Sách The largest component in an inhomogeneous random intersection graph with clustering

Let Q be a probability measure on {0, 1, . ,m}, and let S1, . , Sn be random subsets of a set W = {w1, . ,wm} drawn independently from the probability distribution P(Si = A) = m |A|
ư1 Q(|A|), A ⊂ W, for i = 1, . , n. A random intersection graph G(n,m,Q) with vertex set V = {v1, . , vn} is defined as follows. Every vertex vi is prescribed the set S(vi) = Si, and two vertices vi and vj are declared adjacent (denoted vi ∼ vj) whenever S(vi)∩S(vj) = ∅. The elements of W are sometimes called attributes, and S(vi) is called the set of attributes of vi.Random intersection graphs G(n,m,Q) with the binomial distribution Q ∼ Bi(m, p) were introduced in Singer-Cohen [15] and Karo´nski et al. [13], see also [10] and [16]. The emergence of a giant connected component in a sparse binomial random intersection graph was studied by Behrish [2] for m = ⌊nα⌋, α = 1, and by Lager˚as and Lindholm [14] for m = ⌊βn⌋, where β > 0 is a constant.