Poisson approximation of counts of subgraphs in random intersection graphs

TL;DR

This paper investigates the conditions under which the count of specific subgraphs in random intersection graphs converges to a Poisson distribution, focusing on the case where the number of objects scales polynomially with vertices.

Contribution

It introduces the strictly alpha-balanced condition that guarantees Poisson convergence for subgraph counts in random intersection graphs.

Findings

Identifies the strictly alpha-balanced condition for Poisson convergence.

Provides theoretical criteria for subgraph count distributions.

Analyzes the impact of parameters n, m, and p on graph structure.

Abstract

Random intersection graphs are characterized by three parameters: $n$, $m$ and $p$, where $n$ is the number of vertices, $m$ is the number of objects, and $p$ is the probability that a given object is associated with a given vertex. Two vertices in a random intersection graph are adjacent if and only if they have an associated object in common. When $m=\lfloor n^\alpha\rfloor$ for constant $\alpha$, we provide a condition, called {\em strictly $\alpha$-balanced}, for the Poisson convergence of the number of induced copies of a fixed subgraph.