A note on the component structure in random intersection graphs with tunable clustering

TL;DR

This paper investigates the component structure of random intersection graphs with adjustable clustering, revealing a phase transition at average degree one where a giant component emerges.

Contribution

It introduces a threshold-based analysis of the component sizes in random intersection graphs with tunable clustering, highlighting the phase transition phenomenon.

Findings

Below average degree one, largest component is logarithmic in size.

Above average degree one, a linear-sized giant component appears.

Second largest component remains logarithmic in size when a giant exists.

Abstract

We study the component structure in random intersection graphs with tunable clustering, and show that the average degree works as a threshold for a phase transition for the size of the largest component. That is, if the expected degree is less than one, the size of the largest component is a.a.s. of logarithmic order, but if the average degree is greater than one, a.a.s. a single large component of linear order emerges, and the size of the second largest component is at most of logarithmic order.