The largest component in an inhomogeneous random intersection graph with clustering

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

2010-02-24

arxiv.org/abs/1002.4649v1

Mathematics

Probability

14 pages

Scientific paper

Given b>0, integers n, m=bn and a probability measure Q on {0, 1,..., m}, consider the random intersection graph on the vertex set [n]={1, ..., n}, where i and j are declared adjacent whenever S(i) and S(j) intersect. Here S(1), ..., S(n) denote iid random subsets of [m] such that P(|S(i)|=k)=Q(k). For sparse random intersection graphs we establish a first order asymptotic for the order of the largest connected component N=n(1-Q(0))g+o(n) in probability. Here g is an average of nonextinction probabilities of a related multi-type Poisson branching process.

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