Random intersection graphs when m=ω(n): An equivalence theorem relating the evolution of the G(n, m, p) and G(n, p) models

Abstract

When the random intersection graph G n m p proposed by Karoński, Scheinerman, and Singer-Cohen [Combin Probab Comput 8 (1999), 131-159] is compared with the independent-edge G n p , the evolutions are different under some values of m and equivalent under others. In particular, when m = n α and α > 6, the total variation distance between the graph random variables has limit 0.

Poisson Approximation for the Number of Isolated Complete Graphs in a Random Intersection Graph

“…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.…”

Poisson Approximation of the Number of Cliques in Random Intersection Graphs

“…In light of the equivalence theorem proved in [8], it is no surprise that the above result agrees with the well-known result for Erdős-Rényi graphs G(n,p) when α > 6 andp ∼ mp 2 , since the abovementioned equivalence theorem says that, for α > 6 andp ∼ mp 2 , the graphs G(n, m, p) and G(n,p) have asymptotically the same properties. Moreover, this result is consistent with the hypothesis from [8] of the equivalence of the models for α > 3 and their inequivalence for α < 3. From the applied point of view, it would be interesting to obtain similar results for the general random intersection graph models defined as in [9].…”

“…In comparison, for Erdős-Rényi graphs G(n,p), in which edges appear independently and with probabilityp, the bound obtained by Stein's method (presented in Section 2) for the total variation distance between the number of cliques of size h and the Poisson distribution with parameter λ = n h p ( h 2 ) is of order O(n −1 ) for all fixed h ≥ 3 when λ is bounded. In light of the equivalence theorem proved in [8], it is no surprise that the above result agrees with the well-known result for Erdős-Rényi graphs G(n,p) when α > 6 andp ∼ mp 2 , since the abovementioned equivalence theorem says that, for α > 6 andp ∼ mp 2 , the graphs G(n, m, p) and G(n,p) have asymptotically the same properties. Moreover, this result is consistent with the hypothesis from [8] of the equivalence of the models for α > 3 and their inequivalence for α < 3.…”

On Vertex, Edge, and Vertex-Edge Random Graphs (Extended Abstract)

“…For example, when S is the set of real intervals, one obtains a random interval graph [5,14,21,22]; see Example 2.12 for more. In [10,15,23] one takes S to consist of discrete (finite) sets. Random chordal graphs can be defined by selecting random subtrees of a tree [18].…”