Connectivity of the Uniform Random Intersection Graph

TL;DR

This paper establishes the threshold for connectivity in uniform random intersection graphs, which model secure wireless sensor networks, based on parameters like number of nodes, colours, and their relationships.

Contribution

It determines the precise asymptotic threshold for connectivity in $G(n,m,k)$ graphs as the number of nodes grows, linking parameters $k$, $m$, and $n$.

Findings

Graph is almost surely connected when $rac{k^2 n}{m} o ext{above } 1/\log n$

Graph is almost surely disconnected when $rac{k^2 n}{m} o ext{below } 1/\log n$

Threshold depends on the ratio $k^2 n / m$ and the logarithm of $n$.

Abstract

A \emph{uniform random intersection graph} $G(n,m,k)$ is a random graph constructed as follows. Label each of $n$ nodes by a randomly chosen set of $k$ distinct colours taken from some finite set of possible colours of size $m$. Nodes are joined by an edge if and only if some colour appears in both their labels. These graphs arise in the study of the security of wireless sensor networks. Such graphs arise in particular when modelling the network graph of the well known key predistribution technique due to Eschenauer and Gligor. The paper determines the threshold for connectivity of the graph $G(n,m,k)$ when $n\to \infty$ with $k$ a function of $n$ such that $k\geq 2$ and $m=\lfloor n^\alpha\rfloor$ for some fixed positive real number $\alpha$. In this situation, $G(n,m,k)$ is almost surely connected when \[ \liminf k^2n/m\log n>1, \] and $G(n,m,k)$ is almost surely disconnected when \[ \limsup k^2n/m\log n<1. \]