Large random intersection graphs inside the critical window and triangle counts

TL;DR

This paper studies the behavior of large random intersection graphs within their critical phase, revealing their scaling limits and triangle counts, and connecting them to continuum Erdős–Rényi graphs.

Contribution

It identifies the scaling limits of random intersection graphs in critical windows and establishes their relation to continuum Erdős–Rényi graphs across different clustering regimes.

Findings

Scaling limits vary with clustering regimes

Graphs coincide with continuum Erdős–Rényi in two regimes

Limit theorems for triangle counts in large components

Abstract

We identify the scaling limit of random intersection graphs inside their critical windows. The limit graphs vary according to the clustering regimes, and coincide with the continuum Erdos--Renyi graph in two out of the three regimes. Our approach to the scaling limit relies upon the close connection of random intersection graphs with binomial bipartite graphs, as well as a graph exploration algorithm on the latter. This further allows us to prove limit theorems for the number of triangles in the large connected components of the graphs.