,3] contrasted with the well known random graph model G(n, p), in which vertices are made adjacent to each other independently and with probability p, and showed that for a fixed α > 0, the number of elements m is taken to be m = ⌊n α ⌋.In 1999, Karonski, Scheinerman and Singer-Cohen [2] showed that the total variation distance between the distribution of G(n, m, p) and G(n, p) converges to 0 when α > 6 and p is defined appropriately.Without loss of generality we consider the independent set V. For i = 1, 2, 3, ..., n, let Clearly, X is the number of isolated vertices in G(n, m, p).
In 2011, Yilum Shang [6] proved that the distribution function of X can be approximated by Poisson distribution with parameter In 2013, M. Donganont [9] showed the another proof of Poisson approximation for the number of isolated vertices in G(n, m, p) by Stein-Chen and coupling method.The results as the following, Theorem 1.1.Let W be the number of isolated vertices in a random intersection graph G(n, m, p) .For A ⊆ {0, 1, 2, ...n} and m = ⌊n α ⌋ for some α > 0, we have and When C w = {0, 1, ...., w} Corollary 1.1.Let W be the number of isolated vertices in a random intersection graph G(n, m, p).Let A ⊆ {0, 1, 2, ...n}, m = ⌊n α ⌋ for some α > 0, q = 1 − p, and p = 1 n γ for any γ ∈ R + \ {1} ,then be the set of all possible combinations of k vertices.we note that T k is a tree of order k in G(n, m, p) and say that T k is isolated in G(n, m, p) if there is no edge in G(n, m, p) between a vertex in the tree and the other outside of the tree.
For each i ∈ Γ n,k , we define the indicator random variable there is an isolated tree in G(n, m, p) that spans the vertices i = (i 1 , . . ., i k ), 0 otherwise, and set Then W n,k is the number of isolated trees in G(n, m, p).
In 2013, Mana [10] shows that if m = ⌊n α ⌋ ; α > 0 , then W n,k can be approximated by Poisson approximation with parameter By using Stein-Chen and Coupling Method.The result is following, Theorem 1.2.Let W be the number of isolated trees in a random intersection graph G(n, m, p) .For A ⊆ {0, 1, 2, ...n} and m = ⌊n α ⌋ for some α > 0, we have where and In this paper, we use this idea to show that the number of isolated complete graphs can be approximated by Poisson distribution.Now, define, be the set of all possible combinations of r vertices.we note that K r is a complete graph of order r in G(n, m, p) and say that K r is isolated if there is no edge between a vertex in the complete graph and the other outside of that one.
For each i ∈ Γ n,r , we define the indicator random variable there is an isolated complete graph in G(n, m, p) that spans the vertices i = (i 1 , . . ., i r ), 0 otherwise, and set Thus, W n,r is the number of isolated complete graphs in G(n, m, p).

Stein-Chen and Coupling Method
In 1972, Stein [1] gave a new technique to find a bound in the normal approximation to a distribution of a sum of dependent random variables.His technique was relied instead on the elementary differential equation.In 1975, Chen [4,5] applied Stein's idea to the Poisson case.The central idea of the Stein-Chen method is the difference equation where λ > 0 and A ⊆ N ∪ {0} and I A : N ∪ {0} → R is defined by The equation ( 3) is called Stein's equation for Poisson distribution function and its solution is [7] By substituting j and λ in (3) by any integer-valued random variable W and λ = E(W ), we have So far W could be i∈Γ In 1992, Barbour, Holst and Janson [7] constructed coupling random variable W i and used Stein-Chen method to find the bound in Poisson approximation of W .They assumed that for each i the distribution L(W i ) of W i equals to the conditional distribution L(W − X i |X i = 1) and gave the fundamental theorem as follows: Theorem 2.1.If W and W i are defined as above, then where g λ,A := sup In 2006, Santiwipanont and Teerapabolarn [8] proved that for any subset A of {0, 1, . . ., n}, where and In next section, we will use Theorem 2.1 and (6) to prove our main result by constructing the random variable W i .

Proof of Theorem 1.3
Let A ⊆ N. By (5), it is enough to bound E|W − W i | for i ∈ Γ n,r where the distribution of W i equals to the conditional distribution of W −X i given X i = 1.
Let W i be the number of isolated complete graphs of order r in a random intersection graph G(n, m, p) − i, G(n, m, p) − i obtained from G(n, m, p) by dropping the set i ⊆ V and all the edges containing any of these vertices.Then for w 0 ∈ {0, 1, . . ., ⌊ n−r r ⌋}, we have and From ( 7) and ( 8), the distribution of W i equals to the conditional distribution of (W − X i |X i = 1).For i, j ∈ Γ n,r such that i = j, we define the indicator random variable X (i) j and E ij , as follow there is an isolated complete graph in G(n, m, p) − i that spans the vertices i = (i 1 , . . ., i r ), 0 otherwise, and 1 if there exists adjacent between i r ∈ i and j l ∈ j, 0 otherwise.
We observe that in case X i = 1, that is we have an isolated complete graph in G(n, m, p) that span vertices i = {i 1 , . . ., i r }.Thus the number of isolated complete graphs in a random intersection graph G(n, m, p) − i equals to the number of isolated complete graphs in a random intersection graph G(n, m, p) minus 1, that is In case X i = 0.For j ∈ Γ n,r such that j ∩ i = ∅ and j = i , then that is the number of isolated complete graphs in G(n, m, p) − i equals to the sum of the number of isolated complete graphs in G(n, m, p) and the number of isolated complete graphs in G(n, m, p) − i which are connected to i.We know that Form ( 9) and ( 10), we have We note that, and j∈Γn,r,j∩i=∅ From ( 11), (12) and use the fact that 1 − p ≤ 1 e p , we have