## Abstract

Let (V) and (U) be the point sets of two independent homogeneous Poisson processes on (ℝ). A graph (G)(V) with vertex set (V) is constructed by first connecting pairs of points (v, u) with (V) and (U) independently with probability g(v-u), where g is a non-increasing radial function, and then connecting two points v1,v2(V) if and only if they have a joint neighbor (U). This gives rise to a random intersection graph on (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.

Original language | English |
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Number of pages | 18 |

Journal | Journal of applied probability |

DOIs | |

Publication status | E-pub ahead of print/First online - 22 May 2024 |

## Keywords

- 2024 OA procedure
- Spatial random graphs
- Complex networks
- Degree distribution
- Percolation phase transition
- AB percolation