Large cliques in sparse random intersection graphs

TL;DR

This paper analyzes the size of the largest cliques in sparse random intersection graphs, revealing different asymptotic behaviors depending on degree distributions, and provides algorithms to find these cliques efficiently.

Contribution

It characterizes the asymptotic order of the clique number in sparse random intersection graphs with various degree distributions and offers polynomial-time algorithms to find maximum cliques.

Findings

Power-law degree distribution leads to polynomial-sized maximum cliques.

Bounded degree variance results in maximum cliques of size (ln n)/(ln ln n).

Efficient algorithms can find near-maximum cliques in these graphs.

Abstract

Given positive integers n and m, and a probability measure P on {0, 1, ..., m} the random intersection graph G(n,m,P) on vertex set V = {1,2, ..., n} and with attribute set W = {w_1, w_2, ..., w_m} is defined as follows. Let S_1, S_2, ..., S_n be independent random subsets of W such that for any v \in V and any S \subseteq W we have \pr(S_v = S) = P(|S|) / \binom (m, |S|). The edge set of G(n,m,P) consists of those pairs {u,v} V for which S_u and S_v intersect. We study the asymptotic order of the clique number \omega(G(n,m,P)) in random intersection graphs with bounded expected degrees. For instance, in the case m = \Theta(n) we show that if the vertex degree distribution is power-law with exponent \alpha \in (1;2), then the maximum clique is of a polynomial size, while if the variance of the degrees is bounded, then the maximum clique has (ln n)/(ln ln n) (1 + o_P(1)) vertices whp. In each case there is a polynomial algorithm which finds a clique of size \omega(G(n,m,P)) (1-o_P(1)).