The largest component in an inhomogeneous random intersection graph with clustering

TL;DR

This paper analyzes the size of the largest component in a sparse inhomogeneous random intersection graph with clustering, providing asymptotic results based on a branching process approximation.

Contribution

It introduces a first order asymptotic for the largest component size in inhomogeneous random intersection graphs with clustering.

Findings

Largest component size asymptotically proportional to n(1-Q(0))g

Derived g as an average of nonextinction probabilities of a branching process

Established probabilistic bounds for the component size in sparse graphs

Abstract

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.