Connectivity of the Uniform Random Intersection Graph

Details Connectivity of the Uniform Random Intersection Graph Connectivity of the Uniform Random Intersection Graph

2008-05-19

arxiv.org/abs/0805.2814v2

Mathematics

Combinatorics

19 pages New version with rewritten intro, and a discussion section added. The results and proofs are unchanged from the previ

Scientific paper

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 \[ \limsup k^2n/m\log n<1. \]

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