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Geometric random intersection graphs with general connection probabilities

Part of: Special processes Graph theory

Published online by Cambridge University Press:  22 May 2024

Maria Deijfen  and
Riccardo Michielan
Maria Deijfen*
Affiliation:
Stockholm University
Riccardo Michielan*
Affiliation:
University of Twente
*
*Postal address: Stockholm University, Department of Mathematics, Matematiska institutionen 106 91 Stockholm, Sweden. Email: mia@math.su.se
**Postal address: University of Twente, Department of Electrical Engineering, Mathematics and Computer Science, Hallenweg 19 7522NH Enschede, Netherlands. Email: r.michielan@utwente.nl
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Abstract

Let $\mathcal{V}$ and $\mathcal{U}$ be the point sets of two independent homogeneous Poisson processes on $\mathbb{R}^d$. A graph $\mathcal{G}_\mathcal{V}$ with vertex set $\mathcal{V}$ is constructed by first connecting pairs of points (v, u) with $v\in\mathcal{V}$ and $u\in\mathcal{U}$ independently with probability $g(v-u)$, where g is a non-increasing radial function, and then connecting two points $v_1,v_2\in\mathcal{V}$ if and only if they have a joint neighbor $u\in\mathcal{U}$. This gives rise to a random intersection graph on $\mathbb{R}^d$. Local properties of the graph, including the degree distribution, are investigated and quantified in terms of the intensities of the underlying Poisson processes and the function g. Furthermore, the percolation properties of the graph are characterized and shown to differ depending on whether g has bounded or unbounded support.

Keywords

Spatial random graphscomplex networksAB percolationdegree distributionpercolation phase transition

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 05C80: Random graphs
Type
Original Article
Information
Journal of Applied Probability , First View , pp. 1 - 18
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

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